\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
\begin{equation}
  u(t)=T(t)u_{0}+T(t)g(u)+\int_{0}^{t}T(t-s)f(s,u(s))ds
  \end{equation}
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
\begin{equation} \label{e2.1}
\int_{0}^{b}a(s)ds<\int_{NM}^{+\infty}\frac{ds}{N\Omega(s)},
\end{equation}
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
\begin{equation} \label{e2.2}
\int_{0}^{b}a(s)ds<\liminf_{T\to\infty}
\frac{T-N\alpha(T)}{N\Omega(T)},
\end{equation}
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
\begin{equation} \label{e2.3}
\int_{0}^{b}a(s)ds<\liminf_{T\to\infty}\frac{T-NkT}{N\Omega(T)}.
\end{equation}
\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
\begin{equation} \label{e2.4}
\int_{0}^{b}a(s)ds<\liminf_{T\to\infty}\frac{T-NkT}{N\Omega(T)}.
\end{equation}
\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
\begin{equation} \label{e2.5}
Nce^{N\|m\|_{1}}<1,
\end{equation}
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
\begin{equation} \label{e2.6}
Nke^{N\|m\|_{1}}<1.
\end{equation}
\end{thm}

\section{Proofs of Main Results}

We define $K:C([0,b];X)\to C([0,b];X)$ by
\begin{equation}
(Ku)(t)=\int_{0}^{t}T(t-s)f(s,u(s))ds
\end{equation} 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
 \begin{equation}
\int_{0}^{b}a(s)ds\leq\frac{r-N\alpha(r)}{N\Omega(r)}.
\end{equation}
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
 \begin{equation}
\int_{0}^{b}a(s)ds\leq\frac{r-Nkr-N\|g(0)\|}{N\Omega(r)}.
\end{equation}
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.

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\end{document}