\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 26, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/26\hfil Solutions to boundary-value problems] {Solutions to boundary-value problems for nonlinear differential equations of\\ fractional order} \author[X. Su, S. Zhang\hfil EJDE-2009/26\hfilneg] {Xinwei Su, Shuqin Zhang} % in alphabetical order \address{Department of Mathematics, China University of Mining and Technology, Beijing, 100083, China} \email[Xinwei Su]{kuangdasuxinwei@163.com} \email[Shuqin Zhang]{zhangshuqin@tsinghua.org.cn} \thanks{Submitted June 25, 2008. Published February 3, 2009.} \subjclass[2000]{34B05, 26A33} \keywords{Boundary value problem; fractional derivative; fixed-point theorem; \hfill\break\indent Green's function; existence and uniqueness; continuous dependence} \begin{abstract} we discuss the existence, uniqueness and continuous dependence of solutions for a boundary value problem of nonlinear fractional differential equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Fractional differential equations have gained considerable popularity and importance during the past three decades or so, due mainly to their varied applications in many fields of science and engineering. Analysis of fractional differential equations has been carried out by various authors. As for the research in solutions and also many real applications for factional differential equations, we refer to the book by Kilbas, Srivastava and Trujillo \cite{k1} and references therein. Boundary-value problems for fractional differential equations have been discussed in \cite{a1,b1,g1,n1,s1,z1,z2}. Bai and L\"u \cite{b1} used fixed-point theorems on a cone to obtain the existence and multiplicity of positive solutions for a Dirichlet-type problem of the nonlinear fractional differential equation \begin{gather*} D_{0^{+}}^{\alpha}u(t)+f(t,u(t))=0,\quad 00$,$\alpha-\nu\geq 1$,$\beta-\mu\geq 1$,$f,g:[0,1]\times\mathbb{R} \times \mathbb{R}\to\mathbb{R}$are given functions and$D$is the standard Riemann-Liouville differentiation. Motivated by the previous results, we present in this paper analysis of a boundary value problem for the fractional differential equation involving more general boundary conditions and a nonlinear term dependent on the fractional derivative of the unknown function \begin{gathered} {^CD_{0^{+}}^{\alpha}}u(t)=f(t,u(t),{^CD_{0^{+}}^{\beta}}u(t)), \ \ 00$, ${^CD_{0^{+}}^{\alpha}}$ and ${^CD_{0^{+}}^{\beta}}$ are the Caputo's fractional derivatives and $f:[0,1]\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ is continuous. We impose a growth condition on the function $f$ to prove an existence result for \eqref{e1}. For $f$ Lipschitz in the second and third variables, the uniqueness of solution and the solution's dependence