\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 18, 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-2014/18\hfil Existence and uniqueness of strong solutions]
{Existence and uniqueness of strong solutions for
 nonlocal evolution equations}

\author[P. Chen, Y. Li \hfil EJDE-2014/18\hfilneg]
{Pengyu Chen, Yongxiang Li}  % in alphabetical order

\address{Pengyu Chen \newline
Department of Mathematics, Northwest Normal University,
Lanzhou 730000,  China}
\email{chpengyu123@163.com}

\address{Yongxiang Li  \newline
Department of Mathematics, Northwest Normal University,
Lanzhou 730000,  China}
\email{liyx@nwnu.edu.cn}

\thanks{Submitted April 28, 2013. Published January 10, 2014.}
\subjclass[2000]{34G20, 34K30, 35D35, 47D06}
\keywords{Evolution equation; nonlocal initial condition;
strong solution; \hfill\break\indent
analytic semigroups; existence and uniqueness}

\begin{abstract}
 The aim of this article is to study the existence and
 uniqueness of strong solutions for a class of semilinear evolution
 equations with nonlocal initial conditions. The discussions are based
 on analytic semigroup theory and fixed point theorems.
 An example illustrates the main results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

The nonlocal Cauchy problem for abstract evolution equation was
first investigated by Byszewski and Lakshmikantham \cite{b4}, where, by
using the Banach fixed point theorem, the authors obtained the
existence and uniqueness of mild solutions for nonlocal differential
equations. The nonlocal problem was motivated by physical problems.
Indeed, it is demonstrated that the nonlocal problems have better
effects in applications than the classical Cauchy problems. For
example, it is used to represent mathematical models for evolution
of various phenomena, such as nonlocal neural networks, nonlocal
pharmacokinetics, nonlocal pollution and nonlocal combustion (see
McKibben \cite{m1}). For this reason, differential or integro-differential
equations with nonlocal initial conditions were studied by many
authors and some basic results on nonlocal problems have been
obtained, see the references in this article and their references.
Particularly, in 1999, Byszewski \cite{b7} obtained the existence 
and uniqueness of
classical solution to a class of abstract functional differential
equations with nonlocal conditions of the form
\begin{gather}
u'(t)= f(t,u(t),u(a(t))),\quad t\in I,\label{e1.1}\\
u(t_0)+\sum_{k=1}^{p}c_ku(t_k)=x_0, \label{e1.2}
\end{gather}
where $I:=[t_0,t_0+T]$, $t_0<t_1<\dots<t_p\leq t_0+T$, $T>0$;
$f:I\times E^2\to E$ and $a:I\to I$ are given
functions satisfying some assumptions; $E$ is a Banach space,
$x_0\in E$, $c_k\neq 0$ $(k=1,2,\dots,p)$ and $p\in \mathbb{N}$.
The author pointed out that if $c_k\neq 0$, $k=1,2,\dots,p$, then
the results of the paper can be applied to kinematics to determine
the location evolution $t\to u(t)$ of a physical object for
which we do not know the positions $u(0),u(t_1),\dots,u(t_p)$, but
we know that the nonlocal condition \eqref{e1.2} holds. The nonlocal
condition of type \eqref{e1.2} has also been used by
Deng \cite{d2} to describe
the diffusion phenomenon of a small amount of gas in a transparent
tube. In this case, condition \eqref{e1.2} allows the additional
measurements at $t_k$, $k=1,2,\dots,p$, which is more precise than
the measurement just at $t=t_0$. Consequently, to describe some
physical phenomena, the nonlocal condition can be more useful than
the standard initial condition.

Recently, Vrabie \cite{v1} studied the  existence of global
$C^0$-solutions for a class of nonlinear functional differential
evolution inclusions of the form
\begin{equation}
\begin{gathered}
   u'(t)\in Au(t)+f(t),\quad t\geq 0, \\
f(t)\in F(t,u(t),u_t),\quad t\geq 0,\\
u(t)=g(u)(t),\quad t\in[-\tau,0],
 \end{gathered}  \label{e1.3}
\end{equation}
where $X$ is a real Banach space, $A$ is the infinitesimal generator
of a nonlinear compact semigroup,
$\tau\geq 0$, $F:[0,+\infty)\times X\times C([-\tau,+\infty);\overline{D(A)})\to X$ is a
nonempty convex and weakly compact value multi-function and
$g:C_b([-\tau,+\infty);\overline{D(A)})\to C([-\tau,0);\overline{D(A)})$.

