(Created page with "==Abstract== In this paper, definitions of <math display="inline">L_{\alpha }^1</math> function space, <math display="inline">\beta </math>-well function space and quasi-Cara...")
 
 
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Latest revision as of 14:37, 7 October 2016

Abstract

In this paper, definitions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function space, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well function space and quasi-Carathéodory function are given, existence results for solutions of a class of Sturm–Liouville boundary value problems of fractional differential equations with multiple order derivatives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }[\Phi (\rho (t)D_{0^+}^{\beta }x(t))]} 

are established. The analysis relies on a well known fixed point theorem. Some examples are given to illustrate the efficiency of main theorems.

Keywords

Lα1 function space; ββ-well function; Multiple order fractional differential equation; Sturm–Liouville boundary value problems; Leray–Schauder Alternative principle

2000 MR subject classification

92D25; 34A37; 34K15

1. Introduction

It is well known that the following boundary value problem (BVP for short) for second order ordinary differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} x^{{''}}(t)+f(t,x(t),x^{{'}}(t))=0\mbox{,}\quad t\in (0,1)\mbox{,}\\ ax(0)-bx^{{'}}(0)=0\mbox{,}\\ cx^{{'}}(1)+dx(1)=0\mbox{,} \end{array}
(1)

is called a Sturm–Liouville boundary value problem  [5] and [6], where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(t,u,v)} 

is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]\times R\times R}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a\geq 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b\geq 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c\geq 0} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle d\geq 0}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ac+ad+bc>0}

. BVP(1) comes from a situation involving nonlinear elliptic problems in annular regions, one may see  [5].

In order to generalize the results obtained in  [5] and [6] to boundary value problems for fractional differential equations, in recent years, there exist many papers concerned with the existence of positive solutions of boundary value problems for fractional differential equations (see  [1], [2], [3], [7], [8], [9], [12], [13], [14], [16], [10] and [15]).

In  [8], Kaufmann and Mboumi studied the following boundary value problem for the fractional differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} D_{0^+}^{\alpha }x(t)+a(t)f(x(t))=0\mbox{,}\quad 0<t<1\mbox{,}\\ x(0)=0\mbox{,}\quad x^{{'}}(1)=0\mbox{,} \end{array}
(2)

by using the properties of Green’s function of the corresponding BVP, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f:[0,+\infty )\rightarrow [0,+\infty )} 

is continuous, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a:[0,1]\rightarrow (0,+\infty )}
is a continuous function and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }}
is the standard Riemann–Liouville fractional differential derivative of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \in (1,2)}

. Using the Leggett–Williams fixed point theorem and Krasnoselskii fixed point theorem, the authors in [8] proved that BVP(2) has at least one or three positive solutions.

Dehghant and Ghanbari  [4] studied the existence of solutions of the following boundary value problem for the nonlinear fractional differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} D_{0^+}^{\alpha }x(t)+a(t)f(t,x(t),x^{{'}}(t))=0\mbox{,}\quad 0<t<1\mbox{,}\\ x(0)=0\mbox{,}\quad x^{{'}}(1)=0\mbox{,} \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }} 

is the standard Riemann–Liouville fractional differential derivative of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \in (1,2)}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]\times [0,+\infty )\times R}

, continuous and nonnegative, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a(t)} 

does not identically vanish on any subinterval of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,1)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a(t)}
satisfies the following inequality:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0<\int _0^1[t(1-t)]^{\alpha }a(t)dt<+\infty \mbox{.}

It is easy to see that the following boundary value problem for the ordinary differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x^{{''}}(t)=-1\mbox{,}\quad t\in (0,1)\mbox{,}\quad x^{{'}}(0)=0\mbox{,}\quad x(1)=0

has a unique continuous solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u(t)=\frac{1}{2}(1-t^2)} 

on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]}

. However, we find that the similar boundary value problem for the multiple order fractional differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle D_{0^+}^{\alpha }D_{0^+}^{\beta }x(t)=-1\mbox{,}\quad t\in (0\\\displaystyle 1)\mbox{,}\underset{t\rightarrow 0}{lim}t^{1-\alpha }\underset{0^+}{\overset{\beta }{D}}x(t)=0\mbox{,}\underset{t\rightarrow 1}{lim}t^{1-\beta }x(t)=0\mbox{,}\alpha ,\beta \in (0,1)\end{array}

has a unique solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): x(t)=[-t^{\alpha +\beta }+t^{\beta -1}]\frac{1}{\Gamma (\alpha +1)\Gamma (\beta )}B(\beta ,\alpha +1)\mbox{,}

which is not continuous at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=0} . This fact shows that there exist differences between BVPs for ordinary differential equations and BVPs for fractional differential equations.

Liu  [11] studied the existence of solutions of the following boundary value problem for the fractional differential equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} D_{0^+}^{\alpha }x(t)+f(t,x(t))=0\mbox{,}\quad 0<t<1\mbox{,}\\ \underset{t\rightarrow 0}{lim}t^{2-\alpha }x(t)=0\mbox{,}\quad \underset{0^+}{\overset{\alpha -1}{D}}x(1)=0\mbox{,} \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }} 

is the standard Riemann–Liouville fractional differential derivative of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \in (1,2)}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]\times [0,+\infty )}

, nonnegative and is a Carathéodory function. Note that the boundary condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x(0)=0} 

used in  [4] and [8] is replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 0}t^{2-\alpha }x(t)=0}

. Hence the solutions obtained in  [11] may be unbounded or singular at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=0} .

In this paper, we discuss the existence of solutions to the following Sturm–Liouville boundary value problem for the nonlinear fractional differential equation with the nonlinearity depending on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\beta }x}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c}D_{0^+}^{\alpha }[\Phi (\rho (t)D_{0^+}^{\beta }x(t))]+f(t,x(t)\\\displaystyle D_{0^+}^{\beta }x(t))=0\mbox{,}t\in (0,1)\mbox{,}\\ a\underset{t\rightarrow 0}{lim}t^{1-\beta }x(t)-b\underset{t\rightarrow 0}{lim}\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x(t)=0\mbox{,}\\ c\underset{t\rightarrow 1}{lim}\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x(t)+d\underset{t\rightarrow 1}{lim}t^{1-\beta }x(t)=0\mbox{,} \end{array}
(3)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a,b,c,d\in R}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }}
(or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\beta }}

) is the left-side Riemann–Liouville fractional derivative of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha } 

(or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

) with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha ,\beta \in (0,1)} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho :(0,1)\rightarrow [0,+\infty )}
is continuous and may be singular at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=1}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f}
defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,1)\times R\times R}
is a quasi-Carathéodory function (see Definition 2.4) that may be singular at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=1}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi (s)=\vert s\vert ^{p-2}s}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p>1}
is called a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}

-Laplacian and its inverse function denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}(x)=\vert x\vert ^{q-2}x} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{p}+\frac{1}{q}=1}

.

We obtain some results on the existence of solutions of BVP(3) (see Definition 2.5). Two examples are given to illustrate the efficiency of the main theorems. It is obvious that BVP(3) is more general than those studied in the mentioned papers. Our results generalize and enrich known literatures.

The remainder of this paper is as follows: in Section  2, we present preliminary results and main theorems. In Section  3, two examples are given to illustrate the main results.

2. Main results

For convenience, we present some necessary definitions from fractional calculus theory. These definitions and results can be found in the literatures [3], [7], [12] and [14]. Let the Gamma function and Beta function be defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} \Gamma (\alpha )=\int _0^{+\infty }x^{\alpha -1}e^{-x}dx\mbox{,}\quad B(p,q)=\int _0^1x^{p-1}(1-x)^{q-1}dx\mbox{.} \end{array}

Definition 2.1 [14].