on the order $\alpha$ of the differential operator, the boundary values $A$ and $B$, and the nonlinear term $f$ are also discussed. Throughout this work, we denote by $I_{0^{+}}^{\alpha}$ and $D_{0^{+}}^{\alpha}$ the Riemann-Liouville fractional integral and derivative respectively. The definitions and some properties of fractional integrals and fractional derivatives of different types can be found in \cite{k1,p1}. In order to proceed, we recall some fundamental facts of fractional calculus theory. \begin{remark} \label{rmk1.1} \rm If $\alpha=n$ is an integer, the Riemann-Liouville fractional derivative of order $\alpha$ is the usual derivative of order $n$. The following properties are well known: $I_{0^{+}}^{\alpha}I_{0^{+}}^{\beta}f(t)=I_{0^{+}}^{\alpha+\beta}f(t)$, $D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}f(t)=f(t), \alpha>0, \beta>0, f\in L^{1}(0,1)$; $I_{0^{+}}^{\alpha}: C[0,1]\to C[0,1]$, $\alpha>0$. \end{remark} \begin{remark} \label{rmk1.2} \rm For $\alpha=n$, the Caputo's fractional derivative of order $\alpha$ becomes a conventional $n$-th derivative. The Caputo's fractional derivative is defined in \cite{k1} as follows: $^{C}D_{0^{+}}^{\alpha}f(t) =D_{0^{+}}^{\alpha}(f(t)-\sum_{k=0}^{n-1}\frac{f^{(k)}(0^{+})}{k!}t^{k})$, provided that the right-side derivative exists. In particular, ${^{C}D_{0^{+}}^{\alpha}}C=0$ for any constant $C\in \mathbb{R}$, $\alpha>0$. Moreover, we can derive the following useful properties from \cite[Lemmas 2.21 and 2.22]{k1}: ${^{C}D_{0^{+}}^{\alpha}}I_{0^{+}}^{\alpha}f(t)=f(t), \alpha>0, f(t)\in C[0,1]; I_{0^{+}}^{\alpha}{^{C}D_{0^{+}}^{\alpha}}f(t)=f(t)-f(0), 0<\alpha\leq1, f(t)\in C[0,1]$. \end{remark} Similar composition relation below between $I_{0^{+}}^{\alpha}$ and ${^{C}D_{0^{+}}^{\alpha}}$ can be found in \cite[Lemma 2.3]{z1}, but the author did not point out the space to which $u(t)$ belongs. Besides, the subscript $n$ of the coefficient $c_{n}$ is wrong. \begin{lemma} \label{lem1.1} Assume that $u(t)\in C(0,1)\cap L^{1}(0,1)$ with a derivative of order $n$ that belongs to $C(0,1)\cap L^{1}(0,1)$. Then $$I_{0^{+}}^{\alpha}{^{C}D_{0^{+}}^{\alpha}}u(t) =u(t)+c_{0}+c_{1}t+c_{2}t^{2}+\dots+c_{n-1}t^{n-1}$$ for some $c_{i}\in \mathbb{R}$, $i=0,1,2,\dots,n-1$, where $n$ is the smallest integer greater than or equal to $\alpha$. \end{lemma} We can also prove this lemma using \cite[Lemma 2.2]{b1} and Remark \ref{rmk1.1}. This proof is obvious and we omit it here. \section{Existence and uniqueness results} In this section, we first impose a growth condition on $f$ which allows us to establish an existence result of solution, and then utilize the Lipschitz condition on $f$ to prove a uniqueness theorem for the problem \eqref{e1}. Our approaches are based on the fixed-point theorems