In \cite{z1}, by using the approach of geometry of Banach space,
Hausdroff metric, the measure of noncompactness and fixed point
theorem, Zhu, Huang and Li studied the existence of integral
solutions for the following nonlinear set-valued differential
inclusion with nonlocal initial conditions
\begin{equation}
\begin{gathered}
   u'(t)\in Au(t)+F(t,u(t)),\quad 0<t\leq T, \\
u(0)=g(u),
 \end{gathered}\label{e1.4}
\end{equation}
where $A:D(A)\subseteq X\to X$ is a nonlinear m-dissipative
operator which generates a contraction semigroup $T(t)$ and $F$ is
weakly upper semi-continuous multifunction with respect to its
second variable in a real Banach space $X$.

In most of the existing articles, such as 
\cite{b5,b1,b2,b3,b6,e1,f1,f2,l1,l2,x2,x3,x4}, the existence
of mild solutions for nonlocal evolution equations have been studied
extensively, but there are very few paper studied the regularity of
nonlocal evolution equations. Motivated by the above-mentioned
aspects, in this work we discuss the existence and uniqueness of
strong solutions for a class of semilinear evolution equations with
nonlocal initial conditions
\begin{gather}
u'(t)+Au(t)= f(t,u(t)),\quad t\geq 0,\label{e1.5}\\
u(0)=\sum_{k=1}^{p}c_ku(t_k),\label{e1.6}
\end{gather}
where $H$ is a Hilbert space, $A:D(A)\subset H\to H$ is a positive
definite self-adjoint operator, $J=[0,K]$,  $K>0$ is a constant,
$f: J\times H\to H$ is a given function satisfying some
assumptions, $0<t_1<t_2<\dots<t_p\leq K$, $p\in\mathbb{N}$, $c_k$
are real numbers, $c_k\neq 0$, $k=1,2,\dots,p$.

In the following section we first introduce some notation and
preliminaries which are used throughout this paper, at the same time
the existence of strong solution for linear evolution equation
nonlocal problem has been obtained. In section 3 we state and prove
the existence and uniqueness of strong solutions for nonlinear
evolution equation nonlocal problem. In the last paragraph we give
an example to illustrate our main results.

\section{Preliminaries}

Let $H$ be a Hilbert space with inner product $(\cdot,\cdot)$, then
$\|\cdot\|=\sqrt{(\cdot,\cdot)}$ is the norm on $H$ induced by inner
product. We denote by $C(J,H)$ the Banach space of
all continuous functions from $J$ to $H$ endowed with the maximum
norm $\| u\|_{C}=\max_{t\in J}\| u(t)\|$
and by $\mathcal {L}(H)$ the Banach space of all linear and bounded
operators on $H$.

Let $A:D(A)\subset H\to H$ be a positive definite self-adjoint
operator in Hilbert space $H$ and it have compact resolvent. By the
spectral resolution theorem of self-adjoint operator, the spectrum
$\sigma(A)$ only consists of real eigenvalues and it can be arrayed
in sequences as
\begin{equation}
\lambda_1<\lambda_2<\dots<\lambda_n<\dots,\quad
\lambda_n\to\infty\text{ as } n\to\infty.\label{e2.1}
\end{equation}
By the positive definite property of $A$, the first eigenvalue
$\lambda_1>0$. From \cite{d1,h1,p1}, we know that $-A$ generates an analytic
operator semigroup $T(t)(t\geq0)$ on $H$, which is exponentially
stable and satisfies
\begin{equation}
\| T(t)\|\leq e^{-\lambda_1t},\quad \forall t\geq0.\label{e2.2}
\end{equation}
Since the positive definite self-adjoint operator $A$ has compact
resolvent, the embedding $D(A)\hookrightarrow H$ is compact, and
therefore $T(t)(t\geq0)$ is also a compact semigroup.