The left-side Riemann–Liouville fractional integral of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0} 

of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g:(0,+\infty )\rightarrow R}
is given by
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): I_{0^+}^{\alpha }g(t)=\frac{1}{\Gamma (\alpha )}\int _0^t(t-s)^{\alpha -1}g(s)ds\mbox{,}

provided that the right-hand side exists.

Definition 2.2 [14].

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n} 

be a positive integer. The left-side Riemann–Liouville fractional derivative of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha >0}
of a continuous function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle g:(0,+\infty )\rightarrow R}
is given by
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_{0^+}^{\alpha }g(t)=\frac{1}{\Gamma (n-\alpha )}\frac{d^n}{dt^n}\int _0^t\frac{g(s)}{(t-s)^{\alpha -n+1}}ds\mbox{,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n-1\leq \alpha <n} , provided that the right-hand side exists.

Lemma 2.1 [14].

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n-1\leq \mu <n} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in C^0(0,+\infty )\bigcap L^1(0,+\infty )} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): I_{0^+}^{\mu }D_{0^+}^{\mu }x(t)=x(t)+C_1t^{\mu -1}+C_2t^{\mu -2}+\cdots +C_nt^{\mu -n}\mbox{,}

where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C_i\in R} ,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i=1,2,\ldots ,n} .

Definition 2.3.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma >0} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x:(0,1)\rightarrow R} 

is called a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\sigma }^1}
function if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow t^{1-\sigma }\int _0^t(t-s)^{\sigma -1}\vert x(s)\vert ds}
is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]}

. The set of all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\sigma }^1} 

functions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,1)}
is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\sigma }^1(0,1)}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x:(0,1)\rightarrow R} 

is called a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma }

-well function if it is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\sigma }^1} 

function and for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0\leq t_2\leq t_1\leq 1}

, it holds that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t_1^{1-\sigma }\int _0^{t_1}(t_1-s)^{\sigma -1}x(s)ds+t_2^{1-\sigma }\int _0^{t_2}(t_2-\\\displaystyle -s)^{\sigma -1}x(s)ds-2t_1^{1-\sigma }\int _0^{t_2}(t_1-s)^{\sigma -1}x(s)ds\rightarrow 0\end{array}

as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_1\rightarrow t_2} .

Remark 2.1.

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\sigma }^1(0,1)} 

is a generalization of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L^1(0,1)}

. It is easy to check that all bounded functions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]} 

are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma }

-well functions and the functions that satisfy that there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu >-1} 

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert x(t)\vert \leq t^{\mu }}
for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in [0,1]}
are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma }

-well functions too.

Definition 2.4.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f:(0,1)\times R^2\rightarrow R} 

is called a quasi-Carathéodory function if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)}
is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L^{\alpha }}
function for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (u,v)\in R^2}
(see Definition 2.3),

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (u,v)\rightarrow f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)}
is continuous for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in (0,1)}

,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 

  for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r>0}
there exists a nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function (see Definition 2.3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)\vert \leq \phi _r(t)\mbox{,}\quad t\in (0,+\infty )\mbox{,}\quad (u,v)\in [-r\\\displaystyle r]\times [-r,r]\mbox{.}\end{array}

Definition 2.5.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x:(0,1)\rightarrow R} 

is called a solution of BVP(3) if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in C(0,1)\bigcap L^1(0,1)}
satisfies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\alpha }[\Phi (\rho D_{0^+}^{\beta }x)]\in L_{\alpha }^1(0,1)}
and all equations in (3) are satisfied.

For our construction, we let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): X=\lbrace x:(0,1)\rightarrow R\begin{array}{c} x\in C(0,1),\quad D_{0^+}^{\beta }x\in C(0,1)\mbox{,}\\ \mbox{ the following limits are finite  }\\ \underset{t\rightarrow 0}{lim}t^{1-\beta }x(t),\quad \underset{t\rightarrow 1}{lim}x(t)\mbox{,}\\ \underset{t\rightarrow 0}{lim}\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta}{D}}x(t),\quad \underset{t\rightarrow 1}{lim}\rho (t)\underset{0^+}{\overset{\beta}{D}}x(t) \end{array}\rbrace \mbox{.}

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X} , let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert x\Vert =max\lbrace \underset{t\in (0,1)}{sup}t^{1-\beta }\vert x(t)\vert \\\displaystyle \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}x(t)\vert \rbrace \mbox{.}\end{array}

Lemma 2.2.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X} is a Banach space with the norm  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert \cdot \Vert } defined.

Proof.

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X} 

is a normed linear space. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace x_u\rbrace }
be a Cauchy sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert x_u-x_v\Vert \rightarrow 0,\quad u,v\rightarrow +\infty } . It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}x_u\in C^0(0,1)\mbox{,}\quad u\in N\mbox{,}\\ \underset{t\rightarrow 0}{lim}t^{1-\beta }x_u(t),\underset{t\rightarrow 1}{lim}x_u(t)\mbox{  exist ,}\quad u\in N\mbox{,}\\ \underset{t\rightarrow 0}{lim}\Phi (t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t),\underset{t\rightarrow 1}{lim}\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t)\mbox{  exist ,}\quad u\in N\mbox{,}\\ \underset{t\in (0,1)}{sup}t^{1-\beta }\vert x_u(t)-x_v(t)\vert \rightarrow 0\mbox{,}\quad u,v\rightarrow +\infty \mbox{,}\\ \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}x(t)-D_{0^+}^{\beta }x_v(t)\vert \rightarrow 0\mbox{,}\quad u\\\displaystyle v\rightarrow +\infty \mbox{.} \end{array}

Define

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}t^{1-\beta }x_u(t)=\lbrace \begin{array}{c}\underset{t\rightarrow 0}{lim}t^{1-\beta }x_u(t)\mbox{,}\quad t=0\mbox{,}\\ t^{1-\beta }x_u(t)\mbox{,}\quad t\in (0,1)\mbox{,}\\ \underset{t\rightarrow 1}{lim}t^{1-\beta }x_u(t) \end{array}\\ \Phi (t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x_u(t)=\\\displaystyle =\lbrace \begin{array}{c}\underset{t\rightarrow 0}{lim}\Phi (t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t)\mbox{,}\quad t=0\mbox{,}\\ \Phi (t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x_u(t)\mbox{,}\quad t\in (0,1)\mbox{,}\\ \underset{t\rightarrow 1}{lim}\Phi (t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t)\mbox{.} \end{array} \end{array}

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow \Phi (t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x_u(t)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow t^{1-\beta }x_u(t)}
are continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]}

. Thus there exist two functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_0,y_0} 

defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]}
such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}\underset{u\rightarrow +\infty }{lim}t^{1-\beta }x_u(t)=x_0(t)\mbox{,}\quad \underset{u\rightarrow +\infty }{lim}\Phi (t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t)=\\\displaystyle =y_0(t)\mbox{.} \end{array}

It follows from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert x_u-x_v\Vert \rightarrow 0,\quad u,v\rightarrow +\infty } 

that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} \underset{t\in (0,1)}{sup}\vert t^{1-\beta }x_u(t)-x_0(t)\vert \rightarrow 0\mbox{,}\quad u\rightarrow +\infty \mbox{,}\\ \underset{t\in (0,1)}{sup}\vert \Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta}{D}}x(t)-y_0(t)\vert \rightarrow 0\mbox{,}\quad u\rightarrow +\infty \mbox{.} \end{array}

Denote Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z_0(t)=t^{\beta -1}x_0(t)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_0(t)=\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}}
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in (0,1)}

. This means that functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z_0,w_0:(0,1)\rightarrow R} 

are well defined.

Step 1.   Prove that the limits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 0}t^{1-\beta }z_0(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 1}t^{1-\beta }z_0(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 0}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 1}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)}
exist.

Since both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow t^{1-\beta }z_0(t)=x_0(t)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)=y_0(t)}
are continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,1]}

, the limits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 0}t^{1-\beta }z_0(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 1}t^{1-\beta }z_0(t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 0}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{t\rightarrow 1}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)}
exist.

