Solvability of Sturm–Liouville boundary value problems for multiple order fractional differential equations

==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-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 <math display="inline">D_{0^+}^{\alpha }[\Phi (\rho (t)D_{0^+}^{\beta }x(t))]</math> 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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (1)
|}

is called a Sturm–Liouville boundary value problem  [[#br000025|[5]]] and [[#br000030|[6]]], where <math display="inline">f(t,u,v)</math> is continuous on <math display="inline">[0,1]\times R\times R</math>, <math display="inline">a\geq 0</math>, <math display="inline">b\geq 0</math>, <math display="inline">c\geq 0</math> and <math display="inline">d\geq 0</math> with <math display="inline">ac+ad+bc>0</math>. BVP(1) comes from a situation involving nonlinear elliptic problems in annular regions, one may see  [[#br000025|[5]]].

In order to generalize the results obtained in  [[#br000025|[5]]] and [[#br000030|[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  [[#br000005|[1]]], [[#br000010|[2]]], [[#br000015|[3]]], [[#br000035|[7]]], [[#br000040|[8]]], [[#br000045|[9]]], [[#br000060|[12]]], [[#br000065|[13]]], [[#br000070|[14]]], [[#br000080|[16]]], [[#br000050|[10]]] and [[#br000075|[15]]]).

In  [[#br000040|[8]]], Kaufmann and Mboumi studied the following boundary value problem for the fractional differential equation

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (2)
|}

by using the properties of Green’s function of the corresponding BVP, where <math display="inline">f:[0,+\infty )\rightarrow [0,+\infty )</math> is continuous, <math display="inline">a:[0,1]\rightarrow (0,+\infty )</math> is a continuous function and <math display="inline">D_{0^+}^{\alpha }</math> is the standard Riemann–Liouville fractional differential derivative of order <math display="inline">\alpha \in (1,2)</math>. Using the Leggett–Williams fixed point theorem and Krasnoselskii fixed point theorem, the authors in [[#br000040|[8]]] proved that BVP(2) has at least one or three positive solutions.

Dehghant and Ghanbari  [[#br000020|[4]]] studied the existence of solutions of the following boundary value problem for the nonlinear fractional differential equation

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">D_{0^+}^{\alpha }</math> is the standard Riemann–Liouville fractional differential derivative of order <math display="inline">\alpha \in (1,2)</math>, <math display="inline">f</math> is defined on <math display="inline">[0,1]\times [0,+\infty )\times R</math>, continuous and nonnegative, <math display="inline">a(t)</math> does not identically vanish on any subinterval of <math display="inline">(0,1)</math> and <math display="inline">a(t)</math> satisfies the following inequality:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>0<\int _0^1[t(1-t)]^{\alpha }a(t)dt<+\infty \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>x^{{''}}(t)=-1\mbox{,}\quad t\in (0,1)\mbox{,}\quad x^{{'}}(0)=0\mbox{,}\quad x(1)=0</math>
|}
| style="text-align: right;" | 
|}

has a unique continuous solution <math display="inline">u(t)=\frac{1}{2}(1-t^2)</math> on <math display="inline">[0,1]</math>. However, we find that the similar boundary value problem for the multiple order fractional differential equation

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

has a unique solution

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>x(t)=[-t^{\alpha +\beta }+t^{\beta -1}]\frac{1}{\Gamma (\alpha +1)\Gamma (\beta )}B(\beta ,\alpha +1)\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

which is not continuous at <math display="inline">t=0</math>. This fact shows that there exist differences between BVPs for ordinary differential equations and BVPs for fractional differential equations.

Liu  [[#br000055|[11]]] studied the existence of solutions of the following boundary value problem for the fractional differential equation

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">D_{0^+}^{\alpha }</math> is the standard Riemann–Liouville fractional differential derivative of order <math display="inline">\alpha \in (1,2)</math>, <math display="inline">f</math> is defined on <math display="inline">[0,1]\times [0,+\infty )</math>, nonnegative and is a Carathéodory function. Note that the boundary condition <math display="inline">x(0)=0</math> used in  [[#br000020|[4]]] and [[#br000040|[8]]] is replaced by <math display="inline">lim_{t\rightarrow 0}t^{2-\alpha }x(t)=0</math>. Hence the solutions obtained in  [[#br000055|[11]]] may be unbounded or singular at <math display="inline">t=0</math>.