We recall some concepts and conclusions on the fractional powers of
$A$. For $\alpha>0$, $A^{-\alpha}$ is defined by
\begin{equation}
A^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int_0^{\infty}
s^{\alpha-1}T(s)ds,\label{e2.3}
\end{equation}
where $\Gamma(\cdot)$ is the Euler gamma function.
$A^{-\alpha}\in \mathcal {L}(H)$ is injective, and $A^\alpha$
 can be defined by
$A^\alpha=(A^{-\alpha})^{-1}$ with the domain
$D(A^\alpha)=A^{-\alpha}(H)$. For $\alpha=0$, let $A^\alpha=I$. We
endow an inner product
$(\cdot,\cdot)_\alpha=(A^\alpha\cdot,A^\alpha\cdot)$ to
$D(A^\alpha)$. Since $A^\alpha$ is a closed linear operator, it
follows that $(D(A^\alpha),(\cdot,\cdot)_\alpha)$ is a Hilbert
space. We denote by $H_\alpha$ the Hilbert space
$(D(A^\alpha),(\cdot,\cdot)_\alpha)$. Especially, $H_0=H$ and
$H_1=D(A)$. For $0\leq\alpha<\beta$, $H_\beta$ is densely embedded
into $H_\alpha$ and the embedding $H_\beta\hookrightarrow H_\alpha$
is compact. For the details of the properties of the fractional
powers, we refer to \cite{h1,x1}.

It is well known \cite[Chapter 4, Theorem 2.9]{p1} that for any
 $u_0\in D(A)$ and $h\in C^1(J,H)$, the initial value problem of linear
evolution equation (LIVP)
\begin{equation}
\begin{gathered}
  u'(t)+Au(t)= h(t),\quad  t\in J, \\
   u(0)=u_0,
 \end{gathered}  \label{e2.4}
\end{equation}
has a unique classical solution $u\in C^1(J,H)\cap C(J,D(A))$
expressed by
\begin{equation}
u(t)=T(t)u_0+\int_0^{t}T(t-s)h(s)ds.\label{e2.5}
\end{equation}
If $u_0\in H$ and $h\in L^1(J,H)$, the function $u$ given by \eqref{e2.5}
belongs to $C(J,H)$, which is known as a mild solution of
\eqref{e2.4}. If a mild solution $u$ of \eqref{e2.4} belongs to
$W^{1,1}(J,H)\cap L^1(J,D(A))$ and satisfies the equation for a.e.
$t\in J$, we call it a strong solution.


Throughout this paper, we assume that
\begin{itemize}
\item[(P0)] $\sum_{k=1}^{p}|c_k|<e^{\lambda_1t_1}$.
\end{itemize}
From this assumption,
$\|\sum _{k=1}^{p}c_kT(t_k)\|\leq \sum _{k=1}^{p}| c_k|
e^{-\lambda_1t_1}<1$. By operator spectrum theorem, we know that the
operator
\begin{equation}
B:=\Big(I-\sum _{k=1}^{p}c_kT(t_k)\Big)^{-1}\label{e2.6}
\end{equation}
exists and it is bounded.
Furthermore, by Neumann expression, $B$ can be written as
\begin{equation}
B=\sum_{n=0}^{\infty}\Big(\sum
_{k=1}^{p}c_kT(t_k)\Big)^n.\label{e2.7}
\end{equation}
Therefore,
\begin{equation}
\| B\|\leq \sum
_{n=0}^{\infty}\|\sum
_{k=1}^{p}c_kT(t_k)\|^n=\frac{1}{1-\|\sum
_{k=1}^{p}c_kT(t_k)\|}\leq\frac{1}{1-e^{-\lambda_1t_1}\sum
_{k=1}^{p}| c_k|}.\label{e2.8}
\end{equation}
To prove our main results, for any $h\in C(J,H)$, we consider the
linear evolution equation nonlocal problem (LNP)
\begin{gather}
u'(t)+Au(t)= h(t),\quad t\in J,\label{e2.9}\\
u(0)=\sum_{k=1}^{p}c_ku(t_k).  \label{e2.10}
\end{gather}