due to Schauder and Banach. Let $I=[0,1]$ and $C(I)$ be the space of all continuous real functions defined on $I$. Define the space $X=\{u(t)\mid u(t)\in C(I)\mbox{ and } {^{C}D_{0^{+}}^{\beta}}u(t)\in C(I), 0<\beta\leq 1\}$ endowed with the norm $\|u\|=\max_{t\in I}|u(t)|+\max_{t\in I}|{^{C}D_{0^{+}}^{\beta}}u(t)|$. Then by the method in \cite[Lemma 3.2]{s1} and Remark \ref{rmk1.2} we can know that $(X,\|\cdot\|)$ is a Banach space. Now we present the Green's function for boundary value problem of fractional differential equation. \begin{lemma} \label{lem2.1} Let $1<\alpha\leq2$. Assume that $g:[0,1]\to \mathbb{R}$ is a continuous function. Then the unique solution of \begin{gather*} ^{C}D_{0^{+}}^{\alpha}u(t)=g(t),\ \ 00$,$f:[0,1]\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$is a given continuous function. \end{itemize} \begin{lemma} \label{lem2.3} Assume that {\rm (H)} holds. Then \eqref{e1} is equivalent to the nonlinear integral equation $$u(t)=\int_{0}^{1}G(t,s)f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s +\varphi(t). \label{e2}$$ In other words, every solution of \eqref{e1} is also a solution of \eqref{e2} and vice versa. \end{lemma} \begin{proof} Let$u\in X$be a solution of \eqref{e1}, applying the method used to prove Lemma \ref{lem2.1}, we can obtain that$u$is a solution of \eqref{e2}. Conversely, let$u\in X$be a solution of \eqref{e2}. We denote the right-hand side of the equation \eqref{e2} by$w(t); i.e., \begin{align*} w(t)&= I_{0^{+}}^{\alpha}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t)) +\frac{-a_{2}b_{1}-a_{1}b_{1}t}{l}I_{0^{+}}^{\alpha}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1))\\ &\quad +\frac{-a_{2}b_{2}-a_{1}b_{2}t}{l}I_{0^{+}}^{\alpha-1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +\varphi(t). \end{align*} Using Remarks \ref{rmk1.1} and \ref{rmk1.2}, we have \begin{align*} w'(t) &= D_{0^{+}}^{1}I_{0^{+}}^{1}I_{0^{+}}^{\alpha-1}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t)) -\frac{a_{1}b_{1}}{l}I_{0^{+}}^{\alpha}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1))\\ &\quad -\frac{a_{1}b_{2}}{l}I_{0^{+}}^{\alpha-1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +\frac{a_{1}B-b_{1}A}{l}\\ &= I_{0^{+}}^{\alpha-1}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))- \frac{a_{1}b_{1}}{l}I_{0^{+}}^{\alpha}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1))\\ &\quad -\frac{a_{1}b_{2}}{l}I_{0^{+}}^{\alpha-1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +\frac{a_{1}B-b_{1}A}{l}, \end{align*} \begin{align*} {^{C}D_{0^{+}}^{\alpha}}w(t)&= D_{0^{+}}^{\alpha}(w(t)-w(0^{+})-w'(0^{+})t) =D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))\\ &= f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t)), \end{align*} namely,{^{C}D_{0^{+}}^{\alpha}}u(t)=f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))$. One