\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 54, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/54\hfil Existence of solutions] {Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions} \author[G. Wang, W. Liu, C. Ren \hfil EJDE-2012/54\hfilneg] {Gang Wang, Wenbin Liu, Can Ren} \address{Gang Wang \newline Department of mathematics, University of Mining and Technology, Xuzhou 221008, China} \email{wangg0824@163.com} \address{Wenbin Liu \newline Department of mathematics, University of Mining and Technology, Xuzhou 221008, China} \email{wblium@163.com} \address{Can Ren \newline Department of mathematics, University of Mining and Technology, Xuzhou 221008, China} \email{rencan0502@163.com} \thanks{Submitted November 27, 2011. Published April 5, 2012.} \subjclass[2000]{34B15} \keywords{Fractional differential equation; boundary value problem; \hfill\break\indent fixed point theorem; existence and uniqueness} \begin{abstract} In this article, we study the multi-point boundary-value problem of nonlinear fractional differential equation \begin{gather*} D^\alpha_{0+}u(t)=f(t,u(t)),\quad 1<\alpha\leq 2,\; t\in[0,T],\; T>0,\\ I_{0+}^{2-\alpha}u(t)|_{t=0}=0,\quad D_{0+}^{\alpha-2}u(T)=\sum_{i=1}^ma_i I_{0+}^{\alpha-1}u(\xi_i), \end{gather*} where $D_{0^+}^\alpha$ and $I_{0^+}^\alpha$ are the standard Riemann-Liouville fractional derivative and fractional integral respectively. Some existence and uniqueness results are obtained by applying some standard fixed point principles. Several examples are given to illustrate the results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Fractional differential equations appear naturally in a number of fields such as physics, polymer rheology, regular variation in thermodynamics, biophysics,blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode¡¯s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, etc. An excellent account in the study of fractional differential equations can be found in \cite{k1,l1,s1,s2}. Boundary value problems for fractional differential equations have been discussed in \cite{a1,b1,d1,h1,l2,s4,w1,w2}. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermo-elasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper \cite{a6}. For more details of nonlocal and integral boundary conditions, see \cite{a7,b2,c1} and references therein. Ahmada and Nieto \cite{a1} considered the anti-periodic fractional boundary value problem given \begin{gather*} ^cD^qu(t)=f(t,u(t)),\quad 1<\alpha\leq 2,\\ u(0)=-u(T),\quad ^cD^pu(0)=^cD^pu(T), \end{gather*} where $^cD^q$ is the standard Caputo fractional derivative. Using of some existence and uniqueness results are obtained by applying some standard fixed point principles. Ahmada and Nieto \cite{a3} considered the fractional integro-differential equation with integral boundary conditions \begin{gather*} ^cD^qx(t)=f(t,x(t),(\chi x)(t)),\quad 10, \\ \label{e1.2} I_{0+}^{2-\alpha}u(t)|_{t=0}=0,\quad D_{0+}^{\alpha-2}u(T)=\sum_{i=1}^ma_i I_{0+}^{\alpha-1}u(\xi_i), \end{gather} where $0<\xi_i0$, $a_i\in \mathbb{R}$, $m\geq 2$, $D_{0^+}^\alpha$ and $I_{0^+}^\alpha$ are the standard Riemann-Liouville fractional derivative and fractional integral respectively, $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ is continuous. \section{Preliminaries} For the convenience of the reader, we present here some necessary basic knowledge and definitions for fractional calculus theory, that can be found in the recent literature. \begin{definition} \label{def2.1} \rm The fractional integral of order $\alpha>0$ of a function $y:(0,\infty)\to R$ is given by $$I_{0+}^\alpha y(t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}y(s)ds,$$ provided the right side is pointwise defined on $(0,\infty)$, where $\Gamma(\cdot)$ is the Gamma function. \end{definition} \begin{definition} \label{def2.2} \rm The fractional derivative of order $\alpha>0$ of a function $y:(0,\infty)\to R$ is given by $$D_{0+}^\alpha y(t)=\frac{1}{\Gamma(n-\alpha)}(\frac{d}{dt})^n \int_0^t\frac{y(s)}{(t-s)^{\alpha-n+1}}ds,$$ where $n=[\alpha]+1$, provided the right side is pointwise defined on $(0,\infty)$. \end{definition} \begin{lemma} \label{lem2.1} Let $\alpha>0$ and $u\in C(0,1)\cap L^1(0,1)$.Then fractional differential equation $D^\alpha_{0+}u(t)=0$ has $$u(t)=c_1t^{\alpha-1}+c_2t^{\alpha-2}+\dots +c_Nt^{\alpha-N},\quad c_i\in \mathbb{R},\; N=[\alpha]+1,$$ as unique solution. \end{lemma} \begin{lemma} \label{lem2.2} Assume that $u\in C(0,1)\cap L^1(0,1)$ with a fractional derivative of order $\alpha>0$ that belongs to $C(0,1)\cap L^1(0,1)$. Then $$I_{0+}^\alpha D_{0+}^\alpha u(t)=u(t)+c_1t^{\alpha-1}+c_2t^{\alpha-2}+\dots +c_Nt^{\alpha-N},$$ for some $c_i\in \mathbb{R},i=1,2,\dots ,N$, where $N$ is the smallest integer grater than or equal to $\alpha$. \end{lemma} \begin{definition} \label{def2.3} \rm For $n\in N$, we denote by $AC^n[0,1]$ the space of functions $u(t)$ which have continuous derivatives up to order $n-1$ on $[0,1]$ such that $u^{(n-1)}(t)$ is absolutely continuous: $AC^n[0,1]$ =$\{u|[0,1]\to R$ and $(D^{(n-1)})u(t)$ is absolutely continuous in $[0,1]\}$. \end{definition} \begin{lemma}[\cite{k1}] \label{lem2.3} Let $\alpha>0$, $n=[\alpha]+1$. Assume that $u\in L^1(0,1)$ with a fractional integration of order $n-\alpha$ that belongs to $AC^n[0,1]$. Then the equality $$(I_{0+}^{\alpha}D_{0+}^{\alpha}u)(t)=u(t) -\sum_{i=1}^n\frac{((I_{0+}^{n-\alpha}u)(t))^{n-i}|_{t=0}}{\Gamma(\alpha-i+1)}t^{\alpha-i}$$ holds almost everywhere on $[0, 1]$. \end{lemma} \begin{lemma}[\cite{k1}] \label{lem2.4} \begin{itemize} \item[(i)] Let $k\in N,\alpha > 0$. If $D_{a+}^{\alpha}y(t)$ and $(D_{a+}^{\alpha+k}y)(t)$ exist, then $$(D^kD_{a+}^{\alpha})y(t)=(D_{a+}^{\alpha+k}y)(t);$$ \item[(ii)] If $\alpha>0, \beta > 0, \alpha + \beta > 1$, then $$(I_{a+}^\alpha I_{a+}^{\alpha})y(t)=(I_{a+}^{\alpha+\beta}y)(t)$$ satisfies at any point on $[a,b]$ for $y\in L_p(a,b)$ and $1\leq p \leq \infty$; \item[(iii)] Let $\alpha > 0$ and $y\in C[a,b]$. Then $(D_{a+}^\alpha I_{a+}^{\alpha})y(t)=y(t)$ holds on $[a,b]$; \item[(iv)] Note that for $\lambda > -1, \lambda \neq \alpha-1,\alpha-2,\dots ,\alpha-n$, we have \begin{gather*} D^\alpha t^\lambda=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda-\alpha+1)}t^{\lambda-\alpha},\\ D^\alpha t^{\alpha-i}=0,i=1,2,\dots ,n \end{gather*} \end{itemize} \end{lemma} \begin{lemma} \label{lem2.5} For any $y(t)\in C[0,1]$, the linear fractional boundary-value problem $$\label{e2.1} \begin{gathered} D^\alpha_{0+}u(t)=y(t),\quad 1<\alpha\leq 2,\; t\in[0,T],\\ I_{0+}^{2-\alpha}u(t)|_{t=0}=0,\quad D_{0+}^{\alpha-2}u(T)=\sum_{i=1}^ma_i I_{0+}^{\alpha-1}u(\xi_i), \end{gathered}$$ has unique solution $$\label{e2.2} \begin{split} u(t)&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds\\ &\quad+\frac{t^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big[\frac{\sum_{i=1}^ma_i}{\Gamma(2\alpha-1)}\int_0^{\xi_i}(\xi_i-s)^{2\alpha-2}y(s)ds -\int_0^T(T-s)y(s)ds\Big], \end{split}$$ where $A=\sum_{i=1}^ma_i\xi_i^{2\alpha-2}/\Gamma(2\alpha-1)$ and $T\neq A$. \end{lemma} \begin{proof} By Lemma \ref{lem2.2}. the solution of \eqref{e2.1} can be written as $$u(t)=c_1t^{\alpha-1}+c_2t^{\alpha-2} +\frac{1}{\Gamma(\alpha)}\int^t_0(t-s)^{\alpha-1}y(s)ds.