Theory and Modern Applications

# Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

## Abstract

By using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results.

## 1 Introduction

Fractional calculus has been recognized as an effective modeling methodology by researchers. Fractional differential equations are generalizations of classical differential equations to an arbitrary order. They have broad application in engineering and sciences such as physics, mechanics, chemistry, economics and biology, etc.. For some recent development on the topic, see  and the references therein.

In , Ahmad et al. considered the following anti-periodic fractional boundary value problems:

$\left\{\begin{array}{c}{}^{c}D^{q}x\left(t\right)=f\left(t,x\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \left[0,T\right],T>0,1
(1)

where ${}^{c}D^{q}$ denotes the Caputo fractional derivative of order q, and f is a given continuous function. The results are based on some standard fixed point principles.

In recent years, there has been a great deal of research into the questions of existence and uniqueness of solutions to anti-periodic boundary value problems for differential equations. First, second and higher-order differential equations with anti-periodic boundary value conditions have been considered in papers . The existence of solutions for anti-periodic boundary value problems for fractional differential equations was studied in .

In this paper, we investigate the existence and uniqueness of solutions for an anti-periodic fractional boundary value problem given by

$\left\{\begin{array}{c}{}^{c}D^{\alpha }x\left(t\right)=f\left(t,x\left(t\right){,}^{c}{D}^{q}x\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \left[0,T\right],\hfill \\ x\left(0\right)=-x\left(T\right),{\phantom{\rule{2em}{0ex}}}^{c}{D}^{p}x\left(0\right)={-}^{c}{D}^{p}x\left(T\right),\hfill \end{array}$
(2)

where ${}^{c}D^{\alpha }$ denotes the Caputo fractional derivative of order α, T is a positive constant, $1<\alpha \le 2$, $0, $\alpha -q\ge 1$ and f is a given continuous function.

## 2 Preliminaries

Theorem 2.1 ()

Let E be a closed, convex and nonempty subset of a Banach space X, let$F:E\to E$be a continuous mapping such that FE is a relatively compact subset of X. Then F has at least one fixed point in E.

Theorem 2.2 ()

Let p and q be two positive numbers such that$\frac{1}{p}+\frac{1}{q}=1$. If${|f\left(x\right)|}^{p}$and${|g\left(x\right)|}^{q}$are Riemann integrable on$\left[a,b\right]$, then

$|{\int }_{a}^{b}f\left(x\right)g\left(x\right)\phantom{\rule{0.2em}{0ex}}dx|\le {\left[{\int }_{a}^{b}{|f\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right]}^{\frac{1}{p}}{\left[{\int }_{a}^{b}{|g\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right]}^{\frac{1}{q}}.$

Lemma 2.1 ()

For any$y\in C\left[0,T\right]$, a unique solution of the linear fractional boundary value problem

$\left\{\begin{array}{c}{}^{c}D^{\alpha }x\left(t\right)=y\left(t\right),\phantom{\rule{1em}{0ex}}t\in \left[0,T\right],T>0,1<\alpha \le 2,\hfill \\ x\left(0\right)=-x\left(T\right),{\phantom{\rule{2em}{0ex}}}^{c}{D}^{p}x\left(0\right)={-}^{c}{D}^{p}x\left(T\right),\hfill \end{array}$
(3)

is

$x\left(t\right)={\int }_{0}^{T}G\left(t,s\right)y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,$
(4)

where $G\left(t,s\right)$ is the Green’s function given by

$G\left(t,s\right)=\left\{\begin{array}{c}\frac{{\left(t-s\right)}^{\alpha -1}-\frac{1}{2}{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right){\left(T-s\right)}^{\alpha -p-1}}{2\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}},\phantom{\rule{1em}{0ex}}s\le t,\hfill \\ -\frac{{\left(T-s\right)}^{\alpha -1}}{2\mathrm{\Gamma }\left(\alpha \right)}+\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right){\left(T-s\right)}^{\alpha -p-1}}{2\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}},\phantom{\rule{1em}{0ex}}t\le s.\hfill \end{array}$
(5)