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 <math display="inline">D_{0^+}^{\beta }x</math>

<span id='fd000045'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (3)
|}

where

<math display="inline">\bullet </math>    <math display="inline">a,b,c,d\in R</math>,

<math display="inline">\bullet </math>    <math display="inline">D_{0^+}^{\alpha }</math> (or <math display="inline">D_{0^+}^{\beta }</math>) is the left-side Riemann–Liouville fractional derivative of order <math display="inline">\alpha </math> (or <math display="inline">\beta </math>) with <math display="inline">\alpha ,\beta \in (0,1)</math>,

<math display="inline">\bullet </math>    <math display="inline">\rho :(0,1)\rightarrow [0,+\infty )</math> is continuous and may be singular at <math display="inline">t=0</math> and <math display="inline">t=1</math>,

<math display="inline">\bullet </math>    <math display="inline">f</math> defined on <math display="inline">(0,1)\times R\times R</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]) that may be singular at <math display="inline">t=0</math> and <math display="inline">t=1</math>,

<math display="inline">\bullet </math>    <math display="inline">\Phi (s)=\vert s\vert ^{p-2}s</math> with <math display="inline">p>1</math> is called a <math display="inline">p</math>-Laplacian and its inverse function denoted by <math display="inline">\Phi ^{-1}</math><math display="inline">\Phi ^{-1}(x)=\vert x\vert ^{q-2}x</math> with <math display="inline">\frac{1}{p}+\frac{1}{q}=1</math>.

We obtain some results on the existence of solutions of BVP(3) (see [[#e000035|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  [[#s000010|2]], we present preliminary results and main theorems. In Section  [[#s000015|3]], two examples are given to illustrate the main results.

<span id='s000010'></span>
==2. Main results==

For convenience, we present some necessary definitions from fractional calculus theory. These definitions and results can be found in the literatures [[#br000015|[3]]], [[#br000035|[7]]], [[#br000060|[12]]] and [[#br000070|[14]]]. Let the Gamma function and Beta function be defined by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

==Definition 2.1 [[#br000070|[14]]].                     ==

The left-side Riemann–Liouville fractional integral of order <math display="inline">\alpha >0</math> of a function <math display="inline">g:(0,+\infty )\rightarrow R</math> is given by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>I_{0^+}^{\alpha }g(t)=\frac{1}{\Gamma (\alpha )}\int _0^t(t-s)^{\alpha -1}g(s)ds\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

provided that the right-hand side exists.

==Definition 2.2 [[#br000070|[14]]].                     ==

Let <math display="inline">n</math> be a positive integer. The left-side Riemann–Liouville fractional derivative of order <math display="inline">\alpha >0</math> of a continuous function <math display="inline">g:(0,+\infty )\rightarrow R</math> is given by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="text-align: right;" | 
|}

where <math display="inline">n-1\leq \alpha <n</math>, provided that the right-hand side exists.

==Lemma 2.1 [[#br000070|[14]]].                     ==

''Let  ''<math display="inline">n-1\leq \mu <n</math>'',  ''<math display="inline">x\in C^0(0,+\infty )\bigcap L^1(0,+\infty )</math>''. Then''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>I_{0^+}^{\mu }D_{0^+}^{\mu }x(t)=x(t)+C_1t^{\mu -1}+C_2t^{\mu -2}+\cdots +C_nt^{\mu -n}\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

''where  ''<math display="inline">C_i\in R</math>'',  ''<math display="inline">i=1,2,\ldots ,n</math>''.''

<span id='e000020'></span>

==Definition 2.3.                     ==

Let <math display="inline">\sigma >0</math>. <math display="inline">x:(0,1)\rightarrow R</math> is called a <math display="inline">L_{\sigma }^1</math> function if <math display="inline">t\rightarrow t^{1-\sigma }\int _0^t(t-s)^{\sigma -1}\vert x(s)\vert ds</math> is continuous on <math display="inline">[0,1]</math>. The set of all <math display="inline">L_{\sigma }^1</math> functions on <math display="inline">(0,1)</math> is denoted by <math display="inline">L_{\sigma }^1(0,1)</math>.