\begin{lemma} \label{lem2.1}
 If condition {\rm (P0)} holds, then
\eqref{e2.9}--\eqref{e2.10} has a unique mild solution $u\in C(J,H)$ given
by
\begin{equation}
u(t)=\sum _{k=1}^{p}c_kT(t)B\int_0^{t_k}
T(t_k-s)h(s)ds+\int_0^{t}T(t-s)h(s)ds,\quad t\in J. \label{e2.11}
\end{equation}
Moreover, $u\in W^{1,2}(J,H)\cap L^2(J,D(A))$ is a strong solution
of  \eqref{e2.9}--\eqref{e2.10}.
\end{lemma}

\begin{proof}
 By \eqref{e2.4} and \eqref{e2.5}, we know that  \eqref{e2.9} has
a unique mild solution $u\in C(J,H)$ which can be expressed as
\begin{equation}
u(t)=T(t)u(0)+\int_0^{t}T(t-s)h(s)ds.\label{e2.12}
\end{equation}
From \eqref{e2.12},
\begin{equation}
u(t_k)=T(t_k)u(0)+\int_0^{t_k}T(t_k-s)h(s)ds,\quad k=1,2,\dots,p.\label{e2.13}
\end{equation}
By \eqref{e2.10} and \eqref{e2.13},
\begin{equation}
u(0)=\sum_{k=1}^{p}c_kT(t_k)u(0)+\sum_{k=1}^{p}c_k
\int_0^{t_k}T(t_k-s)h(s)ds.\label{e2.14}
\end{equation}
Since
$I-\sum_{k=1}^{p}c_kT(t_k)$ has a bounded inverse operator $B$,
\begin{equation}
u(0)=\sum_{k=1}^{p}c_kB\int_0^{t_k}T(t_k-s)h(s)ds.\label{e2.15}
\end{equation}
From \eqref{e2.12} and \eqref{e2.15}, we know that $u$ satisfies \eqref{e2.11}.

Inversely, we can verify directly that the function $u\in C(J,H)$
given by \eqref{e2.11} is a mild solution of  \eqref{e2.9}--\eqref{e2.10}.

By the maximal regularity of linear evolution equations with
positive definite operator in Hilbert spaces
(see \cite[Chapter II, Theorem 3.3]{t1}),
when $u(0)=u_0\in H_{1/2}$, the mild solution of the
 \eqref{e2.4} has the regularity
\begin{equation}
u\in  W^{1,2}(J,H)\cap L^2(J,D(A))\cap C(J,H_{1/2})\label{e2.16}
\end{equation}
and it is a strong solution.

We note that $u(t)$ defined by \eqref{e2.11} is the mild solution of
\eqref{e2.4} for $u(0)=\sum_{k=1}^{p}c_kB\int_0^{t_k}T(t_k-s)h(s)ds$.
By the representation \eqref{e2.5} of mild solution,
$u(t)=T(t)u(0)+v(t)$,
where $v(t)=\int_0^{t}T(t-s)h(s)ds$. Since the function $v(t)$ is a
mild solution of \eqref{e2.4} with the null initial value
$u(0)=\theta$, $v$ has the regularity \eqref{e2.16}.
By the analytic property of the semigroup $T(t)$,
$T(t_k)u(0)\in D(A)\subset H_{1/2}$. Hence,
$u(0)=\sum_{k=1}^{p}c_kT(t_k)u(0)+\sum_{k=1}^{p}c_kv(t_k)\in H_{1/2}$.
 Using the regularity \eqref{e2.16} again, we obtain that
$u\in W^{1,2}(J,H)\cap L^2(J,D(A))$ and it is a strong solution of
\eqref{e2.9}--\eqref{e2.10}. This completes the proof.
\end{proof}

For any $r>0$, let
$$
\Omega_r=\{u\in C(J,H): \| u\|_{C}\leq r\},
$$
then $\Omega_r$ is a closed ball in $C(J,H)$ with center $\theta$
and radius $r$.