Remark 2.1 For $p\to {1}^{-}$ the solution of the classical anti-periodic problem (${}^{c}D^{\alpha }x\left(t\right)=f\left(t,x\left(t\right){,}^{c}{D}^{q}x\left(t\right)\right)$, $x\left(0\right)=-x\left(T\right)$, ${x}^{\prime }\left(0\right)=-{x}^{\prime }\left(T\right)$, $0\le t\le T$, $1<\alpha \le 2$, $0, $\alpha -q\ge 1$) is given in .

## 3 Main results

Let $J=\left[0,T\right]$ and $C\left(J\right)$ be the space of all continuous real functions defined on J. Define the space endowed with the norm $\parallel x\parallel ={max}_{t\in J}|x\left(t\right){|+{max}_{t\in J}|}^{c}{D}^{q}x\left(t\right)|$. Obviously, $\left(X,\parallel \cdot \parallel \right)$ is a Banach space.

Theorem 3.1 Let$f:J×R×R\to R$be a continuous function. Assume that

(${H}_{1}$) There exist a constant$l\in \left(0,\alpha -1\right)$and a real-valued function$m\left(t\right)\in {L}^{\frac{1}{l}}\left(\left[0,T\right],\left(0,\mathrm{\infty }\right)\right)$such that

$|f\left(t,x,y\right)|\le m\left(t\right)+{d}_{1}{|x|}^{{\rho }_{1}}+{d}_{2}{|y|}^{{\rho }_{2}},$

where${d}_{1},{d}_{2}\ge 0$, $0\le {\rho }_{1},{\rho }_{2}<1$. Then the problem (2) has at least a solution on$\left[0,T\right]$.

Proof Let the condition (${H}_{1}$) be valid. According to Lemma 2.1, the problem (2) is equivalent to the following integral equation:

$\begin{array}{rcl}x\left(t\right)& =& {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right)}{2{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

Define

$\begin{array}{c}\left(Fx\right)\left(t\right)={\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\left(Fx\right)\left(t\right)=}+\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right)}{2{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\hfill \\ {B}_{r}=\left\{x\left(t\right)\in X,\parallel x\parallel \le r,t\in J\right\},\hfill \end{array}$

where

$\begin{array}{c}r\ge max\left\{{\left(3A{d}_{1}\right)}^{\frac{1}{1-{\rho }_{1}}},{\left(3A{d}_{2}\right)}^{\frac{1}{1-{\rho }_{2}}},3K\right\},\hfill \\ K=\frac{3M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{\mathrm{\Gamma }\left(2-p\right)M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}+\frac{M\mathrm{\Gamma }\left(\alpha -l\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-l+1\right)}\hfill \\ \phantom{K=}×{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}+\frac{M\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -p\right)\mathrm{\Gamma }\left(2-q\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l},\hfill \\ A=\frac{{T}^{\alpha -q}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{3{T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha -p+1\right)},\hfill \end{array}$

and $M={\left({\int }_{0}^{T}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}$. Observe that ${B}_{r}$ is a closed, bounded and convex subset of Banach space X. Now, we prove that $F:{B}_{r}\to {B}_{r}$. For any $x\in {B}_{r}$, by Theorem 2.2 (Hölder inequality), we have

$\begin{array}{rcl}|\left(Fx\right)\left(t\right)|& =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right)}{2{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right){T}^{p}}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{2\mathrm{\Gamma }\left(\alpha \right)}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}}{2\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{p}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}m\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right){T}^{p}\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{t}{\left({\left(t-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{t}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\frac{1}{2\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{T}{\left({\left(T-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{T}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\frac{\mathrm{\Gamma }\left(2-p\right){T}^{p}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left({\int }_{0}^{T}{\left({\left(T-s\right)}^{\alpha -p-1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-l}{\left({\int }_{0}^{T}{\left(m\left(s\right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{l}\\ +\left(\frac{3{T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\\ \le & \frac{3M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{\mathrm{\Gamma }\left(2-p\right)M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}\\ +\left(\frac{3{T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\end{array}$