<math display="inline">\bullet </math><math display="inline">x:(0,1)\rightarrow R</math> is called a <math display="inline">\sigma </math>-well function if it is a <math display="inline">L_{\sigma }^1</math> function and for each <math display="inline">0\leq t_2\leq t_1\leq 1</math>, it holds that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

as <math display="inline">t_1\rightarrow t_2</math>.

==Remark 2.1.                     ==

Note that <math display="inline">L_{\sigma }^1(0,1)</math> is a generalization of <math display="inline">L^1(0,1)</math>. It is easy to check that all bounded functions on <math display="inline">[0,1]</math> are <math display="inline">\sigma </math>-well functions and the functions that satisfy that there exists <math display="inline">\mu >-1</math> such that <math display="inline">\vert x(t)\vert \leq t^{\mu }</math> for all <math display="inline">t\in [0,1]</math> are <math display="inline">\sigma </math>-well functions too.

<span id='e000030'></span>

==Definition 2.4.                     ==

<math display="inline">f:(0,1)\times R^2\rightarrow R</math> is called a quasi-Carathéodory function if

<math display="inline">\bullet </math>    <math display="inline">t\rightarrow f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)</math> is a <math display="inline">L^{\alpha }</math> function for every <math display="inline">(u,v)\in R^2</math> (see [[#e000020|Definition 2.3]]),

<math display="inline">\bullet </math>    <math display="inline">(u,v)\rightarrow f(t,t^{\beta -1}u,\frac{1}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}v)</math> is continuous for each <math display="inline">t\in (0,1)</math>,

<math display="inline">\bullet </math>   for each <math display="inline">r>0</math> there exists a nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function (see [[#e000020|Definition 2.3]]) <math display="inline">\phi _r</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

<span id='e000035'></span>

==Definition 2.5.                     ==

<math display="inline">x:(0,1)\rightarrow R</math> is called a solution of BVP(3) if <math display="inline">x\in C(0,1)\bigcap L^1(0,1)</math> satisfies that <math display="inline">D_{0^+}^{\alpha }[\Phi (\rho D_{0^+}^{\beta }x)]\in L_{\alpha }^1(0,1)</math> and all equations in [[#fd000045|(3)]] are satisfied.

For our construction, we let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="text-align: right;" | 
|}

For <math display="inline">x\in X</math>, let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

<span id='e000040'></span>

==Lemma 2.2.                     ==

<math display="inline">X</math>''is a Banach space with the norm  ''<math display="inline">\Vert \cdot \Vert </math>''defined.''

==Proof.                     ==

Note that <math display="inline">X</math> is a normed linear space. Let <math display="inline">\lbrace x_u\rbrace </math> be a Cauchy sequence in <math display="inline">X</math>, then <math display="inline">\Vert x_u-x_v\Vert \rightarrow 0,\quad u,v\rightarrow +\infty </math>. It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Define

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then <math display="inline">t\rightarrow \Phi (t^{1-\alpha })\rho (t)D_{0^+}^{\beta }x_u(t)</math> and <math display="inline">t\rightarrow t^{1-\beta }x_u(t)</math> are continuous on <math display="inline">[0,1]</math>. Thus there exist two functions <math display="inline">x_0,y_0</math> defined on <math display="inline">[0,1]</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows from <math display="inline">\Vert x_u-x_v\Vert \rightarrow 0,\quad u,v\rightarrow +\infty </math> that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Denote <math display="inline">z_0(t)=t^{\beta -1}x_0(t)</math> and <math display="inline">w_0(t)=\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}</math> for <math display="inline">t\in (0,1)</math>. This means that functions <math display="inline">z_0,w_0:(0,1)\rightarrow R</math> are well defined.

'''Step 1.'''   Prove that the limits <math display="inline">lim_{t\rightarrow 0}t^{1-\beta }z_0(t)</math>, <math display="inline">lim_{t\rightarrow 1}t^{1-\beta }z_0(t)</math>, <math display="inline">lim_{t\rightarrow 0}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)</math> and <math display="inline">lim_{t\rightarrow 1}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)</math> exist.

Since both <math display="inline">t\rightarrow t^{1-\beta }z_0(t)=x_0(t)</math> and <math display="inline">t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)=y_0(t)</math> are continuous on <math display="inline">[0,1]</math>, the limits <math display="inline">lim_{t\rightarrow 0}t^{1-\beta }z_0(t)</math>, <math display="inline">lim_{t\rightarrow 1}t^{1-\beta }z_0(t)</math>, <math display="inline">lim_{t\rightarrow 0}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)</math> and <math display="inline">lim_{t\rightarrow 1}\Phi ^{-1}(t^{1-\alpha })\rho (t)w_0(t)</math> exist.

