\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 261, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/261\hfil Solvability of periodic boundary-value problems] {Solvability of periodic boundary-value problems for second-order nonlinear differential equation involving fractional derivatives} \author[T. Chen, W. Liu \hfil EJDE-2014/261\hfilneg] {Taiyong Chen, Wenbin Liu} % in alphabetical order \address{Taiyong Chen (corresponding author)\newline Department of Mathematics, China University of Mining and Technology, \newline Xuzhou 221116, China \newline Phone +86 516 83591530, Fax +86 516 83591591.} \email{taiyongchen@cumt.edu.cn} \address{Wenbin Liu \newline Department of Mathematics, China University of Mining and Technology, \newline Xuzhou 221116, China} \email{wblium@163.com} \thanks{Submitted May 25, 2014. Published December 17, 2014.} \subjclass[2000]{34A08, 34B15} \keywords{Second-order differential equation; fractional derivative; \hfill\break\indent periodic boundary-value problem; coincidence degree} \begin{abstract} This article concerns the existence of solutions to periodic boundary-value problems for second-order nonlinear differential equation involving fractional derivatives. Under certain linear growth condition of the nonlinearity, we obtain solutions, by using coincidence degree theory. An example illustrates our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. This subject, as old as the problem of ordinary differential calculus, can go back to the times when Leibniz and Newton invented differential calculus. As is known to all, the problem for fractional derivative was originally raised by Leibniz in a letter, dated September 30, 1695. A fractional derivative arises from many physical processes, such as a non-Markovian diffusion process with memory \cite{c001}, charge transport in amorphous semiconductors \cite{c002}, propagations of mechanical waves in viscoelastic media \cite{c003}, etc. Moreover, phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order \cite{c004,c005,c006,c007,c008}. For instance, to describe the horizontal vibration of the rigid thin plate with massless spring immersing vertically in ideal fluid, Torvik and Bagley \cite{c009} introduced the well known fractional differential equation \begin{align*} Ax''(t)+BD_{t}^{3/2}x(t)+Cx(t)=f(t). \end{align*} Recently, fractional differential equations have been of great interest due to the intensive development of the theory of fractional calculus itself and its applications. For example, for fractional initial value problems, the existence and multiplicity of solutions (or positive solutions) were discussed in \cite{c010,c011,c012,c013}. On the other hand, for fractional boundary value problems, Agarwal et al. \cite{c014} considered a two-point boundary value problem at nonresonance, and Bai \cite{c015} considered a $m$-point boundary value problem at resonance. Moreover, for fractional periodic boundary value problems, Belmekki et al \cite{c016} discussed the existence of periodic solutions, and Kaslik et al \cite{c017} discussed the no-existence of periodic solutions. For more articles on fractional boundary value problems, see \cite{c019,c018,c020,c021,c022,c023,c024} and the references therein. In the present article, motivated by the works mentioned previously, we investigate the existence of solutions for the periodic boundary-value problem (PBVP for short) \begin{equation}\label{bvp} \begin{gathered} x''(t)=f(t,x(t),D_{0^+}^\alpha x(t)),\quad t\in[0,1],\\ x(0)=x(1),\quad D_{0^+}^\alpha x(0)=D_{0^+}^\alpha x(1), \end{gathered} \end{equation} where $0<\alpha<2$ is a real number, $D_{0^+}^\alpha$ is a Caputo fractional derivative, and $f:[0,1]\times\mathbb{R}^2\to \mathbb{R}$ is continuous. The rest of this article is organized as follows. Section 2 contains some necessary notation, definitions and lemmas. In Section 3, basing on the coincidence degree theory of Mawhin \cite{c025}, we establish a theorem on existence of solutions for PBVP \eqref{bvp} under linear growth restriction of $f$. Finally, in Section 4, an example is given to illustrate the main result. \section{Preliminaries} For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory, which can be found, for instance, in \cite{c026,c027}. \begin{definition} \label{def2.1} \rm The Riemann-Liouville fractional integral operator of order $\alpha >0$ of a function $u:(0,+\infty)\to \mathbb{R}$ is given by \[ I_{0^+}^\alpha u(t)=\frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha -1}u(s)\,ds, \] provided that the right side integral is pointwise defined on $(0,+\infty)$. \end{definition} \begin{definition} \label{def2.2}\rm The Caputo fractional derivative of order $\alpha >0$ of a continuous function $u:(0,+\infty)\to \mathbb{R}$ is given by \[ D_{0^+}^\alpha u(t)=I_{0^+}^{n-\alpha}\frac{d^nu(t)}{d t^n} =\frac{1}{\Gamma(n-\alpha)}\int_0^t(t-s)^{n-\alpha-1}u^{(n)}(s)\,ds, \] where $n$ is the smallest integer greater than or equal to $\alpha$, provided that the right side integral is pointwise defined on $(0,+\infty)$. \end{definition} \begin{lemma}[\cite{c028}] \label{lem2.1} Let $\alpha>0$. Assume that $u,D_{0^+}^\alpha u\in L(0,1)$. Then the following equality holds \[ I_{0^+}^{\alpha}D_{0^+}^{\alpha}u(t)=u(t)+c_0+c_1t+\dots +c_{n-1}t^{n-1}, \] where $c_i\in{\mathbb{R}}$, $i=0,1,\dots ,n-1$, here $n$ is the smallest integer greater than or equal to $\alpha$. \end{lemma} Now, we briefly recall some notation and an abstract existence result, which can be found in \cite{c025}. Let $X,Y$ be real Banach spaces, $L:\operatorname{dom}L\subset X\to Y$ be a Fredholm operator with index zero, and $P:X\to X$, $Q:Y\to Y$ be projectors such that \[ \operatorname{Im}P=\ker L,\quad \ker Q=\operatorname{Im}L,\quad X=\ker L\oplus\ker P,\quad Y=\operatorname{Im}L\oplus\operatorname{Im}Q. \] It follows that \[ L|_{\operatorname{dom}L\cap\ker P}:\operatorname{dom}L\cap \ker P\to \operatorname{Im}L \] is invertible. We denote the inverse by $K_P$. If $\Omega$ is an open bounded subset of $X$ such that $\operatorname{dom}L\cap\overline{\Omega}\neq\emptyset$, then the map $N:X\to Y$ will be called $L$-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_P(I-Q)N:\overline{\Omega}\to X$ is compact. \begin{lemma}[\cite{c025}] \label{lem2.2} Let $L:\operatorname{dom}L\subset X\to Y$ be a Fredholm operator of index zero and $N:X\to Y$ be $L$-compact on $\overline{\Omega}$. Assume that the following conditions are satisfied \begin{itemize} \item[(1)] $Lx\neq\lambda Nx$ for every $(x,\lambda) \in[(\operatorname{dom}L\setminus\ker L)\cap\partial\Omega]\times(0,1)$; \item[(2)] $Nx\notin\operatorname{Im}L$ for every $x\in\ker L\cap\partial\Omega$; \item[(3)] $\deg (QN|_{\ker L},\Omega\cap\ker L,0)\neq0$, where $Q:Y\to Y$ is a projection such that $\operatorname{Im}L=\ker Q$. \end{itemize} Then the equation $Lx=Nx$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$. \end{lemma} In this article, we take $Y=C[0,1]$ with the norm $\|y\|_\infty=\max_{t\in[0,1]}|y(t)|$, and $X=\{x:x,D_{0^+}^\alpha x\in Y\}$ with the norm $\|x\|_X=\max\{\|x\|_\infty,\|D_{0^+}^\alpha x\|_\infty\}$. By means of the linear functional analysis theory, we can prove that $X$ is a Banach space. Define the operator $L:\operatorname{dom}L\subset X\to Y$ by \begin{equation} \label{zsz} Lx=x'', \end{equation} where \begin{align*} \operatorname{dom}L=\{x\in X:x''\in Y,\ x(0)=x(1),D_{0^+}^\alpha x(0)=D_{0^+}^\alpha x(1)\}. \end{align*} Let $N:X\to Y$ be the Nemytskii operator \begin{equation} \label{nsz} Nx(t)=f(t,x(t),D_{0^+}^\alpha x(t)),\quad \forall t\in[0,1]. \end{equation} Then PBVP \eqref{bvp} is equivalent to the operator equation \[ Lx=Nx,\quad x\in \operatorname{dom}L. \] From the definition of $L$, we can obtain that \begin{gather} \ker L=\{x\in X: x(t)=c,\; \forall t\in[0,1],\;c\in\mathbb{R}\}, \\ \operatorname{Im}L=\big\{y\in Y:\int_0^1(1-s)^{1-\alpha}y(s)\,ds=0\big\}. \label{iml} \end{gather} Let us define the linear continuous projector operators $P:X\to X$ and $Q:Y\to Y$ by \begin{gather*} Px(t)=x(0),\quad \forall t\in[0,1],\\ Qy(t)=(2-\alpha)\int_0^1(1-s)^{1-\alpha}y(s)\,ds,\quad \forall t\in[0,1]. \end{gather*} Obviously \begin{align*} \operatorname{Im}P=\ker L,\quad \ker Q=\operatorname{Im}L,\quad X=\ker L\oplus\ker P, \end{align*} and the operator $K_P:\operatorname{Im}L\to \operatorname{dom}L\cap\ker P$ can be written as \[ K_Py(t) =\int_0^t(t-s)y(s)\,ds-\int_0^1(1-s)y(s)\,ds\cdot t,\quad \forall t\in[0,1]. \] \section{Existence result} In this section, a theorem on existence of solutions for PBVP \eqref{bvp} is given. \begin{theorem} \label{thm3.1} Let $f:[0,1]\times\mathbb{R}^2\to \mathbb{R}$ be continuous. Assume that \begin{itemize} \item[(H1)] there exist nonnegative functions $a,b,c\in Y$ such that \[ |f(t,u,v)|\leq a(t)+b(t)|u|+c(t)|v|, \quad \forall t\in[0,1],(u,v)\in\mathbb{R}^2; \] \item[(H2)] there exists a constant $B>0$ such that either \[ uf(t,u,v)>0, \quad \forall t\in[0,1],\;v\in\mathbb{R},\;|u|>B \] or \[ uf(t,u,v)<0, \quad \forall t\in[0,1],\;v\in\mathbb{R},\;|u|>B. \] \end{itemize} Then PBVP \eqref{bvp} has at least one solution, provided that \begin{equation} \label{xsyq} \frac{1}{\Gamma(3-\alpha)} \Big(\frac{2\|b\|_\infty}{\Gamma(\alpha+1)}+\|c\|_\infty\Big)<1. \end{equation} \end{theorem} Next, we introduce some lemmas that are useful in what follows. \begin{lemma} \label{lem3.1} Let $L$ be defined by \eqref{zsz}, then $L$ is a Fredholm operator of index zero. \end{lemma} \begin{proof} For any $y\in Y$, we have \begin{equation} \label{tyq} Q^2y(t)=Qy(t) (2-\alpha)\int_0^1(1-s)^{1-\alpha}\,ds=Qy(t). \end{equation} Let $y_1=y-Qy$, then from \eqref{tyq} we obtain \begin{align*} \int_0^1(1-s)^{1-\alpha}y_1(s)\,ds &=\int_0^1(1-s)^{1-\alpha}y(s)\,ds -\int_0^1(1-s)^{1-\alpha}Qy(s)\,ds\\ &=\frac{1}{2-\alpha}Qy(t)-\frac{1}{2-\alpha}Q^2y(t)=0, \end{align*} which implies $y_1\in\operatorname{Im}L$. Hence $Y=\operatorname{Im}L+\operatorname{Im}Q$. Since $\operatorname{Im}L\cap\operatorname{Im}Q=\{0\}$, we have \[ Y=\operatorname{Im}L\oplus\operatorname{Im}Q. \] Thus, \[ \dim \ker L=\dim\operatorname{Im}Q=\operatorname{codim}\operatorname{Im}L=1. \] This means that $L$ is a Fredholm operator of index zero. The proof is complete. \end{proof} \begin{lemma} \label{lem3.2} Let $L$ be defined by \eqref{zsz} and $N$ be defined by \eqref{nsz}. Assume $\Omega\subset X$ is an open bounded subset such that $\operatorname{dom}L\cap\overline{\Omega}\neq\emptyset$, then $N$ is $L$-compact on $\overline{\Omega}$. \end{lemma} \begin{proof} By the continuity of $f$, we can show that $QN(\overline{\Omega})$ and $K_P(I-Q)N(\overline{\Omega})$ are bounded. Moreover, there exists a constant $T>0$ such that $|(I-Q)Nx|\leq T$ for all $x\in\overline{\Omega}$, $t\in[0,1]$. Thus, in view of the Arzel\`a-Ascoli theorem, we need only prove that $K_P(I-Q)N(\overline{\Omega})\subset X$ is equicontinuous. For $0\leq t_10$ such that \begin{equation} \label{dxfs} \|D_{0^+}^\alpha x\|_\infty\leq M_1. \end{equation} Thus, from \eqref{xfs1}, we get \begin{equation} \label{xfs2} \|x\|_\infty\leq