and

$\begin{array}{rcl}{|}^{c}{D}^{q}\left(Fx\right)\left(t\right)|& =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{\left(Fx\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\\ =& |{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \\ -\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\\ \le & {\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\left({\int }_{0}^{s}{\left(s-\tau \right)}^{\alpha -2}m\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{\int }_{0}^{s}{\left(s-\tau \right)}^{\alpha -2}\phantom{\rule{0.2em}{0ex}}d\tau \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\left({\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha -p-1}m\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha -p-1}\phantom{\rule{0.2em}{0ex}}d\tau \phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{1}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\\ ×\left[{\left({\int }_{0}^{s}{\left({\left(s-\tau \right)}^{\alpha -2}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{1-l}{\left({\int }_{0}^{s}{\left(m\left(\tau \right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{l}\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\\ ×\left[{\left({\int }_{0}^{T}{\left({\left(T-\tau \right)}^{\alpha -p-1}\right)}^{\frac{1}{1-l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{1-l}{\left({\int }_{0}^{T}{\left(m\left(\tau \right)\right)}^{\frac{1}{l}}\phantom{\rule{0.2em}{0ex}}d\tau \right)}^{l}\right]\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{{d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds+\frac{\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p+1\right)}\\ ×\frac{M}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-q\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{s}^{\alpha -l-1}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{M\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -l-1}}{\mathrm{\Gamma }\left(\alpha -p\right)\mathrm{\Gamma }\left(1-q\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\phantom{\rule{0.2em}{0ex}}ds+\frac{\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p+1\right)}\\ \le & \frac{M\mathrm{\Gamma }\left(\alpha -l\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-l+1\right)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{M\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -p\right)\mathrm{\Gamma }\left(2-q\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}\\ +\left[\frac{{T}^{\alpha -q}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p+1\right)}\right]\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right).\end{array}$

Thus,

$\begin{array}{rcl}\parallel \left(Fx\right)\left(t\right)\parallel & =& \underset{t\in J}{max}|\left(Fx\right)\left(t\right)|+\underset{t\in J}{max}{|}^{c}{D}^{q}\left(Fx\right)\left(t\right)|\\ \le & \frac{3M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{\mathrm{\Gamma }\left(2-p\right)M{T}^{\alpha -l}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}\\ +\frac{M\mathrm{\Gamma }\left(\alpha -l\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-l+1\right)}\\ ×{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}+\frac{M\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q-l}}{\mathrm{\Gamma }\left(\alpha -p\right)\mathrm{\Gamma }\left(2-q\right)}{\left(\frac{1-l}{\alpha -p-l}\right)}^{1-l}+\left[\frac{{T}^{\alpha -q}}{\mathrm{\Gamma }\left(\alpha -q+1\right)}\\ +\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p+1\right)}+\frac{3{T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{\mathrm{\Gamma }\left(2-p\right){T}^{\alpha }}{2\mathrm{\Gamma }\left(\alpha -p+1\right)}\right]\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)\\ =& K+\left({d}_{1}{r}^{{\rho }_{1}}+{d}_{2}{r}^{{\rho }_{2}}\right)A\le \frac{r}{3}+\frac{r}{3}+\frac{r}{3}=r.\end{array}$

Notice that $\left(Fx\right)\left(t\right)$, ${D}^{q}\left(Fx\right)\left(t\right)$ are continuous on J; therefore, $F:{B}_{r}\to {B}_{r}$. In view of the continuity of f, it is easy to know that the operator F is continuous. Now, we show that F is a completely continuous operator. For each $x\in {B}_{r}$, we fix $N={max}_{t\in J}|f\left(t,x\left(t\right){,}^{c}{D}^{q}x\left(t\right)\right)|$, for any $\epsilon >0$, setting