'''Step 2.'''   Prove that <math display="inline">sup_{t\in (0,1)}t^{1-\beta }\vert x_u(t)-z_0(t)\vert \rightarrow 0</math> and <math display="inline">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</math> as <math display="inline">u\rightarrow +\infty </math> in <math display="inline">X</math>.

We have that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

The results follow.

'''Step 3.'''   Prove that <math display="inline">y_0(t)=\Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }z_0(t)</math> for all <math display="inline">t\in (0,1)</math>.

We have that there exists constant <math display="inline">c_u\in R</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then <math display="inline">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)}</math>. Hence <math display="inline">t^{\beta -1}[x_0(t)+c_0t^{\beta -1}]=I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}</math>. Thus <math display="inline">z_0(t)+c_0t^{\beta -1}=I_{0^+}^{\beta }\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}</math>. It follows that <math display="inline">D_{0^+}^{\beta }z_0(t)=\frac{y_0(t)}{\Phi ^{-1}(t^{1-\alpha })\rho (t)}</math>.

It follows that <math display="inline">X</math> is a Banach space. The proof is completed.

We list the following assumption:

'''(H).'''    <math display="inline">f</math> is a quasi-Carathéodory function, <math display="inline">\rho </math> is nonnegative and continuous, <math display="inline">a,b,c,d\geq 0</math> are constants that satisfy

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>t\rightarrow \frac{\Phi ^{-1}(t^{\alpha -1})}{\rho (t)}\mbox{  is a  }\beta \mbox{- well  function on  }(0,1)\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

==Lemma 2.3.                     ==

''Suppose that (H) holds. Then for each  ''<math display="inline">x\in X</math>'', there exists a unique  ''<math display="inline">A_x\in R</math>''satisfying''

<span id='fd000135'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (4)
|}

''such that''

<span id='fd000140'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (5)
|}

==Proof.                     ==

For <math display="inline">x\in X</math>, we find <math display="inline">\Vert x\Vert =:r<+\infty </math>. Since <math display="inline">f</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]), there exists nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function <math display="inline">\phi _r</math> such that

<span id='fd000145'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (6)
|}

Let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then <math display="inline">G</math> is well defined on <math display="inline">R</math>. Since <math display="inline">\Delta >0</math>, <math display="inline">G</math> is continuous and strictly increasing on <math display="inline">R</math>. One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>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</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>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{.}</math>
|}
| style="text-align: right;" | 
|}

Hence there exists a unique

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

such that <math display="inline">G(A_x)=0</math>. Furthermore, <math display="inline">A_x</math> satisfies [[#fd000135|(4)]]. Then [[#fd000140|(5)]] is proved.

Define the nonlinear operator <math display="inline">T</math> on <math display="inline">X</math> by

<span id='fd000170'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (7)
|}

for <math display="inline">x\in X</math>, where <math display="inline">A_x</math> satisfies [[#fd000140|(5)]] and <math display="inline">B_x</math> is defined by

<span id='fd000175'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>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}</math>
|}
| style="text-align: right;" | (8)
|}

==Remark 2.2.                     ==

It is easy to see from [[#fd000175|(8)]] that

<span id='fd000180'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (9)
|}

<span id='e000065'></span>

==Lemma 2.4.                     ==

''Suppose that (H) holds. Then  ''<math display="inline">T:X\rightarrow X</math>''is completely continuous.''

==Proof.                     ==

For <math display="inline">x\in X</math>, we find <math display="inline">\Vert x\Vert \leq r<+\infty </math>. Since <math display="inline">f</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]), there exists nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function <math display="inline">\phi _r</math> such that [[#fd000145|(6)]] holds. Hence <math display="inline">A_x</math> is uniquely determined by [[#fd000140|(5)]]. Hence <math display="inline">B_x</math> is uniquely determined by [[#fd000175|(8)]] too. Then <math display="inline">(Tx)(t)</math> is well defined by [[#fd000170|(7)]]. It is easy to see from [[#fd000170|(7)]] that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>D_{0^+}^{\alpha }[\Phi (\rho (t)D_{0^+}^{\beta }(Tx)(t))]=-f(t,x(t),D_{0^+}^{\beta }x(t))\mbox{.}</math>
|}
| style="text-align: right;" | (10)
|}

Furthermore, we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

From [[#fd000140|(5)]] and [[#fd000175|(8)]], we see

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | (11)
|}

Hence <math display="inline">T:X\rightarrow X</math> is well defined and <math display="inline">x\in X</math> is a solution of BVP(3) if and only if <math display="inline">x</math> is a fixed point of <math display="inline">T</math> in <math display="inline">X</math>.