\section{Main results}

\begin{theorem} \label{thm3.1}
 Let $A$ be a positive definite self-adjoint operator
 in Hilbert space $H$, and  having compact resolvent. Let
 $f:J\times H\to H$ be continuous.
If conditions {\rm (P0)} and
\begin{itemize}
\item[(P1)] There exist positive constants $\eta$ and $M$ with
$$
\eta<\frac{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)}{\sum_{k=1}^{p}| c_k|+1}
$$
such that
$$
\| f(t,u)\|\leq \eta \| u\|+M,\quad t\in J,\;u\in H\,,
$$
\end{itemize}
are satisfied
then  \eqref{e1.5}--\eqref{e1.6} has at least one strong
solution $u\in W^{1,2}(J,H)\cap L^2(J,D(A))$.
\end{theorem}

\begin{proof}
We consider the operator $\mathcal {F}$ on $C(J,H)$ defined by
\begin{equation}
\mathcal {F}u(t)=\sum _{k=1}^{p}c_kT(t)B\int_0^{t_k}
T(t_k-s)f(s,u(s))ds+\int_0^{t}T(t-s)f(s,u(s))ds,
\label{e3.1}
\end{equation}
$t\in J$. By condition (P0) and Lemma \ref{lem2.1}, it is easy to see that the mild
solution of problem \eqref{e1.5}-\eqref{e1.6} is equivalent to the
fixed point of the operator $\mathcal {F}$.
In the following, we will prove that
$\mathcal {F}$ has a fixed point by using the  Schauder fixed
point theorem. At first, we can prove that
$\mathcal {F}:C(J,H)\to C(J,H)$ is continuous by condition (P1) and
the usual techniques (see, e.g. \cite{f1,x3}).

Subsequently, we prove that $\mathcal {F}:C(J,H)\to C(J,H)$
is a compact operator. Let $0\leq\alpha<\frac{1}{2}$,
$0<\nu<\frac{1}{2}-\alpha$. By \cite{a1}, we can prove that the operator
$\mathcal {F}$ defined by \eqref{e3.1} maps $C(J,H)$ into
$C^\nu(J,H_\alpha)$. By Arzela-Ascoli's theorem, the embedding
$C^\nu(J,H_\alpha)\hookrightarrow C(J,H)$ is compact. This implies
that $\mathcal {F}: C(J,H)\to C(J,H)$ is a compact operator.
Combining this with the continuity of $\mathcal {F}$ on $C(J,H)$, we
know that $\mathcal {F}: C(J,H)\to C(J,H)$ is a completely
continuous operator.

Next, we prove that there exists a positive constant $R$ big enough,
such that $Q(\Omega_{R})\subset\Omega_{R}$. For any $u\in C(J,H)$,
by the condition (P1), we have
\begin{equation}
\| f(t,u(t))\|
\leq \eta\| u(t)\| +M\leq \eta\| u\|_{C} +M,\quad t\in J.\label{e3.2}
\end{equation}
Choose
\begin{equation}
R\geq\frac{M(1+\sum_{k=1}^{p}| c_k|)}{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)-\eta(1+\sum_{k=1}^{p}| c_k|)}.\label{e3.3}
\end{equation}
For any $u\in \Omega_{R}$ and $t\in J$, we have
\begin{align*}
\| \mathcal {F}u(t)\|
&\leq  \sum_{k=1}^{p}| c_k| e^{-\lambda_1t}\|B\|\int_0^{t_k}
e^{-\lambda_1(t_k-s)}\| f(s,u(s))\| ds\\
&\quad +\int_0^{t} e^{-\lambda_1(t-s)}\| f(s,u(s))\| ds\\
&\leq \frac{\sum_{k=1}^{p}| c_k|
e^{-\lambda_1t}}{1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|}\int_0^{t_k}e^{-\lambda_1(t_k-s)}
\big(\eta\| u\|_{C}+M\big)ds\\
&\quad +\int_0^{t}e^{-\lambda_1(t-s)} \big(\eta\|
u\|_{C}+M\big)ds\\
&\leq \frac{\sum_{k=1}^{p}|
c_k|+1}{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)}\big(\eta R+M\big)
\leq R.
\end{align*}
Thus, $\| \mathcal {F}u\|_{C}\leq R$. Therefore,
$\mathcal {F}(\Omega_{R})\subset\Omega_{R}$. By Schauder fixed point
theorem, we know that $\mathcal {F}$ has at least one fixed point
$u\in\Omega_{R}$. Since $u$ is mild solution of  \eqref{e2.9}--\eqref{e2.10}
for $h(\cdot)=f(\cdot,u(\cdot))$, by Lemma \ref{lem2.1},
 $u\in W^{1,2}(J,H)\cap L^2(J,D(A))$ is a strong solution of the problem
\eqref{e1.5}--\eqref{e1.6}. This completes the proof. 
\end{proof}