$\begin{array}{rcl}\delta & =& min\left\{\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\alpha -p+1\right)\epsilon }{N{T}^{\alpha -1}\left(\mathrm{\Gamma }\left(\alpha -p+1\right)+\mathrm{\Gamma }\left(2-p\right)\mathrm{\Gamma }\left(\alpha \right)\right)},\\ \frac{1}{2}{\left[\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\alpha -p+1\right)\epsilon }{6N{T}^{\alpha -1}\left(\mathrm{\Gamma }\left(\alpha -p+1\right)+\mathrm{\Gamma }\left(2-p\right)\mathrm{\Gamma }\left(\alpha \right)\right)}\right]}^{\frac{1}{1-q}}\right\}.\end{array}$

For each $x\in {B}_{r}$, we will prove that if ${t}_{1},{t}_{2}\in J$ and $0<{t}_{2}-{t}_{1}<\delta$, then

$\parallel \left(Fx\right)\left({t}_{2}\right)-\left(Fx\right)\left({t}_{1}\right)\parallel <\epsilon .$

In fact,

$\begin{array}{rcl}|\left(Fx\right)\left({t}_{2}\right)-\left(Fx\right)\left({t}_{1}\right)|& =& |{\int }_{0}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left({t}_{1}-{t}_{2}\right)\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \le & {\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}-{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}|f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \le & N{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{\alpha -1}-{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds+N{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{1}-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{N\left({t}_{2}-{t}_{1}\right)\mathrm{\Gamma }\left(2-p\right)}{\mathrm{\Gamma }\left(\alpha -p\right){T}^{1-p}}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\phantom{\rule{0.2em}{0ex}}ds\\ =& \frac{N}{\mathrm{\Gamma }\left(\alpha +1\right)}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right)+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\left({t}_{2}-{t}_{1}\right).\end{array}$

By mean value theorem, we have

$\begin{array}{rcl}|\left(Fx\right)\left({t}_{2}\right)-\left(Fx\right)\left({t}_{1}\right)|& \le & \frac{N}{\mathrm{\Gamma }\left(\alpha +1\right)}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right)+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\left({t}_{2}-{t}_{1}\right)\\ \le & \frac{N}{\mathrm{\Gamma }\left(\alpha +1\right)}\alpha {T}^{\alpha -1}\left({t}_{2}-{t}_{1}\right)+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\left({t}_{2}-{t}_{1}\right)\\ \le & \left(\frac{N}{\mathrm{\Gamma }\left(\alpha \right)}{T}^{\alpha -1}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\delta <\epsilon \end{array}$

and

$\begin{array}{c}{|}^{c}{D}^{q}\left(Fx\right)\left({t}_{2}\right){-}^{c}{D}^{q}\left(Fx\right)\left({t}_{1}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{\left(Fx\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{\left(Fx\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{N}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}{\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{N}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{s}^{\alpha -1}\phantom{\rule{0.2em}{0ex}}ds+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right){\int }_{0}^{{t}_{1}}\frac{{\left({t}_{2}-s\right)}^{-q}-{\left({t}_{1}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right){\int }_{{t}_{1}}^{{t}_{2}}\frac{{\left({t}_{2}-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left[2{\left({t}_{2}-{t}_{1}\right)}^{1-q}+\left({t}_{2}^{1-q}-{t}_{1}^{1-q}\right)\right].\hfill \end{array}$

In the following, we will divide the proof into two cases.