To prove that <math display="inline">T</math> is completely continuous, we address the following three steps.

'''Step 1.'''    <math display="inline">T</math> is continuous.

Let <math display="inline">\lbrace x_n\rbrace _{n=0}^{\infty }</math> be a sequence such that <math display="inline">x_n\rightarrow x_0</math> in <math display="inline">X</math>. We will show <math display="inline">Tx_n\rightarrow Tx_0</math> as <math display="inline">n\rightarrow +\infty </math>. We see

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>r=\underset{n=0,1,2,\ldots}{sup}\Vert x_n\Vert <\infty \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">f</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]), there exists nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function <math display="inline">\phi _r</math> such that [[#fd000145|(6)]] holds with <math display="inline">x</math> being replaced by <math display="inline">x_n</math>.

First we prove that

<span id='fd000215'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>A_{x_n}\rightarrow A_{x_0}\quad \mbox{  as  }n\rightarrow +\infty \mbox{.}</math>
|}
| style="text-align: right;" | (12)
|}

From [[#e000040|Lemma 2.2]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So <math display="inline">\lbrace A_{x_n}\rbrace </math> is bounded. If [[#fd000215|(12)]] does not hold, then there exist two subsequences <math display="inline">\lbrace A_{x_{n_k}}^{(1)}\rbrace </math> and <math display="inline">\lbrace A_{x_{n_k}}^{(2)}\rbrace </math> of <math display="inline">\lbrace A_{x_n}\rbrace </math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>A_{x_{n_k}}^{(i)}\rightarrow c_i(i=1,2)\quad \mbox{  as  }n\rightarrow +\infty </math>
|}
| style="text-align: right;" | 
|}

with <math display="inline">c_1\not =c_2</math>. Since <math display="inline">A_{x_{n_k}}^{(i)}</math> satisfies

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

by Lebesgue’s dominant theorem, let <math display="inline">k\rightarrow +\infty </math>, then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Since

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

together with [[#e000040|Lemma 2.2]], we get that <math display="inline">c_1=c_2=A_{x_0}</math>, a contradiction. So [[#fd000215|(12)]] holds.

Now by the definition of <math display="inline">B_x</math>, one has

<span id='fd000250'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>B_{x_n}\rightarrow B_{x_0}\quad \mbox{  as  }n\rightarrow +\infty \mbox{.}</math>
|}
| style="text-align: right;" | (13)
|}

From

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

since <math display="inline">f</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]) and [[#fd000215|(12)]], [[#fd000250|(13)]], <math display="inline">Tx_n\rightarrow Tx_0</math> as <math display="inline">n\rightarrow +\infty </math>. Hence <math display="inline">T</math> is continuous.

'''Step 2.'''    <math display="inline">T</math> maps bounded sets into bounded sets in <math display="inline">X</math>.

It suffices to show that for each <math display="inline">r>0</math>, there exists a positive number <math display="inline">L>0</math> such that for each <math display="inline">x\in M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace </math>, we have <math display="inline">\Vert Ty\Vert \leq L</math>. By the assumption, <math display="inline">f</math> is a quasi-Carathéodory function (see [[#e000030|Definition 2.4]]), there exists a nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function <math display="inline">\phi _r</math> (see [[#e000020|Definition 2.3]]) such that [[#fd000145|(6)]] holds. By the definition of <math display="inline">T</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Now we consider <math display="inline">t^{1-\beta }\vert (Ty)(t)\vert </math>.

'''Case 1.'''    <math display="inline">a\not =0</math>. We have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

'''Case 2.'''    <math display="inline">a=0</math>. We have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows that there exists a constant <math display="inline">M>0</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Vert Ty\Vert \leq M</math>
|}
| style="text-align: right;" | 
|}

for each <math display="inline">y\in \lbrace y\in X:\Vert y\Vert \leq l\rbrace </math>. Then <math display="inline">T</math> maps bounded sets into bounded sets in <math display="inline">X</math>.