can verify easily that$a_{1}u(0)-a_{2}u'(0)=A,b_{1}u(1)+b_{2}u'(1)=B$. Therefore,$u$is a solution of \eqref{e1}, which completes the proof. \end{proof} Lemma \ref{lem2.3} indicates that the solution of the problem \eqref{e1} coincides with the fixed point of the operator$T$defined as $$Tu(t)=\int_{0}^{1}G(t,s)f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s +\varphi(t).$$ Now, we give the main results of this section. \begin{theorem} \label{thm2.1} Let the assumption {\rm (H)} be satisfied. Suppose further that \begin{itemize} \item[(H1)] $\lim_{(|x|+|y|)\to \infty}\frac{\max_{t\in I}|f(t,x,y)|}{|x|+|y|} <\frac{l\Gamma(\alpha)\Gamma(2-\beta)} {(2l+a_{2}b_{2})\Gamma(2-\beta)+2l}=:K.$ \end{itemize} Then there exists at least one solution$u(t)$to the boundary-value problem {\rm\eqref{e1}}. \end{theorem} \begin{proof} For any$t\in I, we find \begin{align*} &\int_{0}^{1}|G(t,s)|\mathrm{d}s\\ &\leq \frac{1}{\Gamma(\alpha)}\Big(\int_{0}^{t}(t-s)^{\alpha-1}\mathrm{d}s + \frac{a_{2}b_{1}}{l}\int_{0}^{1}(1-s)^{\alpha-1}\mathrm{d}s + \frac{a_{1}b_{1}t}{l}\int_{0}^{1}(1-s)^{\alpha-1}\mathrm{d}s\Big) \\ &\quad +\frac{1}{\Gamma(\alpha-1)}\Big(\frac{a_{2}b_{2}}{l} \int_{0}^{1}(1-s)^{\alpha-2}\mathrm{d}s +\frac{a_{1}b_{2}t}{l}\int_{0}^{1}(1-s)^{\alpha-2}\mathrm{d}s\Big)\\ &= \frac{1}{\Gamma(\alpha+1)}\Big(t^{\alpha} +\frac{a_{2}b_{1}+a_{1}b_{1}t}{l}\Big) +\frac{1}{\Gamma(\alpha)}\frac{a_{2}b_{2}+a_{1}b_{2}t}{l}\\ &\leq \frac{1}{\Gamma(\alpha)}\Big(1+\frac{a_{2}b_{1}+a_{1}b_{1}}{l} +\frac{a_{2}b_{2}+a_{1}b_{2}}{l}\Big)\\ &=\frac{2l+a_{2}b_{2}}{l\Gamma(\alpha)} \end{align*} and \begin{align*} &\int_{0}^{1}|G'_{t}(t,s)|\mathrm{d}s\\ &\leq \frac{1}{\Gamma(\alpha-1)}\int_{0}^{t}(t-s)^{\alpha-2}\mathrm{d}s +\frac{a_{1}b_{1}}{l\Gamma(\alpha)}\int_{0}^{1}(1-s)^{\alpha-1}\mathrm{d}s +\frac{a_{1}b_{2}}{l\Gamma(\alpha-1)}\int_{0}^{1}(1-s)^{\alpha-2}\mathrm{d}s\\ &= \frac{t^{\alpha-1}}{\Gamma(\alpha)}+\frac{a_{1}b_{1}}{l\Gamma(\alpha+1)} +\frac{a_{1}b_{2}}{l\Gamma(\alpha)} \\ &\leq\frac{1}{\Gamma(\alpha)}\Big(1+\frac{a_{1}b_{1}}{l}+ \frac{a_{1}b_{2}}{l}\Big)\leq\frac{2}{\Gamma(\alpha)}. \end{align*} Therefore,|G(t,\cdot)|$and$|G'_{t}(t,\cdot)|$are integrable for any$t\in I$. Denote$h(x,y)=\max_{t\in I}|f(t,x,y)|$and Choose$\varepsilon=\frac{1}{2}(K -\lim_{(|x|+|y|)\to \infty}\frac{h(x,y)}{|x|+|y|})$. It follows from the condition (H1) that there exists a constant$d_{1}>0$such that$h(x,y)\leq (K-\varepsilon)(|x|+|y|)$for$|x|+|y|\geq d_{1}$. Let$M=\max\{h(x,y): |x|+|y|\leq d_{1}\}$and choose$d_{2}>d_{1}$such that$M/d_{2}\leq K-\varepsilon$. Then we get$h(x,y)\leq (K-\varepsilon)d_{2}, |x|+|y|\leq d_{2}$. Therefore,$h(x,y)\leq (K-\varepsilon)c$,$|x|+|y|\leq c$for any$c\geq d_{2}$. Let$k_{1}=\max_{t\in I} |\varphi(t)|$,$k_{2}=\max_{t\in I}|\varphi'(t)|$,$k=\max\{k_{1},k_{2}/\Gamma(2-\beta)\}$,$d_{3}=2Kk/\varepsilon$and$d=\max\{d_{2}, d_{3}\}$. Define $$U=\{u(t): u(t)\in X,\|u(t)\|\leq d,\; t\in I\}.