\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 64, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/64\hfil semilinear nonlocal Cauchy problems] {Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces} \author[X. Xue\hfil EJDE-2005/64\hfilneg] {Xingmei Xue} \address{Xingmei Xue \hfill\break Department of Mathematics, Southeast University, Nanjing 210018, China} \email{xmxue@seu.edu.cn} \date{} \thanks{Submitted April 26, 2005. Published June 23, 2005.} \subjclass[2000]{34G10,47D06} \keywords{Semilinear differential equation; nonlocal initial condition; \hfill\break\indent completely continuous operator} \begin{abstract} In this paper, we study a semilinear differential equations with nonlocal initial conditions in Banach spaces. We derive conditions for $f$, $T(t)$, and $g$ for the existence of mild solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \section{Introduction} In this paper we discuss the nonlocal initial value problem (IVP for short) \begin{gather} \label{e1.1} u'(t)=Au(t)+f(t,u(t)),\quad t\in(0,b),\\ u(0)=g(u)+u_{0}, \label{e1.2} \end{gather} where $A$ is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e. $C_{0}$-semigroup) $T(t)$ in Banach space $X$ and $f:[0,b]\times X\to X$, $g:C([0,b];X)\to X$ are given $X$-valued functions. The above nonlocal IVP has been studied extensively. Byszewski and Lasmikanthem \cite{b3,b4,b5} give the existence and uniqueness of mild solution when $f$ and $g$ satisfying Lipschitz-type conditions. Ntougas and Tsamatos \cite{n1,n2} study the case of compactness conditions of $g$ and $T(t)$. In \cite{l1} Lin and Liu discuss the semilinear integro-differential equations under Lipschitz-type conditions. Byszewski and Akca \cite{b6} give the existence of functional-differential equation when $T(t)$ is compact, and $g$ is convex and compact on a given ball of $C([0,b];X)$. In \cite{f1} Fu and Ezzinbi study the neutral functional differential equations with nonlocal initial conditions. Benchohra and Ntouyas \cite{b2} discuss the second order differential equations with nonlocal conditions under compact conditions. Aizicovici and McKibben \cite{a1} give the existence of integral solutions of nonlinear differential inclusions with nonlocal conditions. In references authors give the conditions of Lipschitz continuous of $g$ as $f$ be Lipschitz continuous, and give the compactness conditions of $g$ as $T(t)$ be compact and $g$ be uniformly bounded. In this paper we give the existence of mild solution of IVP \eqref{e1.1} and \eqref{e1.2} under following conditions of $g$, $T(t)$ and $f$: \begin{enumerate} \item $g$ and $f$ are compact, $T(t)$ is a $C_{0}$-semigroup \item $g$ is Lipschitz continuous, $f$ is compact and $T(t)$ is a $C_{0}$-semigroup \item $g$ is Lipschitz continuous and $T(t)$ is compact. \end{enumerate} Also give existence results in above cases without the assumption of uniformly boundedness of $g$. Let $(X,\|\cdot\|)$ be a real Banach space. Denoted by $C([0,b];X)$ the space of $X$-valued continuous functions on $[0,b]$ with the norm $|u|=\sup\{ \|u(t)\|,t\in [0,b]\}$ and denoted by $L(0,b;X)$ the space of $X$-valued Bochner integrable functions on $[0,b]$ with the norm $\|u\|_{1}=\int_{0}^{b} \|u(t)\|dt$. By