'''Step 3.'''   Let <math display="inline">M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace </math>. Prove that both <math display="inline">\lbrace t\rightarrow t^{1-\beta }(Tx)(t):x\in M\rbrace </math> and <math display="inline">\lbrace t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx)(t):x\in M\rbrace </math> are equicontinuous on <math display="inline">(0,1)</math>.

By <math display="inline">M=\lbrace y\in X:\Vert y\Vert \leq r\rbrace </math>, there exists nonnegative <math display="inline">\beta </math>-well <math display="inline">L_{\alpha }^1</math> function <math display="inline">\phi _r</math> such that [[#fd000145|(6)]] holds. Let <math display="inline">t_1,t_2\in (0,1)</math> with <math display="inline">t_2<t_2</math>. We have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">t^{1-\beta }(t-s)^{\beta -1}</math> is decreasing, we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Using [[#e000040|Lemma 2.2]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">\frac{\Phi ^{-1}(s^{\alpha -1})}{\rho (s)}ds</math> is a <math display="inline">\beta </math>-well function, then we get

<span id='fd000300'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | (14)
|}

Similarly, we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (15)
|}

Then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Since <math display="inline">\phi _r</math> is a <math display="inline">\beta </math>-well function, then we get

<span id='fd000315'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (16)
|}

It is easy to see that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

and <math display="inline">\Phi ^{-1}</math> is uniformly continuous on <math display="inline">[-L,L]</math>, then for each <math display="inline">\epsilon >0</math>, there exists <math display="inline">\delta _0>0</math> such that <math display="inline">\vert x_1-x_2\vert <\delta _0</math> with <math display="inline">x_1,x_2\in [-L,L]</math> implies

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert \Phi ^{-1}(x_1)-\Phi ^{-1}(x_2)\vert <\epsilon \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

It follows from [[#fd000315|(16)]] that there exists <math display="inline">\delta >0</math> such that <math display="inline">t_1,t_2\in (0,1)</math> and <math display="inline">\vert t_1-t_2\vert <\delta </math> imply

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Hence <math display="inline">t_1,t_2\in (0,1)</math> and <math display="inline">\vert t_1-t_2\vert <\delta </math> imply

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows that

<span id='fd000340'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (17)
|}

uniformly as <math display="inline">t_1\rightarrow t_2</math>.

From [[#fd000300|(14)]] and [[#fd000340|(17)]], both <math display="inline">\lbrace t\rightarrow t^{1-\beta }(Tx)(t):x\in M\rbrace </math> and <math display="inline">\lbrace t\rightarrow \Phi ^{-1}(t^{1-\alpha })\rho (t)D_{0^+}^{\beta }(Tx)(t):x\in M\rbrace </math> are equicontinuous on <math display="inline">(0,1)</math>.

The Arzelà–Ascoli theorem implies that <math display="inline">TM</math> is relatively compact. Thus, the operator <math display="inline">T:X\rightarrow X</math> is completely continuous.

In order to prove the existence of solution of BVP(3), we use the following topological transversality theorem given by Granas  [[#br000035|[7]]].

==Lemma 2.5.                     ==

''Let  ''<math display="inline">B</math>''be a convex subset of a normed linear space  ''<math display="inline">X</math>''and assume  ''<math display="inline">0\in B</math>''. Let  ''<math display="inline">T:B\rightarrow B</math>''be a completely continuous operator and let  ''<math display="inline">U(T)=\lbrace x:x=\lambda Tx\rbrace </math>''for some  ''<math display="inline">0<\lambda <1</math>''. Then either  ''<math display="inline">U(T)</math>''is unbounded or  ''<math display="inline">T</math>''has a fixed point.''