Case 1. For $\delta \le {t}_{1}<{t}_{2}, by mean value theorem, we have

$\begin{array}{c}{|}^{c}{D}^{q}\left(Fx\right)\left({t}_{2}\right){-}^{c}{D}^{q}\left(Fx\right)\left({t}_{1}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left[2{\left({t}_{2}-{t}_{1}\right)}^{1-q}+\left({t}_{2}^{1-q}-{t}_{1}^{1-q}\right)\right]\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left[2{\delta }^{1-q}+\left(1-q\right){\delta }^{-q}\left({t}_{2}-{t}_{1}\right)\right]\hfill \\ \phantom{\rule{1em}{0ex}}<\left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left(3-q\right){\delta }^{1-q}<{\left(\frac{1}{2}\right)}^{1-q}\frac{\epsilon }{2}<\frac{\epsilon }{2}.\hfill \end{array}$

Case 2. For $0\le {t}_{1}<\delta$, ${t}_{2}<2\delta$, we have

$\begin{array}{rcl}{|}^{c}{D}^{q}\left(Fx\right)\left({t}_{2}\right){-}^{c}{D}^{q}\left(Fx\right)\left({t}_{1}\right)|& \le & \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)\left[2{\left({t}_{2}-{t}_{1}\right)}^{1-q}+\left({t}_{2}^{1-q}-{t}_{1}^{1-q}\right)\right]\\ \le & \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)3{t}_{2}^{1-q}\\ <& \left(\frac{N{T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{N\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha -p+1\right)}\right)3{\left(2\delta \right)}^{1-q}<\frac{\epsilon }{2}.\end{array}$

Hence,

$\parallel \left(Fx\right)\left({t}_{2}\right)-\left(Fx\right)\left({t}_{1}\right)\parallel <\epsilon .$

Therefore, F is equicontinuous and uniformly bounded. The Arzela-Ascoli theorem implies that F is compact on ${B}_{r}$, so the operator F is completely continuous. Thus the conclusion of Theorem 2.1 implies that the anti-periodic boundary value problem (2) has at least one solution on $\left[0,T\right]$. This completes the proof. □

Corollary 3.1 Let$f:J×R×R\to R$be a continuous function. Assume that

(${H}_{2}$) There exist a constant$l\in \left(0,\alpha -1\right)$and a real-valued function$m\left(t\right)\in {L}^{\frac{1}{l}}\left(\left[0,T\right],\left(0,\mathrm{\infty }\right)\right)$such that

$|f\left(t,x,y\right)|\le m\left(t\right)+{d}_{1}|x|+{d}_{2}|y|,$

and$\left({d}_{1}+{d}_{2}\right)A<1$, where${d}_{1},{d}_{2}\ge 0$, A is defined in the proof of Theorem  3.1. Then the problem (2) has at least a solution on$\left[0,T\right]$.

The proof of Corollary 3.1 is similar to Theorem 3.1.

Theorem 3.2 Assume that

(${H}_{3}$) There exist a constant$r\in \left(0,\alpha -1\right)$and a real-valued function$\mu \left(t\right)\in {L}^{\frac{1}{r}}\left(\left[0,T\right],\left(0,\mathrm{\infty }\right)\right)$such that

$|f\left(t,x,y\right)-f\left(t,u,v\right)|\le \mu \left(t\right)\left(|x-u|+|y-v|\right),$

for any$t\in \left[0,T\right]$, $x,y,u,v\in R$, and if (6)

where${\mu }^{\ast }={\left({\int }_{0}^{T}{\left(\mu \left(s\right)\right)}^{\frac{1}{r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{r}$. Then the problem (2) has a unique solution.

Proof Define the mapping $F:X\to X$ by

$\begin{array}{rcl}\left(Fx\right)\left(t\right)& =& {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right)}{2{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(s,x\left(s\right){,}^{c}{D}^{q}x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

For $x,y\in X$ and for each $t\in \left[0,T\right]$, by Theorem 2.2 (Hölder inequality), we obtain