Step 2.   Prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle sup_{t\in (0,1)}t^{1-\beta }\vert x_u(t)-z_0(t)\vert \rightarrow 0} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle sup_{t\in (0,1)}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}^{\beta }x_u(t)-w_0(t)\vert \rightarrow 0}
as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\rightarrow +\infty }
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

.

We have that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} \underset{t\in (0,1)}{sup}\vert t^{1-\beta }x_u(t)-x_0(t)\vert \rightarrow 0\mbox{,}\quad u\rightarrow +\infty \mbox{,}\\ \underset{t\in (0,1)}{sup}\vert \Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta}{D}}x(t)-y_0(t)\vert \rightarrow 0\mbox{,}\quad u\rightarrow +\infty \mbox{.} \end{array}

The results follow.

Step 3.   Prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y_0(t)=\Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }z_0(t)} 

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in (0,1)}

.

We have that there exists constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_u\in R} 

such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert x_u(t)+c_ut^{\beta -1}-I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}\vert =\vert I_{0^+}^{\beta }D_{0^+}^{\beta }x_u(t)-\\\displaystyle -I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}\vert =I_{0^+}^{\beta }\vert D_{0^+}^{\beta }x_u(t)-\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}\vert =\\\displaystyle =I_{0^+}^{\beta }\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}\vert \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x_u(t)-\\\displaystyle -y_0(t)\vert \leq \underset{t\in (0,1)}{sup}\vert \Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x_u(t)-\\\displaystyle -y_0(t)\vert \underset{0^+}{\overset{\beta }{I}}\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}\rightarrow 0\mbox{  as  }u\rightarrow +\infty \mbox{.}\end{array}

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle lim_{u\rightarrow +\infty }[x_u(t)+c_ut^{\beta -1}]=I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}} . Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t^{\beta -1}[x_0(t)+c_0t^{\beta -1}]=I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle z_0(t)+c_0t^{\beta -1}=I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}} . It follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_{0^+}^{\beta }z_0(t)=\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}} .

It follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X} 

is a Banach space. The proof is completed.

We list the following assumption:

(H).    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
is nonnegative and continuous, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a,b,c,d\geq 0}
are constants that satisfy
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t\rightarrow \frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}\mbox{  is a  }\beta \mbox{- well  function on  }(0,1)\mbox{,}

Lemma 2.3.

Suppose that (H) holds. Then for each  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X} , there exists a unique  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A_x\in R} satisfying

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}\vert A_x\vert \leq \underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\vert f(s,x(s)\\\displaystyle D_{0^+}^{\beta }x(s))\vert ds \end{array}
(4)

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle bd\Phi ^{-1}(A_x)+ac\Phi ^{-1}(A_x-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x(s)\\\displaystyle D_{0^+}^{\beta }x(s))ds)+\\\displaystyle +ad\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )A_x-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu )}{\rho (s)}ds=0\mbox{.}\end{array}
(5)

Proof.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X} , we find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert x\Vert =:r<+\infty } . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function (see Definition 2.4), there exists nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,x(t),D_{0^+}^{\beta }x(t))\vert =\vert f(t,t^{\beta -1}[t^{1-\beta }x(t)]\\\displaystyle \frac{\Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)})\vert \leq \phi _r(t)\mbox{.}\end{array}
(6)

Let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle G(u)=bd\Phi ^{-1}(u)+ac\Phi ^{-1}(u-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x(s)\\\displaystyle D_{0^+}^{\beta }x(s))ds)+\\\displaystyle +ad\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\times \frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )u-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu )}{\rho (s)}ds\mbox{.}\end{array}

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G} 

is well defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R}

. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Delta >0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G} 

is continuous and strictly increasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R}

. One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): G(\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\vert f(t,x(t),D_{0^+}^{\beta }x(t))\vert )\geq 0

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): G(-\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\vert f(t,x(t),D_{0^+}^{\beta }x(t))\vert )\leq 0\mbox{.}

Hence there exists a unique

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}A_x\in [-\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t),D_{0^+}^{\beta }x(t))\vert \\\displaystyle \underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t),D_{0^+}^{\beta }x(t))\vert ] \end{array}

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G(A_x)=0} . Furthermore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A_x} 

satisfies (4). Then (5) is proved.

Define the nonlinear operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}
by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle (Tx)(t)=\\\displaystyle =\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\Gamma (\alpha )I_{0^+}^{\alpha }f(s,x(s),D_{0^+}^{\beta }x(s))+\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +B_xt^{\beta -1}\end{array}
(7)

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A_x} 

satisfies (5) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_x}
is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B_x=\lbrace \begin{array}{c}\frac{b}{a}\Phi ^{-1}(A_x)\mbox{,}\quad a\not =0\mbox{,}\\ \frac{c}{d}\Phi ^{-1}(\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x(s)\\\displaystyle D_{0^+}^{\beta }x(s))ds)+\\\displaystyle +\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\times \frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )s^{1-\alpha }I_{0^+}^{\alpha }f(s,x(s),D_{0^+}^{\beta }x(s)))}{\rho (s)}ds\mbox{,}a=0\mbox{.} \end{array}
(8)

Remark 2.2.

It is easy to see from (8) that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert B_x\vert \leq \lbrace \begin{array}{c}\frac{b}{a}\Phi ^{-1}(\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t(t-s)^{\alpha -1}\vert f(s,x(s)\\\displaystyle D_{0^+}^{\beta }x(s))\vert ds)\mbox{if  }a\not =0\mbox{,}\\[] [\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds+\frac{c}{d}]\times \Phi ^{-1}(\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t(t-\\\displaystyle -s)^{\alpha -1}\vert f(s,x(s),D_{0^+}^{\beta }x(s))\vert ds)\mbox{  if  }a=0\mbox{.} \end{array}
(9)

Lemma 2.4.

Suppose that (H) holds. Then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T:X\rightarrow X} is completely continuous.

Proof.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X} , we find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert x\Vert \leq r<+\infty } . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function (see Definition 2.4), there exists nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
such that (6) holds. Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A_x}
is uniquely determined by (5). Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_x}
is uniquely determined by (8) too. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (Tx)(t)}
is well defined by (7). It is easy to see from (7) that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle D_{0^+}^{\beta }(Tx)(t)=\\\displaystyle =\frac{\Phi ^{-1}(-\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x(s),D_{0^+}^{\beta }x(s))ds+A_xt^{\alpha -1})}{\rho (t)}\mbox{.}\end{array}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx)(t)=\Phi ^{-1}(-t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}f(s\\\displaystyle x(s),D_{0^+}^{\beta }x(s))ds+A_x)\mbox{.}\end{array}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_{0^+}^{\alpha }[\Phi (\rho (t)D_{0^+}^{\beta }(Tx)(t))]=-f(t,x(t),D_{0^+}^{\beta }x(t))\mbox{.}
(10)

Furthermore, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}(Tx)\in C^0(0,1)\mbox{,}\quad D_{0^+}^{\beta }(Tx)\in C^0(0,1)\mbox{,}\\ \underset{t\rightarrow 0}{lim}t^{1-\beta }(Tx)(t)=B_x\mbox{,}\\ \underset{t\rightarrow 0}{lim}\quad \Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}(Tx)(t)=\Phi ^{-1}(A_x)\mbox{,}\\ \underset{t\rightarrow 1}{lim}\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}(Tx)(t)=\Phi ^{-1}(-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s\\\displaystyle x(s),D_{0^+}^{\beta }x(s))ds+A_x)\mbox{,}\\ \underset{t\rightarrow 1}{lim}t^{1-\beta }(Tx)(t)=B_x+\\\displaystyle +\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu +\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\mbox{.} \end{array}

From (5) and (8), we see

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} a\quad \underset{t\rightarrow 0}{lim}t^{1-\beta }(Tx)(t)-b\quad \underset{t\rightarrow 0}{lim}\quad \Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta}{D}}(Tx)(t)=0\mbox{,}\\ c\quad \underset{t\rightarrow 1}{lim}\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta}{D}}(Tx)(t)+d\quad \underset{t\rightarrow 1}{lim}t^{1-\beta }(Tx)(t)=0\mbox{.} \end{array}
(11)

Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T:X\rightarrow X} 

is well defined and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X}
is a solution of BVP(3) if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
is a fixed point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

.

To prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

is completely continuous, we address the following three steps.

Step 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

is continuous.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace x_n\rbrace _{n=0}^{\infty }} 

be a sequence such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_n\rightarrow x_0}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

. We will show Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Tx_n\rightarrow Tx_0} 

as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\rightarrow +\infty }

. We see

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): r=\underset{n=0,1,2,\ldots}{sup}\Vert x_n\Vert <\infty \mbox{.}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function (see Definition 2.4), there exists nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
such that (6) holds with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}
being replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_n}

.

First we prove that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): A_{x_n}\rightarrow A_{x_0}\quad \mbox{  as  }n\rightarrow +\infty \mbox{.}
(12)

From Lemma 2.2, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert A_{x_n}\vert \leq \underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\vert f(s,x_n(s)\\\displaystyle D_{0^+}^{\beta }x_n(s))\vert ds\leq \underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(s)ds\mbox{.}\end{array}

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace A_{x_n}\rbrace } 

is bounded. If (12) does not hold, then there exist two subsequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace A_{x_{n_k}}^{(1)}\rbrace }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace A_{x_{n_k}}^{(2)}\rbrace }
of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace A_{x_n}\rbrace }
such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): A_{x_{n_k}}^{(i)}\rightarrow c_i(i=1,2)\quad \mbox{  as  }n\rightarrow +\infty

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_1\not =c_2} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A_{x_{n_k}}^{(i)}} 

satisfies
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle bd\Phi ^{-1}(A_{x_n_k}^{(i)})+ac\Phi ^{-1}(A_{x_n_k}^{(i)}-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s\\\displaystyle x_{n_k^(i)}(s)\\\displaystyle D_{0^+}^{\beta }x_{n_k^(i)}(s))ds)+\\\displaystyle +ad\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )A_{x_n_k}^{(i)}-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x_{n_k^(i)}(\nu ),D_{0^+}^{\beta }x_{n_k^(i)}(\nu ))d\nu )}{\rho (s)}ds=0\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert \int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )A_{x_n_k}^{(i)}-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x_{n_k^(i)}(\nu ),D_{0^+}^{\beta }x_{n_k^(i)}(\nu ))d\nu )}{\rho (s)}ds\vert \leq \int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds\Phi ^{-1}((\Gamma (\alpha )+\\\displaystyle +1)\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t(t-s)^{\alpha -1}\phi _r(s)ds)\mbox{,}\end{array}

by Lebesgue’s dominant theorem, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k\rightarrow +\infty } , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle bd\Phi ^{-1}(c_i)+ac\Phi ^{-1}(c_i-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x_0(s)\\\displaystyle D_{0^+}^{\beta }x_0(s))ds)+\\\displaystyle +ad\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\times \frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )c_i-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x_0(\nu ),D_{0^+}^{\beta }x_0(\nu ))d\nu )}{\rho (s)}ds=0\mbox{.}\end{array}

Since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle bd\Phi ^{-1}(A_{x_0})+ac\Phi ^{-1}(A_{x_0}-\int _0^1\frac{(1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x_0(s)\\\displaystyle D_{0^+}^{\beta }x_0(s))ds)+\\\displaystyle +ad\int _0^1\frac{(1-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\times \frac{\Phi ^{-1}(s^{\alpha -1})\Phi ^{-1}(\Gamma (\alpha )A_{x_0}-s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x_0(\nu ),D_{0^+}^{\beta }x_0(\nu ))d\nu )}{\rho (s)}ds=0\mbox{,}\end{array}

together with Lemma 2.2, we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle c_1=c_2=A_{x_0}} , a contradiction. So (12) holds.

Now by the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_x} , one has

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B_{x_n}\rightarrow B_{x_0}\quad \mbox{  as  }n\rightarrow +\infty \mbox{.}
(13)

From

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx_n)(t)=\Phi ^{-1}(-\\\displaystyle -t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,x_n(s),D_{0^+}^{\beta }x_n(s))ds+A_{x_n})\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t^{1-\beta }(Tx_n)(t)=\\\displaystyle =t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x_n(\nu ),D_{0^+}^{\beta }x_n(\nu ))d\nu +\Gamma (\alpha )A_{x_n}s^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +B_{x_n}\mbox{,}\end{array}

since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function (see Definition 2.4) and (12), (13), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Tx_n\rightarrow Tx_0}
as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\rightarrow +\infty }

. Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

is continuous.

Step 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

maps bounded sets into bounded sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

.

It suffices to show that for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle r>0} , there exists a positive number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L>0} 

such that for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace }

, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert Ty\Vert \leq L} . By the assumption, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f} 

is a quasi-Carathéodory function (see Definition 2.4), there exists a nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
(see Definition 2.3) such that (6) holds. By the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}

, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}^{\beta }(Ty)(t)\vert =\vert \Phi ^{-1}(-\\\displaystyle -t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))ds+A_{x_n})\vert \leq \vert \Phi ^{-1}(t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(s)ds+\\\displaystyle +\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-u)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(u)du)\vert \leq \Phi ^{-1}(2\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(s)ds)\mbox{.}\end{array}

Now we consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t^{1-\beta }\vert (Ty)(t)\vert } .

Case 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a\not =0} . We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t^{1-\beta }\vert (Ty)(t)\vert =\\\displaystyle =\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +B_y\vert \leq \vert B_y\vert +\\\displaystyle +\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq (\frac{b}{a}+\\\displaystyle +\underset{t\in [0,1]}{max}t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds)\Phi ^{-1}(2\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(s)ds)\mbox{.}\end{array}

Case 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=0} . We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t^{1-\beta }\vert (Ty)(t)\vert =\\\displaystyle =\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +B_y\vert \leq \vert B_y\vert +\\\displaystyle +\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq (\frac{c}{d}+\\\displaystyle +2\underset{t\in [0,1]}{max}t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds)\Phi ^{-1}(2\underset{t\in [0,1]}{max}t^{1-\alpha }\int _0^t\frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}\phi _r(s)ds)\mbox{.}\end{array}

It follows that there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M>0} 

such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert Ty\Vert \leq M

for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle y\in \lbrace y\in X:\Vert y\Vert \leq l\rbrace } . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

maps bounded sets into bounded sets in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

.

Step 3.   Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace } . Prove that both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace t\rightarrow t^{1-\beta }(Tx)(t):x\in M\rbrace } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx)(t):x\in M\rbrace }
are equicontinuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,1)}

.

By Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace } , there exists nonnegative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta } -well Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} 

function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r}
such that (6) holds. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_1,t_2\in (0,1)}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_2<t_2}

. We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert t_1^{1-\beta }(Ty)(t_1)-t_2^{1-\beta }(Ty)(t_2)\vert =\\\displaystyle =\vert t_1^{1-\beta }\int _0^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds-\\\displaystyle -t_2^{1-\beta }\int _0^{t_2}\frac{(t_2-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert =\\\displaystyle =\vert t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +\int _0^{t_2}\frac{t_1^{1-\beta }(t_1-s)^{\beta -1}-t_2^{1-\beta }(t_2-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \mbox{.}\end{array}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t^{1-\beta }(t-s)^{\beta -1}} 

is decreasing, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert t_1^{1-\beta }(Ty)(t_1)-\\\displaystyle -t_2^{1-\beta }(Ty)(t_2)\vert \leq t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\vert \Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})\vert }{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds+\\\displaystyle +\int _0^{t_2}\frac{t_2^{1-\beta }(t_2-s)^{\beta -1}-t_1^{1-\beta }(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\vert \Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,y(\nu ),D_{0^+}^{\beta }y(\nu ))d\nu +\Gamma (\alpha )A_ys^{\alpha -1})\vert }{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\mbox{.}\end{array}

Using Lemma 2.2, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert t_1^{1-\beta }(Ty)(t_1)-\\\displaystyle -t_2^{1-\beta }(Ty)(t_2)\vert \leq \frac{1}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\vert t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds+\\\displaystyle +\int _0^{t_2}\frac{t_2^{1-\beta }(t_2-s)^{\beta -1}-t_1^{1-\beta }(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds\vert \Phi ^{-1}(2\underset{s\in [0,1]}{max}s^{1-\alpha }\int _0^s(s-\nu )^{\alpha -1}\vert f(\nu \\\displaystyle y(\nu )\\\displaystyle D_{0^+}^{\beta }y(\nu ))\vert d\nu )\leq \frac{1}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\vert t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds+\\\displaystyle +\int _0^{t_2}\frac{t_2^{1-\beta }(t_2-s)^{\beta -1}-t_1^{1-\beta }(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds\vert \Phi ^{-1}(2\underset{s\in [0,1]}{max}s^{1-\alpha }\int _0^s(s-\\\displaystyle -u)^{\alpha -1}\phi _r(u)du)=\frac{1}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\vert t_1^{1-\beta }\int _0^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds+\\\displaystyle +t_2^{1-\beta }\int _0^{t_2}\frac{(t_2-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds-\\\displaystyle -2t_1^{1-\beta }\int _0^{t_2}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds\vert \Phi ^{-1}(2\underset{s\in [0,1]}{max}s^{1-\alpha }\int _0^s(s-\\\displaystyle -\nu )^{\alpha -1}\phi _r(\nu )d\nu )\mbox{.}\end{array}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds} 

is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well function, then we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert t_1^{1-\beta }(Ty)(t_1)-t_2^{1-\beta }(Ty)(t_2)\vert \rightarrow 0\quad \mbox{  uniformly as  }t_1\rightarrow t_2\mbox{.}
(14)

Similarly, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert \Phi ^{-1}(t_1^{1-\alpha })\rho (t_1)D_{0^+}^{\beta }(Ty)(t_1)-\\\displaystyle -\Phi ^{-1}(t_2^{1-\alpha })\rho (t_2)D_{0^+}^{\beta }(Ty)(t_2)\vert =\vert \Phi ^{-1}(-\\\displaystyle -t_1^{1-\alpha }\int _0^{t_1}\frac{(t_1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))ds+A_y)-\Phi ^{-1}(-t_2^{1-\alpha }\int _0^{t_2}\frac{(t_2-s)^{\alpha -1}}{\Gamma (\alpha )}f(s\\\displaystyle y(s),D_{0^+}^{\beta }y(s))ds+A_y)\vert \mbox{.}\end{array}
(15)

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert (-t_1^{1-\alpha }\int _0^{t_1}\frac{(t_1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))ds+A_y)-(-t_2^{1-\alpha }\int _0^{t_2}\frac{(t_2-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))ds+A_y)\vert =\vert -t_1^{1-\alpha }\int _0^{t_1}\frac{(t_1-s)^{\alpha -1}}{\Gamma (\alpha )}f(s\\\displaystyle y(s),D_{0^+}^{\beta }y(s))ds+t_2^{1-\alpha }\int _0^{t_2}\frac{(t_2-s)^{\alpha -1}}{\Gamma (\alpha )}f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))ds\vert \leq \vert t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\vert f(s,y(s)\\\displaystyle D_{0^+}^{\beta }y(s))\vert ds+\int _0^{t_2}\frac{t_2^{1-\beta }(t_2-s)^{\beta -1}-t_1^{1-\beta }(t_1-s)^{\beta -1}}{\Gamma (\beta )}\vert f(s\\\displaystyle y(s)\\\displaystyle D_{0^+}^{\beta }y(s))\vert ds\vert \leq \vert t_1^{1-\beta }\int _{t_2}^{t_1}\frac{(t_1-s)^{\beta -1}}{\Gamma (\beta )}\phi _r(s)ds+\\\displaystyle +\int _0^{t_2}\frac{t_2^{1-\beta }(t_2-s)^{\beta -1}-t_1^{1-\beta }(t_1-s)^{\beta -1}}{\Gamma (\beta )}\phi _r(s)ds\vert =\\\displaystyle =\vert t_1^{1-\beta }I_{0^+}^{\beta }\phi _r(t_1)+t_2^{1-\beta }I_{0^+}^{\beta }\phi _r(t_2)-\\\displaystyle -2t_1^{1-\beta }I_{0^+}^{\beta }\phi _r(t_2)\vert \mbox{.}\end{array}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi _r} 

is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \beta }

-well function, then we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert (-t_1^{1-\alpha }I_{0^+}^{\alpha }f(t_1,y(t_1)\\\displaystyle D_{0^+}^{\beta }y(t_1))+A_y)-(-t_2^{1-\alpha }I_{0^+}^{\alpha }f(t_2,y(t_2)\\\displaystyle D_{0^+}^{\beta }y(t_2))+A_y)\vert \rightarrow 0\mbox{  uniformly as  }t_1\rightarrow t_2\mbox{.}\end{array}
(16)

It is easy to see that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}\vert (-t^{1-\alpha }I_{0^+}^{\alpha }f(t,y(t)\\\displaystyle D_{0^+}^{\beta }y(t))+A_y)\vert \leq 2\Gamma (\alpha )\underset{s\in [0,1]}{max}s^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi _r(t)=:L\mbox{,} \end{array}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}} 

is uniformly continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [-L,L]}

, then for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon >0} , there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta _0>0} 

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert x_1-x_2\vert <\delta _0}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_1,x_2\in [-L,L]}
implies
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert \Phi ^{-1}(x_1)-\Phi ^{-1}(x_2)\vert <\epsilon \mbox{.}

It follows from (16) that there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta >0} 

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_1,t_2\in (0,1)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert t_1-t_2\vert <\delta }
imply
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert (-t_1^{1-\alpha }I_{0^+}^{\alpha }f(t_1,y(t_1)\\\displaystyle D_{0^+}^{\beta }y(t_1))+A_y)-(-t_2^{1-\alpha }I_{0^+}^{\alpha }f(t_2,y(t_2)\\\displaystyle D_{0^+}^{\beta }y(t_2))+A_y)\vert <\delta _0\mbox{.}\end{array}

Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_1,t_2\in (0,1)} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert t_1-t_2\vert <\delta }
imply
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert \Phi ^{-1}(t_1^{1-\alpha })\rho (t_1)D_{0^+}^{\beta }(Ty)(t_1)-\\\displaystyle -\Phi ^{-1}(t_2^{1-\alpha })\rho (t_2)D_{0^+}^{\beta }(Ty)(t_2)\vert =\vert \Phi ^{-1}(-\\\displaystyle -t_1^{1-\alpha }I_{0^+}^{\alpha }f(t_1,y(t_1)\\\displaystyle D_{0^+}^{\beta }y(t_1))+A_y)-\Phi ^{-1}(-t_2^{1-\alpha }I_{0^+}^{\alpha }f(t_2,y(t_2)\\\displaystyle D_{0^+}^{\beta }y(t_2))+A_y)\vert <\epsilon \mbox{.}\end{array}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}\vert \Phi ^{-1}(t_1^{1-\alpha })\rho (t_1)D_{0^+}^{\beta }(Ty)(t_1)-\\\displaystyle -\Phi ^{-1}(t_2^{1-\alpha })\rho (t_2)D_{0^+}^{\beta }(Ty)(t_2)\vert \rightarrow 0 \end{array}
(17)

uniformly as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_1\rightarrow t_2} .