B+\frac{2M_1}{\Gamma(\alpha+1)}:=M_2. \end{equation} Combining \eqref{dxfs} with \eqref{xfs2}, we have \begin{align*} \|x\|_X=\max\{\|x\|_\infty,\|D_{0^+}^\alpha x\|_\infty\} \leq\max\{M_1,M_2\}:=M. \end{align*} Therefore, $\Omega_1$ is bounded. The proof is complete. \end{proof} \begin{lemma} \label{lem3.4} Suppose {\rm (H2)} holds, then the set $\Omega_2=\{x\in\ker L:Nx\in\operatorname{Im}L\}$ is bounded. \end{lemma} \begin{proof} For $x\in\Omega_2$, we have $x(t)=c$, $c\in\mathbb{R}$ and $Nx\in\operatorname{Im}L$. Then \[ \int_0^1(1-s)^{1-\alpha}f(s,c,0)\,ds=0, \] which together with (H2) implies $|c|\leq B$. Thus, we have \[ \|x\|_X\leq\max\{B,0\}=B. \] Hence, $\Omega_2$ is bounded. The proof is complete. \end{proof} \begin{lemma} \label{lem3.5} Suppose the first part of {\rm (H2)} holds, then the set \[ \Omega_3=\{x\in\ker L:\lambda x+(1-\lambda)QNx=0,\; \lambda\in[0,1]\} \] is bounded. \end{lemma} \begin{proof} For $x\in\Omega_3$, we have $x(t)=c$, $c\in\mathbb{R}$ and \begin{equation} \label{djjf} \lambda c+(1-\lambda)(2-\alpha)\int_0^1(1-s)^{1-\alpha}f(s,c,0)\,ds=0. \end{equation} If $\lambda=0$, then $|c|\leq B$ because of the first part of (H2). If $\lambda\in(0,1]$, we can also obtain $|c|\leq B$. Otherwise, if $|c|>B$, in view of the first part of (H2), one has \[ \lambda c^2+(1-\lambda)(2-\alpha)\int_0^1(1-s)^{1-\alpha}cf(s,c,0)\,ds>0, \] which contradicts \eqref{djjf}. Therefore, $\Omega_3$ is bounded. The proof is complete. \end{proof} \begin{remark} \label{rmk3.1} \rm If the second part of {\rm (H2)} holds, then the set \[ \Omega'_3=\{x\in\ker L:-\lambda x+(1-\lambda)QNx=0,\; \lambda\in[0,1]\} \] is bounded. \end{remark} \begin{proof}[Proof of Theorem \ref{thm3.1}] Set \[ \Omega=\{x\in X:\|x\|_X<\max\{M,B\}+1\}. \] Obviously, $\Omega_1\cup\Omega_2\cup\Omega_3\subset\Omega$ (or $\Omega_1\cup\Omega_2\cup\Omega'_3\subset\Omega$). It follows from Lemma \ref{lem3.1} and Lemma \ref{lem3.2} that $L$ (defined by \eqref{zsz}) is a Fredholm operator of index zero and $N$ (defined by \eqref{nsz}) is $L$-compact on $\overline{\Omega}$. By Lemma \ref{lem3.3} and Lemma \ref{lem3.4}, the following two conditions are satisfied \begin{itemize} \item[(1)] $Lx\neq\lambda Nx$ for every $(x,\lambda)\in[(\operatorname{dom}L\setminus\ker L)\cap\partial\Omega]\times(0,1)$; \item[(2)] $Nx\notin\operatorname{Im}L$ for every $x\in\ker L\cap\partial\Omega$. \end{itemize} It remains verifying condition (3) of Lemma \ref{lem2.2}. To do that, let \[ H(x,\lambda)=\pm\lambda x+(1-\lambda)QNx. \] Based on Lemma \ref{lem3.5} (or Remark \ref{rmk3.1}), we have \begin{align*} H(x,\lambda)\neq0,\quad \forall x\in\partial\Omega\cap\ker L. \end{align*} Thus, by the homotopy property of degree, we have \begin{align*} \deg (QN|_{\ker L},\Omega\cap\ker L,0) &=\deg (H(\cdot,0),\Omega\cap\ker L,0)\\ &=\deg (H(\cdot,1),\Omega\cap\ker L,0)\\ &=\deg (\pm I,\Omega\cap\ker L,0)\neq0. \end{align*} So that condition (3) of Lemma \ref{lem2.2} is satisfied. Consequently, by using Lemma \ref{lem2.2}, the operator equation $Lx=Nx$ has at least one solution in $\operatorname{dom}L\cap\overline{\Omega}$. Namely, PBVP \eqref{bvp} has at least one solution in $X$. The proof is complete. \end{proof} \section{An example} To illustrate our main result, we consider the periodic boundary-value problem \begin{equation}\label{ex} \begin{gathered} x''(t) =-2+\frac{1}{2}x(t)+te^{-(D_{0^+}^{3/2}x(t))^2},\quad t\in[0,1],\\ x(0)=x(1),\quad D_{0^+}^{3/2}x(0)=D_{0^+}^{3/2}x(1). \end{gathered} \end{equation} Corresponding to PBVP \eqref{bvp}, we have $\alpha=3/2$ and \begin{align*} f(t,u,v)=-2+\frac{1}{2}u+te^{-v^2}. \end{align*} Choose $a(t)=3$, $b(t)=1/2$, $c(t)=0$, $B=4$. 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