$\begin{array}{c}|\left(Fx\right)\left(t\right)-\left(Fy\right)\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\mu \left(s\right)\left(|x\left(s\right)-y\left(s\right)|+{|}^{c}{D}^{q}x\left(s\right){-}^{c}{D}^{q}y\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{1}{2}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\mu \left(s\right)\left(|x\left(s\right)-y\left(s\right)|+{|}^{c}{D}^{q}x\left(s\right){-}^{c}{D}^{q}y\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(2-p\right)\left(T-2t\right)}{2{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-s\right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}\mu \left(s\right)\left(|x\left(s\right)-y\left(s\right)|+{|}^{c}{D}^{q}x\left(s\right){-}^{c}{D}^{q}y\left(s\right)|\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\parallel x-y\parallel }{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{\parallel x-y\parallel }{2\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -1}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel \mathrm{\Gamma }\left(2-p\right){T}^{p}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{T}{\left(T-s\right)}^{\alpha -p-1}\mu \left(s\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\parallel x-y\parallel }{\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{t}{\left({\left(t-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-r}{\left({\int }_{0}^{t}{\left(\mu \left(s\right)\right)}^{\frac{1}{r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{r}\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel }{2\mathrm{\Gamma }\left(\alpha \right)}{\left({\int }_{0}^{T}{\left({\left(T-s\right)}^{\alpha -1}\right)}^{\frac{1}{1-r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-r}{\left({\int }_{0}^{T}{\left(\mu \left(s\right)\right)}^{\frac{1}{r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{r}\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel \mathrm{\Gamma }\left(2-p\right){T}^{p}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left({\int }_{0}^{T}{\left({\left(T-s\right)}^{\alpha -p-1}\right)}^{\frac{1}{1-r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1-r}{\left({\int }_{0}^{T}{\left(\mu \left(s\right)\right)}^{\frac{1}{r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{r}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left[\frac{3{\mu }^{\ast }{T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-r}{\alpha -r}\right)}^{1-r}+\frac{{\mu }^{\ast }\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -p-r}\right)}^{1-r}\right]\parallel x-y\parallel \hfill \end{array}$

and

$\begin{array}{c}{|}^{c}{D}^{q}\left(Fx\right)\left(t\right){-}^{c}{D}^{q}\left(Fy\right)\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{\left(Fx\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}{\left(Fy\right)}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\hfill \\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\hfill \\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}f\left(\tau ,y\left(\tau \right){,}^{c}{D}^{q}y\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \hfill \\ \phantom{\rule{2em}{0ex}}-\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}f\left(\tau ,y\left(\tau \right){,}^{c}{D}^{q}y\left(\tau \right)\right)\phantom{\rule{0.2em}{0ex}}d\tau \right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\left({\int }_{0}^{s}\frac{{\left(s-\tau \right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)-f\left(\tau ,y\left(\tau \right){,}^{c}{D}^{q}y\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}}{\int }_{0}^{t}\frac{{\left(t-s\right)}^{-q}}{\mathrm{\Gamma }\left(1-q\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({\int }_{0}^{T}\frac{{\left(T-\tau \right)}^{\alpha -p-1}}{\mathrm{\Gamma }\left(\alpha -p\right)}|f\left(\tau ,x\left(\tau \right){,}^{c}{D}^{q}x\left(\tau \right)\right)-f\left(\tau ,y\left(\tau \right){,}^{c}{D}^{q}y\left(\tau \right)\right)|\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\parallel x-y\parallel }{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-q\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\left({\int }_{0}^{s}{\left(s-\tau \right)}^{\alpha -2}\mu \left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel \mathrm{\Gamma }\left(2-p\right)}{{T}^{1-p}\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\left({\int }_{0}^{T}{\left(s-\tau \right)}^{\alpha -p-1}\mu \left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\parallel x-y\parallel {\mu }^{\ast }}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(1-q\right)}{\left(\frac{1-r}{\alpha -r-1}\right)}^{1-r}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{s}^{a-r-1}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel {\mu }^{\ast }\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -r-1}}{\mathrm{\Gamma }\left(1-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -r-p}\right)}^{1-r}{\int }_{0}^{t}{\left(t-s\right)}^{-q}\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{\parallel x-y\parallel {\mu }^{\ast }{T}^{\alpha -q-r}\mathrm{\Gamma }\left(\alpha -r\right)}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-r+1\right)}{\left(\frac{1-r}{\alpha -r-1}\right)}^{1-r}\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\parallel x-y\parallel {\mu }^{\ast }\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -r-p}\right)}^{1-r}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left[\frac{{\mu }^{\ast }{T}^{\alpha -q-r}\mathrm{\Gamma }\left(\alpha -r\right)}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-r+1\right)}{\left(\frac{1-r}{\alpha -r-1}\right)}^{1-r}\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{{\mu }^{\ast }\mathrm{\Gamma }\left(2-p\right){T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -r-p}\right)}^{1-r}\right]\parallel x-y\parallel .\hfill \end{array}$