From (14) and (17), both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace t\rightarrow t^{1-\beta }(Tx)(t):x\in M\rbrace } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx)(t):x\in M\rbrace }
are equicontinuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,1)}

.

The Arzelà–Ascoli theorem implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle TM} 

is relatively compact. Thus, the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T:X\rightarrow X}
is completely continuous.

In order to prove the existence of solution of BVP(3), we use the following topological transversality theorem given by Granas  [7].

Lemma 2.5.

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B} be a convex subset of a normed linear space  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X} and assume  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0\in B} . Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T:B\rightarrow B} be a completely continuous operator and let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(T)=\lbrace x:x=\lambda Tx\rbrace } for some  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 0<\lambda <1} . Then either  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(T)} is unbounded or  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} has a fixed point.

Theorem 2.1.

Suppose that (H) holds and there exist nonnegative  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} functions  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A,B} and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{cc} & \vert f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)\vert \leq A(t)\Phi (\vert u\vert )+B(t)\Phi (\vert v\vert )+C(t) \end{array}
(18)

holds for all  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in (0,1),u,v\in R} . Then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle BVP(3)} has at least one solution if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \delta _0=:\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}A(t))+\\\displaystyle +\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}B(t))<\frac{1}{C_q\mu _0}\mbox{   for   }a\not =0\mbox{,}\end{array}
(19)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} C_q=\lbrace \begin{array}{c} 1\mbox{,}\quad q\in (1,2]\mbox{,}\\ 2^{q-2}\mbox{,}\quad q>2\mbox{.} \end{array}\quad \frac{1}{p}+\frac{1}{q}=1,p,q\mbox{   are defined in Section   }1\mbox{,}\\ \mu _0=\frac{2}{\phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{sup}t^{1-\beta }\underset{0^+}{\overset{\beta}{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\frac{c}{d}+1\mbox{,}\\ \mu _1=\frac{1}{\phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{sup}t^{1-\beta }\underset{0^+}{\overset{\beta}{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\frac{b}{a}+1\mbox{.} \end{array}

Proof.

We shall prove that under the assumptions (18) and (19), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

has at least one fixed point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}
by using Leray–Schauder Alternative principle. Let
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): U(T)=\lbrace x:x=\lambda Tx\mbox{  for some  }0<\lambda <1\rbrace \mbox{.}

Case 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a\not =0} . Indeed, by the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in U(T)}

, by using (4) and (9), we have the estimates

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t^{1-\beta }\vert (Tx)(t)\vert =\vert B_x+\\\displaystyle +t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu +\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq \vert B_x\vert +\\\displaystyle +\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\times \frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu +\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t)\\\displaystyle D_{0^+}^{\beta }x(t))\vert )\leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (u^{1-\beta }\vert x(t)\vert )+\\\displaystyle +B(t)\Phi (\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}x(t)\vert )+\\\displaystyle +C(t)])\leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}^{\beta }(Tx)(t)\vert =\vert \Phi ^{-1}(-t^{1-\alpha }I_{0^+}^{\alpha }f(t\\\displaystyle x(t)\\\displaystyle D_{0^+}^{\beta }x(t))+A_x)\vert \leq \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\mbox{.}\end{array}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert Tx\Vert =max\lbrace \underset{t\in (0,1)}{sup}\vert t^{1-\beta }\vert (Tx)(t)\vert \\\displaystyle \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}(Tx)(t)\vert \rbrace \leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\mbox{.}\end{array}

One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Phi ^{-1}(a+b+c)\leq C_q[\Phi (a)+\Phi (b)+\Phi (c)]\mbox{,}\quad a,b,c\geq 0,\quad C_q=\lbrace \begin{array}{c} 1\mbox{,}\quad q\in (1,2]\mbox{,}\\ 2^{q-2}\mbox{,}\quad q>2\mbox{.} \end{array}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert x\Vert =\lambda \Vert Tx\Vert \leq \Vert Tx\Vert \leq C_q\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}A(t))\Vert x\Vert +\\\displaystyle +C_q\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}B(t))\Vert x\Vert +\\\displaystyle +C_q\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}C(t))\mbox{.}\end{array}

From (19), we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert x\Vert \leq \frac{C_q\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}C(t))}{1-C_q\mu _0\delta }\mbox{.}

It follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(T)} 

is bounded. Hence Lemma 2.4 implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}
has a fixed point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

. Then BVP(3) has at least one solution.

Case 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=0} . Indeed, by the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in X}

, by using (4) and (9), we have the estimates

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle t^{1-\beta }\vert (Tx)(t)\vert =\vert B_x+\\\displaystyle +t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(u,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu +\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq \vert B_x\vert +\\\displaystyle +\vert t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )}\frac{\Phi ^{-1}(-\int _0^s(s-\nu )^{\alpha -1}f(\nu ,x(\nu ),D_{0^+}^{\beta }x(\nu ))d\nu +\Gamma (\alpha )A_xs^{\alpha -1})}{\Phi ^{-1}(\Gamma (\alpha ))\rho (s)}ds\vert \leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{b}{a}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t)\\\displaystyle D_{0^+}^{\beta }x(t))\vert )\leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{b}{a}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (t^{1-\beta }\vert x(t)\vert )+\\\displaystyle +B(t)\Phi (\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}x(t)\vert )+\\\displaystyle +C(t)])\leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{b}{a}]\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Phi ^{-1}(t^{1-\alpha })\rho (t)\vert D_{0^+}^{\beta }(Tx)(t)\vert =\vert \Phi ^{-1}(-t^{1-\alpha }I_{0^+}^{\alpha }f(t\\\displaystyle x(t)\\\displaystyle D_{0^+}^{\beta }x(t))+A_x)\vert \leq \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\mbox{.}\end{array}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert Tx\Vert =max\lbrace \underset{t\in (0,1)}{sup}\vert t^{1-\beta }\vert (Tx)(t)\vert \\\displaystyle \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}(Tx)(t)\vert \rbrace \leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{b}{a}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}[A(t)\Phi (\Vert x\Vert )+\\\displaystyle +B(t)\Phi (\Vert x\Vert )+C(t)])\mbox{.}\end{array}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert x\Vert =\lambda \Vert Tx\Vert \leq \Vert Tx\Vert \leq C_q\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}A(t))\Vert x\Vert +\\\displaystyle +C_q\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}B(t))\Vert x\Vert +\\\displaystyle +C_q\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}C(t))\mbox{.}\end{array}

From (19), we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert x\Vert \leq \frac{C_q\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}C(t))}{1-C_q\mu _1\delta }\mbox{.}

It follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle U(T)} 

is bounded. Hence Lemma 2.4 implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}
has a fixed point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle X}

. Then BVP(3) has at least one solution. From Cases 1 and 2, the proof is completed.

Theorem 2.2.

Suppose that (H) holds and that there exists  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\alpha }^1} function  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi } such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert f(t,t^{\beta -1}u,\frac{v}{\Phi ^{-1}(t^{1-\alpha })\rho (t)})\vert \leq \phi (t)w(\vert u\vert +\vert v\vert )\mbox{,}\quad t\in (0,1],u,v\in R
(20)

with  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w\in C(R,[0,\infty ))} nondecreasing. If

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \underset{\mu >0}{sup}\frac{\mu }{\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\mu ))}>1\quad \mbox{   for   }a\not =0\mbox{,}
(21)

then, BVP(3) has at least one solution, where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mu _0=\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{sup}t^{1-\beta }\underset{0^+}{\overset{\beta}{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\frac{c}{d}+1\mbox{,}

Proof.