a \emph{mild solution} of the nonlocal IVP \eqref{e1.1} and \eqref{e1.2} we mean the function $u\in C([0,b];X)$ which satisfies $$u(t)=T(t)u_{0}+T(t)g(u)+\int_{0}^{t}T(t-s)f(s,u(s))ds$$ for all $t\in[0,b]$. A $C_{0}$-semigroup $T(t)$ is said to be \emph{compact} if $T(t)$ is compact for any $t>0$. If the semigroup $T(t)$ is compact then $t\mapsto T(t)x$ are equicontinuous at all $t>0$ with respect to $x$ in all bounded subsets of $X$; i.e., the semigroup $T(t)$ is $equicontinuous$. To prove the existence results in this paper we need the following fixed point theorem by Schaefer. \begin{lemma}[\cite{s1}] \label{lem1.1} Let $S$ be a convex subset of a normed linear space $E$ and assume $0\in S$. Let $F:S\to S$ be a continuous and compact map, and let the set $\{x\in S:x=\lambda Fx$ for some $\lambda \in(0,1)\}$ be bounded. Then $F$ has at least one fixed point in $S$. \end{lemma} In this paper we suppose that $A$ generates a $C_{0}$ semigroup $T(t)$ on $X$. And, without loss of generality, we always suppose that $u_{0}=0$. \section{Main Results} In this section we give some existence results of the nonlocal IVP \eqref{e1.1} and \eqref{e1.2}. Here we list the following results. \begin{itemize} \item[(Hg)] (1) $g:C([0,b];X)\to X$ is continuous and compact. \\ (2) There exist $M>0$ such that $\|g(u)\|\leq M$ for $u\in C([0,b];X)$. \item[(Hf)] (1) $f(\cdot,x)$ is measurable for $x\in X$,$f(t,\cdot)$ is continuous for a.e. $t\in [0,b]$. \\ (2) There exist a function $a(\cdot)\in L^{1}(0,b,R^{+})$ and an increasing continuous function $\Omega:R^{+}\to R^{+}$ such that $\|f(t,x)\|\leq a(t)\Omega(\|x\|)$ for all $x\in X$ and a.e. $t\in[0,b]$. \\ (3) $f:[0,b]\times X \to X$ is compact. \end{itemize} \begin{thm} \label{thm2.1} If (Hg) and (Hf) are satisfied, then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.1} \int_{0}^{b}a(s)ds<\int_{NM}^{+\infty}\frac{ds}{N\Omega(s)},$$ where $N=\sup\{\|T(t)\|,t\in[0,b]\}$. \end{thm} Next, we give an existence result when $g$ is Lipschitz: \begin{itemize} \item[(Hg')] There exist a constant $k<1/N$ such that $\|g(u)-g(v)\|\leq k|u-v|$ for $u,v\in C([0,b];X)$. \end{itemize} \begin{thm} \label{thm2.2} If (Hg'), (Hg)(2), and (Hf) are satisfied, then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} when \eqref{e2.1} holds. \end{thm} Above we suppose that $g$ is uniformly bounded. Next, we give existence results without the hypothesis (Hg)(2). \begin{thm} \label{thm2.3} If (Hg)(1) and (Hf) are satisfied, then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.2} \int_{0}^{b}a(s)ds<\liminf_{T\to\infty} \frac{T-N\alpha(T)}{N\Omega(T)},$$ where $\alpha(T)=\sup \{\|g(u)\|;|u|\leq T\}$. \end{thm} \begin{thm} \label{thm2.4} If (Hg') and (Hf) are satisfied, then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.3} \int_{0}^{b}a(s)ds<\liminf_{T\to\infty}\frac{T-NkT}{N\Omega(T)}.$$ \end{thm} Next, we give an existence result when $g$ is Lipschitz and the semigroup $T(t)$ is compact. \begin{thm} \label{thm2.5} Assume that (Hg'), (Hf)(1), (Hf)(2) are satisfied, and assume that $T(t)$ is compact. Then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.4} \int_{0}^{b}a(s)ds<\liminf_{T\to\infty}\frac{T-NkT}{N\Omega(T)}.