$$ From $I_{0+}^{2-\alpha}u(t)|_{t=0}=0$, and by Lemmas \ref{lem2.3} and \ref{lem2.4}, we know that $c_2=0$, and \begin{gather*} D_{0+}^{\alpha-2}u(t)=c_1t\Gamma(\alpha)+I_{0+}^2y(t),\\ I_{0+}^{\alpha-1}u(t)=c_1\frac{\Gamma(\alpha)}{\Gamma(2\alpha-1)}t^{2\alpha-2} +I_{0+}^{\alpha-1}I_{0+}^\alpha y(t), \end{gather*} from $D_{0+}^{\alpha-2}u(T)=\sum_{i=1}^ma_i I_{0+}^{\alpha-1}u(\xi_i)$, we have $$c_1=\frac{1}{\Gamma(\alpha)(T-A)} \Big[\frac{\sum_{i=1}^ma_i}{\Gamma(2\alpha-1)}\int_0^{\xi_i}(\xi_i-s) ^{2\alpha-2}y(s)ds -\int_0^T(T-s)y(s)ds\Big],$$ where $A=\sum_{i=1}^ma_i\xi_i^{2\alpha-2}/\Gamma(2\alpha-1)$ and $T\neq A$, so \begin{align*} u(t) &= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}y(s)ds\\ &\quad \frac{t^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big[\frac{\sum_{i=1}^ma_i}{\Gamma(2\alpha-1)} \int_0^{\xi_i}(\xi_i-s)^{2\alpha-2}y(s)ds -\int_0^T(T-s)y(s)ds\Big]. \end{align*} The proof is complete. \end{proof} \section{Existence and uniqueness of solutions} Let $E =C([0,T],R)$ denote the Banach space of all continuous functions from $[0,T]\to R$ endowed with the norm defined by $\|x\|=sup\{|x(t)|,t\in[0,T]\}$. Now we state some known fixed point theorems which are needed to prove the existence of solutions for \eqref{e1.1}--\eqref{e1.2}. \begin{theorem}[\cite{s3}] \label{thm3.1} Let $X$ be a Banach space. Assume that $T:X\to X$ is a completely continuous operator and the set $V=\{u\in X |u=\mu Tu,0<\mu< 1\}$ is bounded. Then $T$ has a fixed point in $X$. \end{theorem} \begin{theorem}\cite{s3} \label{thm3.2} Let $X$ be a Banach space. Assume that $\Omega$ is an open bounded subset of $X$ with $\theta \in \Omega$and let $T:\bar{\Omega}\to X$ be a completely continuous operator such that $$\|Tu\|\leq \|u\|, \forall u\in \partial \Omega.$$ Then $T$ has a fixed point in $\bar{\Omega}$. \end{theorem} We define, in relation to \eqref{e2.2}, an operator $P:E\to E$, as $$\label{e3.1} \begin{split} (Pu)(t) &= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(t,u(s))ds\\ &\quad +\frac{t^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}f(t,u(s))ds\\ &\quad -\int_{0}^{T}(T-s)f(t,u(s))ds\Big). \end{split}$$ Observe that this equation has a solution if and only if the operator $P$ has a fixed point. \begin{theorem} \label{thm3.3} Assume that there exists a positive constant $L_1$ such that $|f(t,u)|\leq L_1$ for $t\in[0,T],u\in E$. Then \eqref{e1.1}-\eqref{e1.2} has at least one solution. \end{theorem} \begin{proof} We show, as a first step, that the operator $P$ is completely continuous. Clearly, continuity of the operator $P$ follows from the continuity of $f$. Let $\Omega\subset E$ be bounded. Then, $\forall u\in \Omega$ together with the assumption $|f(t,u)|\leq L_1$, we obtain \begin{align*} (Pu)(t) &\leq \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(t,u(s))|ds\\ &+\frac{t^{\alpha-1}}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}|f(t,u(s))|ds\\ &\quad -\int_{0}^{T}(T-s)|f(t,u(s))|ds\Big)\\ &\leq L_1\Big[\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}ds\\ &\quad +\frac{t^{\alpha-1}}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}ds -\int_{0}^{T}(T-s)ds\Big)\Big]\\ &\leq L_1\Big[\frac{T^{\alpha}}{\Gamma(\alpha+1)} +\frac{T^{\alpha-1}}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i \xi^{2\alpha-1}}{\Gamma(2\alpha)} -\frac{T^2}{2}\Big)\Big], \end{align*} which implies $$\|Pu\|\leq L_1\Big[\frac{T^{\alpha}}{\Gamma(\alpha+1)} +\frac{T^{\alpha-1}}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i \xi^{2\alpha-1}}{\Gamma(2\alpha)}-\frac{T^2}{2} \Big)\Big]<\infty.