It follows from  and  that there exist constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{\mu }_0,\overline{\mu }_1>0} 

such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\overline{\mu }_0}{\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\overline{\mu }_0))}>1\quad \mbox{  if  }a\not =0\mbox{,}
(23)

Case 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a\not =0} . Let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): U=\lbrace x\in X:\Vert x\Vert \leq \overline{\mu }_0\rbrace \mbox{.}

We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\not =\lambda Tx} 

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. In fact, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=\lambda Tx} 

for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert x\Vert =max\lbrace \underset{t\in (0,1)}{sup}\lambda t^{1-\beta }\vert (Tx)(t)\vert \\\displaystyle \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}\lambda \vert (Tx)(t)\vert \rbrace \leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t)\\\displaystyle \underset{0^+}{\overset{\beta }{D}}x(t))\vert )\leq [2\underset{t\in [0,1]}{max}t^{1-\beta }\int _0^t\frac{(t-s)^{\beta -1}}{\Gamma (\beta )\Phi ^{-1}(\Gamma (\alpha ))}\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t)w(t^{1-\beta }\vert x(t)\vert +\\\displaystyle +\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x(t)\vert ))\leq [\frac{2}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t)w(2\Vert x\Vert ))=\\\displaystyle =\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t))\Phi ^{-1}(w(2\Vert x\Vert ))\mbox{.}\end{array}

So

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\mu }_0\leq \mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\overline{\mu }_0))\mbox{.}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\overline{\mu }_0}{\mu _0\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\overline{\mu }_0))}\leq 1\mbox{,}

which contradicts with (23). From the above discussion, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\not =\lambda Tx} 

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

is completely continuous, by Lemma 2.4, we see that BVP(3) has at least one solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}

.

Case 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=0} . Let

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): U=\lbrace x\in X:\Vert x\Vert \leq \overline{\mu }_1\rbrace \mbox{.}

We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\not =\lambda Tx} 

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. In fact, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=\lambda Tx} 

for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\in \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert x\Vert =max\lbrace \underset{t\in (0,1)}{sup}\lambda t^{1-\beta }\vert (Tx)(t)\vert \\\displaystyle \underset{t\in (0,1)}{sup}\Phi ^{-1}(t^{1-\alpha })\rho (t)\vert \underset{0^+}{\overset{\beta }{D}}\lambda \vert (Tx)(t)\vert \rbrace \leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\vert f(t,x(t)\\\displaystyle \underset{0^+}{\overset{\beta }{D}}x(t))\vert )\leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\frac{c}{d}+\\\displaystyle +1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t)w(t^{1-\beta }\vert x(t)\vert +\\\displaystyle +\Phi ^{-1}(t^{1-\alpha })\rho (t)\underset{0^+}{\overset{\beta }{D}}x(t)\vert ))\leq [\frac{1}{\Phi ^{-1}(\Gamma (\alpha ))}\underset{t\in [0,1]}{max}t^{1-\beta }\underset{0^+}{\overset{\beta }{I}}\frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}+\\\displaystyle +\frac{c}{d}+1]\times \Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t)w(2\Vert x\Vert ))=\\\displaystyle =\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha }{I}}\phi (t))\Phi ^{-1}(w(2\Vert x\Vert ))\mbox{.}\end{array}

So

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \overline{\mu }_1\leq \mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\overline{\mu }_1))\mbox{.}

It follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\overline{\mu }_1}{\mu _1\Phi ^{-1}(2\Gamma (\alpha )\underset{t\in [0,1]}{max}t^{1-\alpha }\underset{0^+}{\overset{\alpha}{I}}\phi (t))\Phi ^{-1}(w(2\overline{\mu }_1))}\leq 1\mbox{,}

which contradicts with (24).

From the above discussion, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x\not =\lambda Tx} 

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda \in (0,1)}

. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

is completely continuous, by Lemma 2.4, we see that BVP(3) has at least one solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x}

. The proof is complete.

3. Two examples

In this section, we give two examples to illustrate the main theorems.

Example 3.1.

Consider the following BVP

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c}D_{0^+}^{\frac{1}{2}}[t^{\frac{1}{2}}(1-t)^{\frac{1}{2}}D_{0^+}^{\frac{1}{2}}x(t)]+f(t,x(t)\\\displaystyle D_{0^+}^{\frac{1}{2}}x(t))=0\mbox{,}t\in (0,1)\mbox{,}\\ \underset{t\rightarrow 0}{lim}t^{\frac{1}{2}}x(t)-\underset{t\rightarrow 0}{lim}t^{\frac{1}{2}}\underset{0^+}{\overset{\frac{1}{2}}{D}}x(t)=0\mbox{,}\\ \underset{t\rightarrow 1}{lim}t^{\frac{1}{2}}\underset{0^+}{\overset{\frac{1}{2}}{D}}x(t)+\underset{t\rightarrow 1}{lim}t^{\frac{1}{2}}x(t)=0\mbox{.} \end{array}
(25)

We consider two cases:

Case 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle f(t,u(t)\\\displaystyle D_{0^+}^{\frac{1}{2}}x(t))=2t^{-\frac{1}{2}}+\lambda (t-\frac{1}{2})^4t^{\frac{1}{2}}x(t)+\mu t(1-t)^{\frac{1}{2}}D_{0^+}^{\frac{1}{2}}x(t)\end{array}} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda ,\mu >0}

.

Corresponding to BVP(3), we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =\beta =\frac{1}{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=b=c=d=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi (x)=x} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}(x)=x}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho (t)=t^{\frac{1}{2}}(1-t)^{\frac{1}{2}}} 

and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(t,u,v)=2t^{-\frac{1}{2}}+\lambda (t-\frac{1}{2})^4t^{\frac{1}{2}}u+\mu t(1-t)^{\frac{1}{2}}v\mbox{.}

Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=\lambda (t-\frac{1}{2})^4} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)=\mu } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)=2t^{-\frac{1}{2}}}

. One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,t^{-\frac{1}{2}}u\\\displaystyle t^{-1}(1-t)^{-\frac{1}{2}}v)\vert \leq A(t)\vert u\vert +B(t)\vert v\vert +C(t)\mbox{,}t\in (0,1),u\\\displaystyle v\in R\mbox{.}\end{array}

It is easy to see that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \mu _0\leq 2\underset{t\in [0,1]}{sup}t^{1/2}\int _0^t\frac{(t-s)^{-1/2}}{[\Gamma (1/2)]^2}\frac{1}{s^{3/4}(t-s)^{1/4}}ds+2=2+\\\displaystyle +\frac{2B(1/4,1/4)}{[\Gamma (1/2)]^2}\mbox{.}\end{array}

Hence Theorem 2.1 implies that BVP(25) has a solution if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle 2\lambda \underset{t\in [0,1]}{max}t^{\frac{1}{2}}\int _0^t(t-s)^{-\frac{1}{2}}(s-1/2)^4ds+\\\displaystyle +2\mu \underset{t\in [0,1]}{max}t^{\frac{1}{2}}\int _0^t(t-s)^{-\frac{1}{2}}ds<\frac{1}{2+\frac{2B(1/4,1/4)}{[\Gamma (1/2)]^2}}\mbox{.}\end{array}

So BVP(25) has a solution for sufficiently small Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }

.

Case 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle f(t,x(t)\\\displaystyle D_{0^+}^{\frac{1}{2}}x(t))=t^{-\frac{1}{2}}(t^{\frac{1}{2}}x(t)+t(1-t)^{\frac{1}{2}}D_{0^+}^{\frac{1}{2}}x(t))^{1/2}\end{array}} .