$$ \end{thm} At last we would like to discuss the IVP \eqref{e1.1} and \eqref{e1.2} under the following growth conditions of $f$ and $g$. \begin{itemize} \item[(Hf)(2')] There exist $m(\cdot),h(\cdot)\in L^{1}(0,b;R^{+})$ such that \begin{equation*} \|f(t,x)\|\leq m(t)\|x\|+h(t), \end{equation*} for a.e. $t\in[0,b]$ and $x\in X$. \item[(Hg)(2')] There exist constant $c,d$ such that for $u\in C([0,b];X)$, $\|g(u)\|\leq c|u|+d$. \end{itemize} Clearly (Hf)(2') is the special case of {\it H(f)(2)} with $a(t)=max\{m(t),h(t)\}$ and $\Omega(s)=s+1$. \begin{thm} \label{thm2.6} Assume (Hg)(1), (Hg)(2'), (Hf)(1), (Hf)(2'), and assume (Hf)(3) is true, or $T(t)$ is compact. Then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.5} Nce^{N\|m\|_{1}}<1,$$ where $\|\cdot\|_{1}$ means the $L^{1}(0,b)$ norm. \end{thm} \begin{thm} \label{thm2.7} Assume (Hg'), (Hf)(1), (Hf)(2'), and assume (Hf)(3) is true or $T(t)$ is compact. Then there is at least one mild solution for the IVP \eqref{e1.1} and \eqref{e1.2} provided that $$\label{e2.6} Nke^{N\|m\|_{1}}<1.$$ \end{thm} \section{Proofs of Main Results} We define $K:C([0,b];X)\to C([0,b];X)$ by $$(Ku)(t)=\int_{0}^{t}T(t-s)f(s,u(s))ds$$ for $t\in[0,b]$. To prove the existence results we need following lemmas. \begin{lemma} \label{lem3.1} If (Hf) holds, then $K$ is continuous and compact; i.e. $K$ is completely continuous. \end{lemma} \begin{proof} The continuity of $K$ is proved as follows. Let $u_{n}\to u$ in $C([0,b];X)$. Then \begin{equation*} |Ku_{n}-Ku|\leq N\int_{0}^{b}\|f(s,u_{n}(s))-f(s,u(s))\|ds. \end{equation*} So $Ku_{n}\to Ku$ in $C([0,b];X)$ by the Lebesgue's convergence theorem. Let $B_{r}=\{u\in C([0,b];X);|u|\leq r\}$. Form the Ascoli-Arzela theorem, to prove the compactness of $K$, we should prove that $KB_{r}\subset C([0,b];X)$ is equi-continuous and $KB_{r}(t)\subset X$ is pre-compact for $t\in[0,b]$ for any $r>0$. For any $u\in B_{r}$ we know \begin{align*} & \|Ku(t+h)-Ku(t)\| \\ &\leq N\int_{t}^{t+h}\|f(s,u(s))\|ds+\int_{0}^{t}\|[T(t+h-s) -T(t-s)]f(s,u(s))\|ds\\ & \leq N\int_{t}^{t+h}a(s)\Omega(r)ds +N\int_{0}^{t}\|[T(h)-I]f(s,u(s))\|ds. \end{align*} Since $f$ is compact, $\|[T(h)-I]f(s,u(s))\|\to 0$ (as $h\to 0$) uniformly for $s\in[0,b]$ and $u\in B_{r}$. This implies that for any $\epsilon>0$ there existing $\delta>0$ such that $\|[T(h)-I]f(s,u(s))\|\leq \epsilon$ for $0\leq h<\delta$ and all $u\in B_{r}$. We know that: \begin{equation*} \|Ku(t+h)-Ku(t)\|\leq N\Omega(r)\int_{t}^{t+h}a(s)ds+N \epsilon \end{equation*} for $0\leq h<\delta$ and all $u\in B_{r}$. So $KB_{r}\subset C([0,b];X)$ is equicontinuous. The set $\{T(t-s)f(s,u(s));t,s\in[0,b], u\in B_{r}\}$ is pre-compact as $f$ is compact and $T(\cdot)$ is a $C_{0}$ semigroup.So $KB_{r}(t)\subset X$ is pre-compact as $$KB_{r}(t)\subset t\ \overline{\mathop{\rm conv}}\{T(t-s)f(s,u(s));s\in[0,t] , u\in B_{r}\}$$ for all $t\in[0,b]$. \end{proof} Define $J:C([0,b];X)\to C([0,b];X)$ by $(Ju)(t)=T(t)g(u)$. So $u$ is the mild solution of IVP \eqref{e1.1}and \eqref{e1.2} if and only if $u$ is the fixed point of $J+K$. We can prove the following lemma easily. \begin{lemma} \label{lem3.2} If (Hg)(1) is true then $J$ is continuous and compact. \end{lemma} \begin{proof}[Proof of theorem \ref{thm2.1}] From above we know that $J+K$ is continuous and compact. To prove the existence, we should only prove that the set of fixed points of $\lambda(J+K)$ is uniformly bounded for $\lambda\in(0,1)$ by the Schaefer's fixed point theorem (Lemma \ref{lem1.1}). Let $u=\lambda(J+K)u$,i.e.,for $t\in[0,b]$ \begin{equation*} u(t)=\lambda T(t)g(u)+\lambda\int_{0}^{t}T(t-s)f(s,u(s))ds. \end{equation*} We have \begin{equation*} \|u(t)\| \leq NM+N\int_{0}^{t} a(s)\Omega(\|u(s)\|ds. \end{equation*} Denoting by $x(t)$ the right-hand side of the above inequality, we know that $x(0)=NM$ and $\|u(t)\|\leq x(t)$ for $t\in[0,b]$, and \begin{equation*} x'(t)=Na(t)\Omega(\|u(t)\|)\leq Na(t)\Omega(x(t)) \end{equation*} for a.e. $t\in[0,b]$. This implies \begin{equation*} \int_{NM}^{x(t)}\frac{ds}{N\Omega(s)}\leq\int_{0}^{t}a(s)ds <\int_{NM}^{\infty}\frac{ds}{N\Omega(s)}, \end{equation*} for $t\in[0,b]$. This implies that there is a constant $r>0$ such that $x(t)\leq r$, where $r$ is independent of $\lambda$. We complete the proof as $\|u(t)\| \leq r$ for $u\in \{u;u=\lambda(J+K)u$ for some $\lambda\in(0,1)\}$. \end{proof} For the next lemma, let $L:C([0,b];X)\to C([0,b];X)$ be defined as $(Lu)(t)=u(t)-T(t)g(u)$. \begin{lemma} \label{lem3.3} If (Hg') holds then $L$ is bijective and $L^{-1}$ is Lipschitz continuous with constant $1/(1-Nk)$. \end{lemma} \begin{proof} For any $v\in C([0,b];X)$, by using the Banach's fixed point theorem, we know that there is unique $u\in C([0,b];X)$ satisfying $Lu=v$. It implies that $L$ is bijective. For any $v_{1},v_{2}\in C([0,b];X)$, \begin{align*} \|L^{-1}v_{1}(t)-L^{-1}v_{2}(t)\| &\leq \|T(t)g(L^{-1}v_{1})-T(t)g(L^{-1}v_{1})\|+\|v_{1}(t)-v_{2}(t)\|\\ & \leq Nk|L^{-1}v_{1}-L^{-1}v_{2}|+\|v_{1}(t)-v_{2}(t)\| \end{align*} for $t\in[0,b]$. This implies \begin{equation*} |L^{-1}v_{1}-L^{-1}v_{2}|\leq \frac{1}{1-Nk}|v_{1}-v_{2}|. \end{equation*} which completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.2}] Clearly $u$ is the mild solution of IVP and \eqref{e1.2} if and only if $u$ is the fixed point of $L^{-1}K$. Similarly with Theorem \ref{thm2.1} we should only prove that the set $\{u;\lambda u=(L^{-1}K)u$ for some $\lambda>1\}$ is bounded as $L^{-1}K$ be continuous and compact due to the fixed point theorem of Schaefer.If $\lambda u=L^{-1}Ku$. Then for any $t\in[0,b]$ \begin{equation*} \lambda u(t)=T(t)g(\lambda u)+\int_{0}^{t}T(t-s)f(s,u(s))ds. \end{equation*} We have \begin{align*} \|u(t)\|&\leq \frac{1}{\lambda}NM+\frac{1}{\lambda}N\int_{0}^{t}a(s)\Omega(\|u(s)\|)ds\\ & \leq NM+N\int_{0}^{t}a(s)\Omega(\|u(s)\|)ds. \end{align*} Just as proved in Theorem \ref{thm2.1} we know there is a constant $r$ which is independent of $\lambda$, such that $|u|\leq r$ for all $u\in\{u;\lambda u=(L^{-1}K)u$ for some $\lambda>1\}$. So we proved this theorem. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.3}] By lemma \ref{lem3.1} and Lemma \ref{lem3.2} we know that $J+K$ is continuous and compact. From \eqref{e2.2} there exists a constant $r>0$ such that $$\int_{0}^{b}a(s)ds\leq\frac{r-N\alpha(r)}{N\Omega(r)}.