<span id='e000080'></span>

==Theorem 2.1.                     ==

''Suppose that (H) holds and there exist nonnegative  ''<math display="inline">L_{\alpha }^1</math>''functions  ''<math display="inline">A,B</math>''and  ''<math display="inline">C</math>''such that''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (18)
|}

''holds for all  ''<math display="inline">t\in (0,1),u,v\in R</math>''. Then  ''<math display="inline">BVP(3)</math>''has at least one solution if''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (19)
|}

''where''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

We shall prove that under the assumptions (18) and (19), <math display="inline">T</math> has at least one fixed point in <math display="inline">X</math> by using Leray–Schauder Alternative principle. Let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>U(T)=\lbrace x:x=\lambda Tx\mbox{  for some  }0<\lambda <1\rbrace \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

'''Case 1.'''    <math display="inline">a\not =0</math>. Indeed, by the definition of <math display="inline">T</math> for <math display="inline">x\in U(T)</math>, by using [[#fd000135|(4)]] and [[#fd000180|(9)]], we have the estimates

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

From [[#fd000360|(19)]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

It follows that <math display="inline">U(T)</math> is bounded. Hence [[#e000065|Lemma 2.4]] implies that <math display="inline">T</math> has a fixed point in <math display="inline">X</math>. Then BVP(3) has at least one solution.

'''Case 2.'''    <math display="inline">a=0</math>. Indeed, by the definition of <math display="inline">T</math> for <math display="inline">x\in X</math>, by using [[#fd000135|(4)]] and [[#fd000180|(9)]], we have the estimates

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

From [[#fd000360|(19)]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

It follows that <math display="inline">U(T)</math> is bounded. Hence [[#e000065|Lemma 2.4]] implies that <math display="inline">T</math> has a fixed point in <math display="inline">X</math>. Then BVP(3) has at least one solution. From Cases 1 and 2, the proof is completed.

<span id='e000090'></span>

==Theorem 2.2.                     ==

''Suppose that (H) holds and that there exists  ''<math display="inline">L_{\alpha }^1</math>''function  ''<math display="inline">\phi </math>''such that''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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</math>
|}
| style="text-align: right;" | (20)
|}

''with  ''<math display="inline">w\in C(R,[0,\infty ))</math>''nondecreasing. If''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{,}</math>
|}
| style="text-align: right;" | (21)
|}

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{,}</math>
|}
| style="text-align: right;" | 
|}

==Proof.                     ==

It follows from  and  that there exist constants <math display="inline">\overline{\mu }_0,\overline{\mu }_1>0</math> such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{,}</math>
|}
| style="text-align: right;" | (23)
|}

'''Case 1.'''    <math display="inline">a\not =0</math>. Let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>U=\lbrace x\in X:\Vert x\Vert \leq \overline{\mu }_0\rbrace \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

We claim that <math display="inline">x\not =\lambda Tx</math> for all <math display="inline">\partial U</math> and <math display="inline">\lambda \in (0,1)</math>. In fact, if <math display="inline">x=\lambda Tx</math> for some <math display="inline">x\in \partial U</math> and <math display="inline">\lambda \in (0,1)</math>. We have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{,}</math>
|}
| style="text-align: right;" | 
|}

which contradicts with [[#fd000470|(23)]]. From the above discussion, we have <math display="inline">x\not =\lambda Tx</math> for all <math display="inline">\partial U</math> and <math display="inline">\lambda \in (0,1)</math>. Since <math display="inline">T</math> is completely continuous, by [[#e000065|Lemma 2.4]], we see that BVP(3) has at least one solution <math display="inline">x</math>.

'''Case 2.'''    <math display="inline">a=0</math>. Let

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>U=\lbrace x\in X:\Vert x\Vert \leq \overline{\mu }_1\rbrace \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

We claim that <math display="inline">x\not =\lambda Tx</math> for all <math display="inline">\partial U</math> and <math display="inline">\lambda \in (0,1)</math>. In fact, let <math display="inline">x=\lambda Tx</math> for some <math display="inline">x\in \partial U</math> and <math display="inline">\lambda \in (0,1)</math>. We have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

It follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{,}</math>
|}
| style="text-align: right;" | 
|}

which contradicts with [[#fd000475|(24)]].

From the above discussion, we have <math display="inline">x\not =\lambda Tx</math> for all <math display="inline">\partial U</math> and <math display="inline">\lambda \in (0,1)</math>. Since <math display="inline">T</math> is completely continuous, by [[#e000065|Lemma 2.4]], we see that BVP(3) has at least one solution <math display="inline">x</math>. The proof is complete.