Hence, we obtain

$\begin{array}{rcl}\parallel Fx-Fy\parallel & \le & \left[\frac{3{\mu }^{\ast }{T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-r}{\alpha -r}\right)}^{1-r}+\frac{\mathrm{\Gamma }\left(2-p\right){\mu }^{\ast }{T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -p-r}\right)}^{1-r}\\ +\frac{\mathrm{\Gamma }\left(\alpha -r\right){\mu }^{\ast }{T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-r+1\right)}\\ ×{\left(\frac{1-r}{\alpha -r-1}\right)}^{1-r}+\frac{\mathrm{\Gamma }\left(2-p\right){\mu }^{\ast }{T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -r-p}\right)}^{1-r}\right]\parallel x-y\parallel .\end{array}$

From the assumption (6), it follows that F is a contraction mapping. Therefore, the Banach fixed point theorem yields that F has a unique fixed point which is the unique solution of the problem (2). □

## 4 Examples

Example 4.1 Let $\alpha =\frac{3}{2}$, $p=q=\frac{1}{2}$, $T=1$. Consider the following anti-periodic fractional boundary value problem:

$\left\{\begin{array}{c}{}^{c}D^{\frac{3}{2}}x\left(t\right)=f\left(t,x\left(t\right){,}^{c}{D}^{\frac{1}{2}}x\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \left[0,1\right],\hfill \\ x\left(0\right)=-x\left(1\right),{\phantom{\rule{2em}{0ex}}}^{c}{D}^{\frac{1}{2}}x\left(0\right)={-}^{c}{D}^{\frac{1}{2}}x\left(1\right).\hfill \end{array}$
(7)

We have

$f\left(t,x\left(t\right){,}^{c}{D}^{\frac{1}{2}}x\left(t\right)\right)=m\left(t\right)+{\left(t-\frac{1}{2}\right)}^{4}\left[{\left(x\left(t\right)\right)}^{{\rho }_{1}}+{\left(}^{c}{D}^{\frac{1}{2}}x\left(t\right){\right)}^{{\rho }_{2}}\right],$

$m\left(t\right)\in {L}^{4}\left(\left[0,1\right],\left(0,\mathrm{\infty }\right)\right)$, $0\le {\rho }_{1},{\rho }_{2}\le 1$.

Since

$\begin{array}{rcl}|f\left(t,x\left(t\right){,}^{c}{D}^{\frac{1}{2}}x\left(t\right)\right)|& \le & |m\left(t\right)|+{\left(t-\frac{1}{2}\right)}^{4}{|x\left(t\right)|}^{{\rho }_{1}}+{\left(t-\frac{1}{2}\right)}^{4}{|{D}^{\frac{1}{2}}x\left(t\right)|}^{{\rho }_{2}}\\ \le & |m\left(t\right)|+\frac{1}{16}{|x\left(t\right)|}^{{\rho }_{1}}+\frac{1}{16}{|}^{c}{D}^{\frac{1}{2}}x\left(t\right){|}^{{\rho }_{2}},\end{array}$

therefore, by Theorem 3.1, the problem (7) has at least a solution on $\left[0,1\right]$.