$$ Then$U$is a convex, closed and bounded subset of$X$. Moreover, for any$u\in U$,$h(u(t),{^{C}D_{0^{+}}^{\beta}}u(t))\leq (K-\varepsilon)d$. Now we prove that the operator$T$maps$U$to itself. For any$u\in U, we can get \begin{align*} |Tu(t)| &\leq |\varphi(t)|+\int_{0}^{1}|G(t,s)h(u(s),{^{C}D_{0^{+}}^{\beta}}u(s) )|\mathrm{d}s \\ &\leq k_{1}+d(K-\varepsilon)\int_{0}^{1}|G(t,s)|\mathrm{d}s \\ &\leq k_{1}+d(K-\varepsilon)\frac{2l+a_{2}b_{2}}{l\Gamma(\alpha)}, \end{align*} \begin{align*} &|{^{C}D_{0^{+}}^{\beta}}(Tu)(t)|\\ &= \big|\frac{1}{\Gamma(1-\beta)} \int_{0}^{t}(t-s)^{-\beta}(Tu)'(s)\mathrm{d}s\big| \\ &\leq \frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta} \Big(\int_{0}^{1}|G'_{s}(s,\tau) f(\tau,u(\tau),{^{C}D_{0^{+}}^{\beta}}u(\tau))|\mathrm{d}\tau +|\varphi'(s)|\Big)\mathrm{d}s \\ &\leq \frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta} \Big(|\varphi'(s)|+\int_{0}^{1}|G'_{s}(s,\tau)h(u(\tau) ,{^{C}D_{0^{+}}^{\beta}}u(\tau))|\mathrm{d}\tau\Big) \mathrm{d}s\\ &\leq \frac{k_{2}}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta}\mathrm{d}s +d(K-\varepsilon) \frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta} \Big(\int_{0}^{1}|G'_{s}(s,\tau)|\mathrm{d}\tau \Big)\mathrm{d}s \\ &\leq \frac{k_{2}}{\Gamma(2-\beta)}+d(K-\varepsilon) \frac{2}{\Gamma(2-\beta)\Gamma(\alpha)} \end{align*} for0<\beta<1, and \begin{align*} |(Tu)'(t)| &= \big| \int_{0}^{1}G'_{t}(t,s) f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s+\varphi'(t)\big| \\ &\leq |\varphi'(t)|+ \int_{0}^{1}|G'_{t}(t,s)h(u(s),{^{C}D_{0^{+}}^{\beta}}u(s))|\mathrm{d}s \\ &\leq k_{2}+d(K-\varepsilon) \int_{0}^{1}|G'_{t}(t,s)|\mathrm{d}s \leq k_{2}+d(K-\varepsilon) \frac{2}{\Gamma(\alpha)} \end{align*} for\beta=1. Hence, $\|Tu\|\leq 2k+d(K-\varepsilon)\frac{(2l+a_{2}b_{2})\Gamma(2-\beta)+2l} {l\Gamma(\alpha)\Gamma(2-\beta)} \leq d\frac{\varepsilon}{K}+d(K-\varepsilon)\frac{1}{K}=d.$ Note also that \begin{align*} Tu(t)&= I_{0^{+}}^{\alpha}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t)) +\frac{-a_{2}b_{1}-a_{1}b_{1}t}{l}I_{0^{+}}^{\alpha}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1))\\ &\quad +\frac{-a_{2}b_{2}-a_{1}b_{2}t}{l}I_{0^{+}}^{\alpha-1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +\varphi(t), \end{align*} \begin{align*} (Tu)'(t)&= I_{0^{+}}^{\alpha-1}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))- \frac{a_{1}b_{1}}{l}I_{0^{+}}^{\alpha}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1))\\ &\quad -\frac{a_{1}b_{2}}{l}I_{0^{+}}^{\alpha-1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +\frac{a_{1}B-b_{1}A}{l}, \end{align*} \begin{align*} {^{C}D_{0^{+}}^{\beta}}(Tu)(t)&= I_{0^{+}}^{1-\beta}(Tu)'(t)\\ &= I_{0^{+}}^{\alpha-\beta}f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))- \frac{a_{1}b_{1}}{l}I_{0^{+}}^{\alpha-\beta+1}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) \\ &\quad -\frac{a_{1}b_{2}}{l}I_{0^{+}}^{\alpha-\beta}f(1,u(1),{^{C}D_{0^{+}}^{\beta}}u(1)) +I_{0^{+}}^{1-\beta}\frac{a_{1}B-b_{1}A}{l}. \end{align*} It is easy to seeTu(t), {^{C}D_{0^{+}}^{\beta}}(Tu)(t)\in C(I)$. Therefore,$T:U\to U$. Claim:$T$is a continuous operator. In fact, for$u_{n}$,$n=0,1,2,\dots$and$u\in U$such that$\lim_{n\to \infty}\|u_{n}-u\|\to 0, we have \begin{align*} &|Tu_{n}(t)-Tu(t)|\\ & =\big| \int_{0}^{1}G(t,s)\left( f(s,u_{n}(s),{^{C}D_{0^{+}}^{\beta}}u_{n}(s)) -f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\right) \mathrm{d}s \big| \\ &\leq \max_{t\in I}|f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \int_{0}^{1}|G(t,s)|\mathrm{d}s \\ &\leq \frac{2l+a_{2}b_{2}}{l\Gamma(\alpha)}\max_{t\in I}| f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))|, \end{align*} \begin{align*} &|{^{C}D_{0^{+}}^{\beta}}(Tu_{n})(t)-{^{C}D_{0^{+}}^{\beta}}(Tu)(t)|\\ &=\big| \frac{1}{\Gamma(1-\beta)} \int_{0}^{t}(t-s)^{-\beta}((Tu_{n})'(s)-(Tu)'(s))\mathrm{d}s\big| \\ &\leq \frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta} \Big(\int_{0}^{1}\big| G'_{s}(s,\tau) \Big( f(\tau,u_{n}(\tau),{^{C}D_{0^{+}}^{\beta}}u_{n}(\tau))\\ &\quad -f(\tau,u(\tau),{^{C}D_{0^{+}}^{\beta}}u(\tau))\Big)\big|\mathrm{d}\tau\Big) \mathrm{d}s \\ &\leq \max_{t\in I} |f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \\ &\quad\times \frac{1}{\Gamma(1-\beta)}\int_{0}^{t}(t-s)^{-\beta} \Big( \int_{0}^{1}|G'_{s}(s,\tau)|\mathrm{d}\tau\Big)\mathrm{d}s \\ &\leq \max_{t\in I}| f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \frac{2}{\Gamma(1-\beta)\Gamma(\alpha)}\int_{0}^{t}(t-s)^{-\beta} \mathrm{d}s \\ &\leq \frac{2}{\Gamma(2-\beta)\Gamma(\alpha)}\max_{t\in I}| f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \end{align*} for0<\beta<1, and \begin{align*} &|(Tu_{n})'(t)-(Tu)'(t)| \\ &=\big| \int_{0}^{1}G'_{t}(t,s) \left(f(s,u_{n}(s),{^{C}D_{0^{+}}^{\beta}}u_{n}(s)) -f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\right) \mathrm{d}s \big| \\ &\leq \max_{t\in I}|f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \int_{0}^{1}|G'_{t}(t,s)|\mathrm{d}s \\ &\leq \frac{2}{\Gamma(\alpha)}\max_{t\in I}|f(t,u_{n}(t),{^{C}D_{0^{+}}^{\beta}}u_{n}(t)) -f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))| \end{align*} for\beta=1$. Then in view of the uniform continuity of the function$f$on$I\times [-d,d]\times [-d,d]$, we obtain that$T$is continuous. \par The last step is to prove that$T$is a completely continuous operator. Let$t, \tau\in I$be such