<span id='s000015'></span>
==3. Two examples==

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

==Example 3.1.                     ==

Consider the following BVP

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (25)
|}

We consider two cases:

'''Case 1.'''    <math display="inline">\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}</math> with <math display="inline">\lambda ,\mu >0</math>.

Corresponding to BVP(3), we find that <math display="inline">\alpha =\beta =\frac{1}{2}</math>, <math display="inline">a=b=c=d=1</math>, <math display="inline">\Phi (x)=x</math> with <math display="inline">\Phi ^{-1}(x)=x</math>, <math display="inline">\rho (t)=t^{\frac{1}{2}}(1-t)^{\frac{1}{2}}</math> and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="text-align: right;" | 
|}

Choose <math display="inline">A(t)=\lambda (t-\frac{1}{2})^4</math>, <math display="inline">B(t)=\mu </math> and <math display="inline">C(t)=2t^{-\frac{1}{2}}</math>. One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Hence [[#e000080|Theorem 2.1]] implies that BVP(25) has a solution if

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So BVP(25) has a solution for sufficiently small <math display="inline">\lambda </math> and <math display="inline">\mu </math>.

'''Case 2.'''    <math display="inline">\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}</math>.

Corresponding to BVP(3), we find that <math display="inline">\alpha =\beta =\frac{1}{2}</math>, <math display="inline">a=b=c=d=1</math>, <math display="inline">\Phi (x)=x</math> with <math display="inline">\Phi ^{-1}(x)=x</math>, <math display="inline">\rho (t)=t^{\frac{1}{2}}(1-t)^{\frac{1}{2}}</math> and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>f(t,t^{-\frac{1}{2}}u,t^{-1}(1-t)^{-\frac{1}{2}}v)=\phi (t)(u+v)^{1/2}\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Choose <math display="inline">w(x)=x^{1/2}</math> and <math display="inline">\phi (t)=t^{-\frac{1}{2}}</math>. We find that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see from [[#e000090|Theorem 2.2]] that

<span id='fd000560'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (26)
|}

implies that BVP(25) has at least one solution. It is easy to see that [[#fd000560|(26)]] is always correct since <math display="inline">w(x)=x^{1/2}</math>. Hence BVP(25) has at least one solution.

==Example 3.2.                     ==

Consider the following BVP

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (27)
|}

We consider two cases:

'''Case 1.'''    <math display="inline">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)</math> with <math display="inline">\lambda ,\mu >0</math>.

Corresponding to BVP(3), we find that <math display="inline">\alpha =\beta =\frac{1}{2}</math>, <math display="inline">a=b=c=d=1</math>, <math display="inline">\Phi (x)=x^3</math> with <math display="inline">\Phi ^{-1}(x)=x^{\frac{1}{3}}</math>, <math display="inline">\rho (t)=1</math> and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>f(t,u,v)=2t^{-\frac{1}{2}}+\lambda t^{\frac{1}{2}}u+\mu t^{\frac{1}{6}}v\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Choose <math display="inline">A(t)=\lambda </math>, <math display="inline">B(t)=\mu </math> and <math display="inline">C(t)=2t^{-\frac{1}{2}}</math>. One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

Hence [[#e000080|Theorem 2.1]] implies that BVP(27) has a solution if

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So BVP(27) has a solution for sufficiently small <math display="inline">\lambda </math> and <math display="inline">\mu </math>.

'''Case 2.'''    <math display="inline">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))</math>, <math display="inline">w:R\rightarrow R</math> is a continuous function.

Corresponding to BVP(3), we find that <math display="inline">\alpha =\beta =\frac{1}{2}</math>, <math display="inline">a=b=c=d=1</math>, <math display="inline">\Phi (x)=x^3</math> with <math display="inline">\Phi ^{-1}(x)=x^{\frac{1}{3}}</math>, <math display="inline">\rho (t)=1</math> and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>f(t,t^{-\frac{1}{2}}u,t^{-\frac{1}{6}}v)=\phi (t)w(u+v)\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Choose <math display="inline">w(x)=x^{1/2}</math> and <math display="inline">\phi (t)=t^{-\frac{1}{2}}</math>. We find that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

One sees that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

It is easy to see from [[#e000090|Theorem 2.2]] that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (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 [[#fd000045|(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 <math display="inline">\alpha \in [n-1,n)</math> with <math display="inline">n\in \lbrace 1,2,3,\ldots )</math> that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

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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