Example 4.2 Consider the following anti-periodic fractional boundary value problem:

$\left\{\begin{array}{c}{}^{c}D^{\frac{3}{2}}x\left(t\right)=\frac{1}{{\left(t+4\right)}^{2}}\left(\frac{|x{+}^{c}{D}^{\frac{1}{2}}x|}{1+|x{+}^{c}{D}^{\frac{1}{2}}x|}+5{t}^{2}\right),\hfill \\ x\left(0\right)=-x\left(1\right),{\phantom{\rule{2em}{0ex}}}^{c}{D}^{\frac{1}{2}}x\left(0\right)={-}^{c}{D}^{\frac{1}{2}}x\left(1\right).\hfill \end{array}$
(8)

We have

$|f\left(t,x{,}^{c}{D}^{\frac{1}{2}}x\right)-f\left(t,y{,}^{c}{D}^{\frac{1}{2}}y\right)|\le \frac{1}{16}\left(|x-y|+{|}^{c}{D}^{\frac{1}{2}}x{-}^{c}{D}^{\frac{1}{2}}y|\right).$

Obviously, $\mu \left(t\right)\equiv \frac{1}{16}\in {L}^{4}\left(\left[0,1\right],\left(0,\mathrm{\infty }\right)\right)$, $r=\frac{1}{4}$ and ${\mu }^{\ast }={\left({\int }_{0}^{T}{\left(\mu \left(s\right)\right)}^{\frac{1}{r}}\phantom{\rule{0.2em}{0ex}}ds\right)}^{r}={\left({\int }_{0}^{1}{\left(\frac{1}{16}\right)}^{4}\phantom{\rule{0.2em}{0ex}}ds\right)}^{\frac{1}{4}}=\frac{1}{16}$. Note that $\mathrm{\Gamma }\left(\frac{3}{2}\right)\approx 0.8862$, $\mathrm{\Gamma }\left(\frac{7}{4}\right)\approx 0.9191$, $\mathrm{\Gamma }\left(\frac{5}{4}\right)\approx 0.9064$, we have

$\begin{array}{c}\frac{3{\mu }^{\ast }{T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha \right)}{\left(\frac{1-r}{\alpha -r}\right)}^{1-r}+\frac{\mathrm{\Gamma }\left(2-p\right){\mu }^{\ast }{T}^{\alpha -r}}{2\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -p-r}\right)}^{1-r}+\frac{\mathrm{\Gamma }\left(\alpha -r\right){\mu }^{\ast }{T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(\alpha -1\right)\mathrm{\Gamma }\left(\alpha -q-r+1\right)}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1-r}{\alpha -r-1}\right)}^{1-r}+\frac{\mathrm{\Gamma }\left(2-p\right){\mu }^{\ast }{T}^{\alpha -q-r}}{\mathrm{\Gamma }\left(2-q\right)\mathrm{\Gamma }\left(\alpha -p\right)}{\left(\frac{1-r}{\alpha -r-p}\right)}^{1-r}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{3{\left(\frac{3}{5}\right)}^{\frac{3}{4}}}{32\mathrm{\Gamma }\left(\frac{3}{2}\right)}+\frac{\mathrm{\Gamma }\left(\frac{3}{2}\right)}{32}+\frac{\mathrm{\Gamma }\left(\frac{5}{4}\right){3}^{\frac{3}{4}}}{16\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{7}{4}\right)}+\frac{1}{16}\hfill \\ \phantom{\rule{1em}{0ex}}\approx 0.7212+0.0277+0.0793+0.0625=0.8907<1.\hfill \end{array}$

Therefore, (8) has a unique solution on $\left[0,1\right]$ by Theorem 3.2.

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## Acknowledgement

The authors are highly grateful for the referee’s careful reading and comments on this note.

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Correspondence to Fang Wang.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The author Zhenhai Liu contributed to each part of this study equally and read and approved the final version of the manuscript.

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Wang, F., Liu, Z. Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv Differ Equ 2012, 116 (2012). https://doi.org/10.1186/1687-1847-2012-116

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• DOI: https://doi.org/10.1186/1687-1847-2012-116

### Keywords

• fractional differential equations
• boundary value problem
• anti-periodic
• fixed point theorem 