that$t<\tau$and$N=\max_{t\in I,u\in U}|f(t,u(t),{^{C}D_{0^{+}}^{\beta}}u(t))|+1. Then we have \begin{align*} &|Tu(t)-Tu(\tau)|\\ &=\Big| \int_{0}^{1}(G(t,s)-G(\tau,s))f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s+\varphi(t) -\varphi(\tau)\Big| \\ &\leq N\Big( \int_{0}^{t}|G(t,s)-G(\tau,s)|\mathrm{d}s+\int_{t}^{\tau} |G(t,s)-G(\tau,s)|\mathrm{d}s \\ &\quad +\int_{\tau}^{1}|G(t,s)-G(\tau,s)|\mathrm{d}s\Big) +|\varphi(t)-\varphi(\tau)|\\ &\leq N\Big[\int_{0}^{t}\Big(\frac{(\tau-s)^{\alpha-1} -(t-s)^{\alpha-1}}{\Gamma(\alpha)} +(\tau-t)\Big(\frac{a_{1}b_{1}}{l}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\\ &\quad + \frac{a_{1}b_{2}}{l}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)} \Big)\Big)\mathrm{d}s\\ &\quad +\int_{t}^{\tau}\Big(\frac{(\tau-s)^{\alpha-1}}{\Gamma(\alpha)} +(\tau-t)\Big(\frac{a_{1}b_{1}}{l}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)} +\frac{a_{1}b_{2}}{l}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}\Big)\Big) \mathrm{d}s\\ &\quad +\int_{\tau}^{1}(\tau-t)\Big(\frac{a_{1}b_{1}}{l} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)} +\frac{a_{1}b_{2}}{l}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}\Big) \mathrm{d}s\Big]+|\varphi(t)-\varphi(\tau)|\\ &= N\Big[\int_{0}^{\tau}\frac{(\tau-s)^{\alpha-1}}{\Gamma(\alpha)}\mathrm{d}s -\int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\mathrm{d}s +(\tau-t)\Big(\frac{a_{1}b_{1}}{l}\int_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\mathrm{d}s \\ &\quad +\frac{a_{1}b_{2}}{l}\int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)} \mathrm{d}s \Big)\Big] +(\tau-t)\frac{|a_{1}B-b_{1}A|}{l}\\ &= N\Big[\frac{\tau^{\alpha}-t^{\alpha}}{\Gamma(\alpha+1)} +(\tau-t)\Big(\frac{a_{1}b_{1}}{l\Gamma(\alpha+1)} +\frac{a_{1}b_{2}}{l\Gamma(\alpha)}\Big)\Big] +(\tau-t)\frac{|a_{1}B-b_{1}A|}{l}, \end{align*} \begin{align*} &|{^{C}D_{0^{+}}^{\beta}}(Tu)(t)-{^{C}D_{0^{+}}^{\beta}}(Tu)(\tau)| \\ &= \frac{1}{\Gamma(1-\beta)}\big| \int_{0}^{t}(t-s)^{-\beta}\Big(\int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta +\varphi'(s)\Big) \mathrm{d}s \\ &\quad -\int_{0}^{\tau}(\tau-s)^{-\beta}\Big( \int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta +\varphi'(s)\Big) \mathrm{d}s \big| \\ &\leq \frac{1}{\Gamma(1-\beta)}\big| \int_{0}^{t}(t-s)^{-\beta}\Big(\int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta\Big) \mathrm{d}s \\ &\quad - \int_{0}^{t}(\tau-s)^{-\beta}\Big( \int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta\Big) \mathrm{d}s \big| \\ &\quad +\frac{1}{\Gamma(1-\beta)}\big| \int_{0}^{t}(\tau-s)^{-\beta}\Big( \int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta\Big) \mathrm{d}s \\ &\quad -\int_{0}^{\tau}(\tau-s)^{-\beta}\Big( \int_{0}^{1}G'_{s}(s,\theta) f(\theta,u(\theta),{^{C}D_{0^{+}}^{\beta}}u(\theta))\mathrm{d}\theta\Big) \mathrm{d}s \big|\\ &\quad +\frac{1}{\Gamma(1-\beta)}\big| \int_{0}^{t}(t-s)^{-\beta}\varphi'(s)\mathrm{d}s- \int_{0}^{t}(\tau-s)^{-\beta}\varphi'(s)\mathrm{d}s\big|\\ &\quad +\frac{1}{\Gamma(1-\beta)}\big|\int_{0}^{t}(\tau-s)^{-\beta}\varphi'(s)\mathrm{d}s- \int_{0}^{\tau}(\tau-s)^{-\beta}\varphi'(s)\mathrm{d}s\big|\\ &\leq \frac{2N}{\Gamma(1-\beta)\Gamma(\alpha)} \int_{0}^{t}( (t-s)^{-\beta}-(\tau-s)^{-\beta}) \mathrm{d}s +\frac{2N}{\Gamma(1-\beta)\Gamma(\alpha)} \int_{t}^{\tau}(\tau-s)^{-\beta}\mathrm{d}s \\ &\quad +\frac{|a_{1}B-b_{1}A|}{l\Gamma(1-\beta)}\int_{0}^{t}( (t-s)^{-\beta}-(\tau-s)^{-\beta}) \mathrm{d}s+\frac{|a_{1}B-b_{1}A|}{l\Gamma(1-\beta)} \int_{t}^{\tau}(\tau-s)^{-\beta}\mathrm{d}s \\ &\leq \Big(\frac{2N}{\Gamma(2-\beta)\Gamma(\alpha)} +\frac{|a_{1}B-b_{1}A|}{l\Gamma(2-\beta)}\Big)( \tau^{1-\beta}-t^{1-\beta}+2(\tau-t)^{1-\beta}) \end{align*} for0<\beta<1, and \begin{align*} &|(Tu)'(t)-(Tu)'(\tau)| \\ &=\big| \int_{0}^{1}G'_{t}(t,s) f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s+\varphi'(t) \\ &\quad -\int_{0}^{1}G'_{\tau}(\tau,s) f(s,u(s),{^{C}D_{0^{+}}^{\beta}}u(s))\mathrm{d}s-\varphi'(\tau) \big| \\ &\leq \frac{N}{\Gamma(\alpha-1)}\Big( \int_{0}^{t}( (t-s)^{\alpha-2} -(\tau-s)^{\alpha-2})\mathrm{d}s +\int_{t}^{\tau}(\tau-s)^{\alpha-2}\mathrm{d}s \Big) \\ &\leq \frac{N}{\Gamma(\alpha)}(\tau^{\alpha-1}-t^{\alpha-1} +2(\tau-t)^{\alpha-1}) \end{align*} for\beta=1$. Now, using the fact that the functions$\tau^{\alpha}-t^{\alpha},\tau^{\alpha-1}-t^{\alpha-1}$and$\tau^{1-\beta}-t^{1-\beta}$are uniformly continuous on the interval$I$, we conclude that$TU$is an equicontinuous set. Obviously it is uniformly bounded since$TU\subseteq U$. Thus,$T$is completely continuous. The Schauder fixed-point theorem asserts the existence of solution in$U$for the problem \eqref{e1} and the theorem is proved. \end{proof} The following corollary is obvious. \begin{corollary} \label{coro2.1} Let the assumption {\rm (H)} be satisfied. Suppose further that there exist two nonnegative functions$a(t),b(t)\in C[0,1]$such that$|f(t,x,y)|\leq a(t)|x|^{\rho}+b(t)|y|^{\theta}$, where$0<\rho, \theta<1$. Then there exists at least one solution for the boundary value problem {\rm\eqref{e1}}. \end{corollary} \begin{example} \label{exa2.1} \rm Consider the problem \begin{gather*} ^CD_{0^{+}}^{3/2}u=(t-\frac{1}{2})^{3}(u(t) +{^CD_{0^{+}}^{1/2}}u(t)), \quad 01/8$), Theorem \ref{thm2.1} ensures the existence of solution for this problem. \end{example} \begin{theorem} \label{thm2.2} Let the assumption {\rm (H)} be satisfied. Furthermore, let the function $f$ fulfill a Lipschitz condition with respect to the second and third variables; i.e., $|f(t,x,y) -f(t,u,v)|\leq L(|x-u|+|v-y|)$ with a Lipschitz constant $L$ such that \$0