$$ Hence, $T(\Omega)$ is uniformly bounded. For any $t_1, t_2\in[0, T], u\in \Omega$, we have \begin{align*} &|(Pu)(t_1)-(Pu)(t_2)|\\ &=\Big|\int_0^{t_1}\frac{(t_1-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,u(s))ds\\ &\quad +\frac{t_1^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}f(s,u(s))ds\\ &\quad -\int_{0}^{T}(T-s)^f(s,u(s))ds\Big) -\int_0^{t_2}\frac{(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}f(s,u(s))ds -\frac{t_2^{\alpha-1}}{\Gamma(\alpha)(T-A)}\\ &\quad\times\Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}f(s,u(s))ds -\int_{0}^{T}(T-s)f(s,u(s))ds\Big)\Big| \\ &\leq L_1\Big|\int_0^{t_1}\frac{(t_1-s)^{\alpha-1} -(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}ds\\ &\quad +\frac{t_1^{\alpha-1}-t_2^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)}\int_{0}^{\xi_i} (\xi_i-s)^{2\alpha-2}ds \\ &\quad -\int_{0}^{T}(T-s)ds\Big)- \int_{t_1}^{t_2}\frac{(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}ds\Big| \\ &\leq L_1\Big[\Big|\int_0^{t_1}\frac{(t_1-s)^{\alpha-1} -(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}ds- \int_{t_1}^{t_2}\frac{(t_2-s)^{\alpha-1}}{\Gamma(\alpha)}ds\Big|\\ &\quad +\Big|\frac{t_1^{\alpha-1}-t_2^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}ds- \int_{0}^{T}(T-s)ds\Big)\Big|\Big]\\ & \to 0 \quad\text{as } t_1\to t_2. \end{align*} Thus, by the Arzela-Ascoli theorem, $P(\Omega)$ is equicontinuous. Consequently, the operator $P$ is compact. Next, we consider the set $V=\{u\in E:u=\mu Pu,0<\mu<1\}$, and show that it is bounded. Let $u\in V;$ then $u=\mu Pu,0 <\mu< 1$. For any $t\in [0,T]$, we have \begin{align*} u(t)&= \int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(t,u(s))ds\\ &\quad +\frac{t^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}f(t,u(s))ds\\ &\quad -\int_{0}^{T}(T-s)f(t,u(s))ds\Big), \end{align*} and \begin{align*} |u(t)| &=\mu|Pu|\\ &\leq\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}|f(t,u(s))|ds\\ &\quad +\frac{t^{\alpha-1}}{\Gamma(\alpha)(T-A)} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}|f(t,u(s))|ds\\ &\quad -\int_{0}^{T}(T-s)|f(t,u(s))|ds\Big) \\ &\leq L_1\Big[\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}ds\\ &\quad +\frac{t^{\alpha-1}}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i }{\Gamma(2\alpha-1)} \int_{0}^{\xi_i}(\xi_i-s)^{2\alpha-2}ds -\int_{0}^{T}(T-s)ds\Big)\Big]\\ &\leq \max_{t\in[0,T]}\Big\{L_1\Big[\frac{|t^{\alpha}|}{\Gamma(\alpha+1)} +\frac{|t^{\alpha-1}|}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i \xi^{2\alpha-1}}{\Gamma(2\alpha)}-\frac{T^2}{2}\Big)\Big] \Big\} =M. \end{align*} Thus, $\|u\|\leq M$. So, the set $V$ is bounded. Thus, by the conclusion of Theorem \ref{thm3.1}, the operator $P$ has at least one fixed point, which implies that \eqref{e1.1}-\eqref{e1.2} has at least one solution. \end{proof} \begin{theorem} \label{thm3.4} Let $lim_{x\to 0}\frac{f(t,x)}{x}=0$. Then \eqref{e1.1}-\eqref{e1.2} has at least one solution. \end{theorem} \begin{proof} Since $\lim_{x\to 0}\frac{f(t,x)}{x}=0$, there exists a constant $r>0$ such that $|f(t,x)|\leq \varepsilon|x|$ for $0<|x|0$ is such that $$\label{e3.2} \max_{t\in[0,T]}\Big\{\frac{|t^{\alpha}|}{\Gamma(\alpha+1)} +\frac{|t^{\alpha-1}|}{\Gamma(\alpha)|T-A|} \Big(\frac{\sum_{i=1}^ma_i \xi^{2\alpha-1}}{\Gamma(2\alpha)}-\frac{T^2}{2}\Big)\Big\} \varepsilon\leq 1,$$ Define \$\Omega_1=\{x\in E:\|x\|