$$ For any $u\in B_{r}$ and $v=Ju+Ku$,we get \begin{equation*} \|v(t)\|\leq N\alpha(r)+N\int_{0}^{t}a(s)\Omega(r)ds\leq r, \end{equation*} for $t\in[0,b]$. It implies that$(J+K)B_{r}\subset B_{r}$. By Schauder's fixed point theorem, we know that there is at least one fixed point $u\in B_{r}$ of the completely continuous map $J+K$, and $u$ is a mild solution. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.4}] By Lemma \ref{lem3.1} and Lemma \ref{lem3.3} we know that $L^{-1}K$ is continuous and compact. From \eqref{e2.3} there exists a constant number $r>0$ such that $$\int_{0}^{b}a(s)ds\leq\frac{r-Nkr-N\|g(0)\|}{N\Omega(r)}.$$ For any $u\in B_{r}$ and $v=L^{-1}Ku$, we get \begin{equation*} \|v(t)\|\leq Nk|v|+N\|g(0)\| +N\int_{0}^{t}a(s)\Omega(r)ds, \end{equation*} for $t\in[0,b]$. It implies that $|v|\leq r$, i.e., $L^{-1}KB_{r}\subset B_{r}$. By Schauder's fixed point theorem, there is at least one fixed point $u\in B_{r}$ of the completely continuous map $L^{-1}K$, and $u$ is a mild solution. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.5}] By the proof of \cite[Theorem 2.1]{n1} we know that $K$ is completely continuous under (Hf)(1), (Hf)(2) and condition of compactness of semigroup $T(t)$. So $L^{-1}K$ is completely continuous. Similarly with the proof of Theorem \ref{thm2.4}, we complete the the proof of this theorem. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.6}] From \cite[Theorem 2.1]{n1}, Lemma \ref{lem3.1} and Lemma \ref{lem3.2} we know that the map $J+K$ is completely continuous. By Lemma \ref{lem1.1} ,we should only prove that the set $\{u;u=\lambda(J+K)u$ for some $\lambda\in(0,1)\}$ is bounded. For any $u\in \{u;u=\lambda (J+K)u$ for some $\lambda\in(0,1)\}$, we have \begin{align*} \|u(t)\|&\leq \lambda (Nc|u|+Nd)+\lambda N\int_{0}^{t}m(s)\|u(s)\|ds +\lambda N\int_{0}^{t}h(s)ds\\ &\leq Nc|u|+N\int_{0}^{t}m(s)\|u(s)\|ds+N(d+\|h\|_{1}). \end{align*} This implies that for $t\in[0,b]$ \begin{equation*} |u|\leq\frac{N(d+\|h\|_{1})\exp(N\|m\|_{1})}{1-Nc\exp(N\|m\|_{1})}, \end{equation*} by Gronwall's inequality. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.7}] From \cite[Theorem 2.1]{n1}, Lemma \ref{lem3.2} and Lemma \ref{lem3.3} we know that the map $L^{-1}K$ is completely continuous. By Schaefer's fixed point theorem (Lemma \ref{lem1.1}), we should only prove that the set $\{u; u=\lambda(L^{-1}K)u$ for some $\lambda\in(0,1)\}$ is bounded. For any $u\in \{u;u=\lambda(L^{-1}K)u$ for some $\lambda\in(0,1)\}$, similarly with the estimation above, we know that \begin{equation*} |u|\leq \frac{N(\|g(0)\|+\|h\|_{1})\exp(N\|m\|_{1})}{1-Nk\exp(N\|m\|_{1})}. \end{equation*} The proof is complete. \end{proof} {\bf Acknowledgements}: The author would like to thank the referee very much for valuable comments and suggestions. \begin{thebibliography}{00} \bibitem{a1} Aizicovici, S. and Mckibben, M.; {\it Existence results for a class of abstract nonlocal Cauchy problems}, Nonlinear Analysis, 39(2000), 649-668. \bibitem{b1} Balachandran, K.; Park, J.; and Chanderasekran; {\it Nonlocal Cauchy problems for delay integrodifferential equations of Sobolev type in Banach spaces}, Appl. Math. Letters, 15(2002), 845-854. \bibitem {b2} Benchohra, M. and Ntouyas, S.; {\it Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces},J. Math. Anal. 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