Corresponding to BVP(3), we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =\beta =\frac{1}{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=b=c=d=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi (x)=x} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}(x)=x}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho (t)=t^{\frac{1}{2}}(1-t)^{\frac{1}{2}}} 

and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(t,t^{-\frac{1}{2}}u,t^{-1}(1-t)^{-\frac{1}{2}}v)=\phi (t)(u+v)^{1/2}\mbox{.}

Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(x)=x^{1/2}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi (t)=t^{-\frac{1}{2}}}

. We find that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,t^{-\frac{1}{2}}u,t^{-1}(1-t)^{-\frac{1}{2}}v)\vert \leq \phi (t)w(\vert u\vert +\vert v\vert )\mbox{,}\quad t\in (0\\\displaystyle 1),u,v\in R\mbox{.}\end{array}

One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \mu _0\leq 2\underset{t\in [0,1]}{sup}t^{1/2}\int _0^t\frac{(t-s)^{-1/2}}{[\Gamma (1/2)]^2}\frac{1}{s^{3/4}(t-s)^{1/4}}ds+2=2+\\\displaystyle +\frac{2B(1/4,1/4)}{[\Gamma (1/2)]^2}\mbox{.}\end{array}

It is easy to see from Theorem 2.2 that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \underset{\mu >0}{sup}\frac{\mu }{(2\underset{t\in [0,1]}{max}t^{\frac{1}{2}}\int _0^t(t-s)^{-\frac{1}{2}}s^{-\frac{1}{2}}ds)w(2\mu )}>2+\frac{2B(1/4,1/4)}{[\Gamma (1/2)]^2}
(26)

implies that BVP(25) has at least one solution. It is easy to see that (26) is always correct since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(x)=x^{1/2}} . Hence BVP(25) has at least one solution.

Example 3.2.

Consider the following BVP

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} D_{0^+}^{\frac{1}{2}}[(D_{0^+}^{\frac{1}{2}}x(t))^3]+f(t,x(t),D_{0^+}^{\frac{1}{2}}x(t))=0\mbox{,}\quad t\in (0,1)\mbox{,}\\ \underset{t\rightarrow 0}{lim}t^{\frac{1}{2}}x(t)-\underset{t\rightarrow 0}{lim}\quad t^{\frac{1}{6}}\underset{0^+}{\overset{\frac{1}{2}}{D}}x(t)=0\mbox{,}\\ \underset{t\rightarrow 1}{lim}t^{\frac{1}{6}}\underset{0^+}{\overset{\beta}{D}}x(t)+\underset{t\rightarrow 1}{lim}t^{\frac{1}{2}}x(t)=0\mbox{.} \end{array}
(27)

We consider two cases:

Case 1.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(t,u(t),D_{0^+}^{\frac{1}{2}}x(t))=2t^{-\frac{1}{2}}+\lambda t^{\frac{1}{2}}x(t)+\mu t^{\frac{1}{6}}D_{0^+}^{\frac{1}{2}}x(t)} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda ,\mu >0}

.

Corresponding to BVP(3), we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =\beta =\frac{1}{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=b=c=d=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi (x)=x^3} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}(x)=x^{\frac{1}{3}}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho (t)=1} 

and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(t,u,v)=2t^{-\frac{1}{2}}+\lambda t^{\frac{1}{2}}u+\mu t^{\frac{1}{6}}v\mbox{.}

Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A(t)=\lambda } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(t)=\mu } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C(t)=2t^{-\frac{1}{2}}}

. One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,t^{-\frac{1}{2}}u,t^{-\frac{1}{6}}v)\vert \leq A(t)\vert u\vert +B(t)\vert v\vert +C(t)\mbox{,}\quad t\in (0\\\displaystyle 1),u,v\in R\mbox{.}\end{array}

It is easy to see that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \mu _0=2+2\underset{t\in [0,1]}{sup}t^{\frac{5}{6}}\int _0^1\frac{(1-u)^{-1/2}}{[\Gamma (1/2)]^{5/3}}u^{-\frac{1}{6}}du\leq 2+\\\displaystyle +\frac{2B(1/2,5/6)}{[\Gamma (1/2)]^{5/3}}\mbox{.}\end{array}

Hence Theorem 2.1 implies that BVP(27) has a solution if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} 4\lambda 3+4\mu 3<\frac{[\Gamma (1/2)]^{5/3}}{2[\Gamma (1/2)]^{5/3}+2B(1/2,5/6)}\mbox{.} \end{array}

So BVP(27) has a solution for sufficiently small Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda } 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }

.

Case 2.    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f(t,x(t),D_{0^+}^{\frac{1}{2}}x(t))=t^{-\frac{1}{2}}w(t^{\frac{1}{2}}x(t)+t^{\frac{1}{6}}D_{0^+}^{\frac{1}{2}}x(t))} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w:R\rightarrow R} 

is a continuous function.

Corresponding to BVP(3), we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =\beta =\frac{1}{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a=b=c=d=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi (x)=x^3} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Phi ^{-1}(x)=x^{\frac{1}{3}}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho (t)=1} 

and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f(t,t^{-\frac{1}{2}}u,t^{-\frac{1}{6}}v)=\phi (t)w(u+v)\mbox{.}

Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w(x)=x^{1/2}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \phi (t)=t^{-\frac{1}{2}}}

. We find that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert f(t,t^{-\frac{1}{2}}u,t^{-\frac{1}{6}}v)\vert \leq \phi (t)w(\vert u\vert +\vert v\vert )\mbox{,}\quad t\in (0,1),u\\\displaystyle v\in R\mbox{.}\end{array}

One sees that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \mu _0\leq 2\underset{t\in [0,1]}{sup}t^{1/2}\int _0^t\frac{(t-s)^{-1/2}}{[\Gamma (1/2)]^{5/3}}s^{-\frac{1}{6}}ds+2=2+\\\displaystyle +\frac{2B(1/2,5/6)}{[\Gamma (1/2)]^{5/3}}\mbox{.}\end{array}

It is easy to see from Theorem 2.2 that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} \underset{u>0}{sup}\frac{u}{2B(1/2,1/2)3w(2u)3}>2+\frac{2B(1/2,5/6)}{[\Gamma (1/2)]^{5/3}}\mbox{.} \end{array}
(28)

implies that BVP(27) has at least one solution.

4. Conclusions

The existence of solutions of a class of boundary value problems for nonlinear fractional differential equations involving Riemann–Liouville fractional derivatives is studied. The fractional differential equation concerned in (3) is a composition of two left-sided Riemann–Liouville fractional derivatives. The investigation shows that these results and methods are helpful for study in the nonlinear area.

As is well known for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \in [n-1,n)} 

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in \lbrace 1,2,3,\ldots )}
that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} D_{a^+}^{\alpha }g(t)=\frac{1}{\Gamma (n-\alpha )}\frac{\partial ^n}{\partial ^nt}\int _a^t(t-s)^{n-\alpha -1}g(s)ds\mbox{,}\quad \mbox{  and  }\\ D_{b^{-}}^{\alpha }g(t)=\frac{1}{\Gamma (n-\alpha )}\frac{\partial ^n}{\partial ^nt}\int _t^b(s-t)^{n-\alpha -1}g(s)ds \end{array}

are called left-sided Riemann–Liouville fractional derivative and right-sided Riemann–Liouville fractional derivative respectively and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c} ^cD_{a^+}^{\alpha }g(t)=\frac{1}{\Gamma (n-\alpha )}\int _a^t(t-s)^{n-\alpha -1}\frac{\partial ^n}{\partial ^ns}g(s)ds\mbox{,}\quad \mbox{  and  }\\ ^cD_{b^{-}}^{\alpha }g(t)=\frac{1}{\Gamma (n-\alpha )}\int _t^b(s-t)^{n-\alpha -1}\frac{\partial ^n}{\partial ^ns}g(s)ds \end{array}

are called left-sided Caputo fractional derivative and right-sided Caputo fractional derivative respectively.

Further studies are also located on seeking solutions of such kind of BVPs in which the fractional differential equations are concerned with the composition of left- and right-sided Riemann–Liouville fractional derivatives or the composition of left- and right-sided Caputo fractional derivatives, or the composition of right- and left-sided Riemann–Liouville fractional derivatives, etc. are involved.

Another important part is to demonstrate the application of powerful mathematical tools (fixed point theorems in Banach spaces) for solving nonlinear fractional differential equations.

Acknowledgments

The authors thank anonymous referees and editors for valuable remarks and suggestions and some useful comments on improving the presentation of this paper.

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