\begin{theorem} \label{thm3.2}
 Let $A$ be a positive definite
self-adjoint operator in Hilbert space $H$ and it have compact
resolvent, $f$: $J\times H\to H$
be continuous. If the condition (P0) and the  condition
\begin{itemize}
\item[(P2)]  There exists a positive constant
$$
\eta<\frac{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)}{\sum_{k=1}^{p}| c_k|+1}
$$
such that
$$
\| f(t,u)-f(t,v)\|\leq \eta \|u-v\|,\quad \forall u,v\in H,
$$
\end{itemize}
holds then  \eqref{e1.5}--\eqref{e1.6} has a unique strong
solution $\widehat{u}\in W^{1,2}(J,H)\cap L^2(J,D(A))$.
\end{theorem}

\begin{proof}
 By the proof of Theorem \ref{thm3.1}, we know that the
operator $\mathcal {F}:C(J,H)\to C(J,H)$ is completely
continuous and the mild solution of problem \eqref{e1.5}--\eqref{e1.6} is
equivalent to the fixed point of $\mathcal {F}$.
For any $u,v\in C(J,H)$, from the assumption (P2) and \eqref{e3.1}, we have
\begin{equation}
\begin{aligned}
\| \mathcal {F}u(t)-\mathcal{F}v(t)\|
&\leq  \sum_{k=1}^{p}| c_k|
e^{-\lambda_1t}\| B\|\int_0^{t_k}
e^{-\lambda_1(t_k-s)}\| f(s,u(s))-f(s,v(s))\| ds\\
&\quad +\int_0^{t}
e^{-\lambda_1(t-s)}\| f(s,u(s))-f(s,v(s))\| ds\\
&\leq \frac{\sum_{k=1}^{p}| c_k|
e^{-\lambda_1t}}{1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|}\int_0^{t_k}e^{-\lambda_1(t_k-s)}
\eta\| u-v\|_{C}ds\\
&\quad +\int_0^{t}e^{-\lambda_1(t-s)} \eta\| u-v\|_{C}ds\\
&\leq \frac{\eta(\sum_{k=1}^{p}|
c_k|+1)}{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)}\| u-v\|_{C}.
\end{aligned}\label{e3.5}
\end{equation}
Therefore, we have
\begin{equation}
\| \mathcal {F}u-\mathcal {F}v\|_{C}
\leq\frac{\eta(\sum_{k=1}^{p}| c_k|+1)}{\lambda_1(1-e^{-\lambda_1t_1}
 \sum_{k=1}^{p}| c_k|)}\| u-v\|_{C}.\label{e3.6}
\end{equation}
Thus, by the assumption (P2) and \eqref{e3.6}, we know that
$\mathcal{F}$ is a contraction operator on $C(J,H)$, and therefore
$\mathcal{F}$ has a unique fixed point $\widehat{u}$ on $C(J,H)$. Since
$\widehat{u}$ is mild solution of  \eqref{e2.9}--\eqref{e2.10} for
$h(\cdot)=f(\cdot,\widehat{u}(\cdot))$, by Lemma \ref{lem2.1},
$\widehat{u}\in W^{1,2}(J,H)\cap L^2(J,D(A))$ is a unique strong
solution of  \eqref{e1.5}--\eqref{e1.6}. This completes the proof of
Theorem \ref{thm3.2}.
\end{proof}

\section{Application}

To illustrate our  results, we consider the
following semilinear heat equation with nonlocal condition
\begin{equation}
\begin{gathered}
\frac{\partial}{\partial t}w(x,t)-\kappa\frac{\partial^2}{\partial x^2}w(x,t)
=g(x,t,w(x,t)),\quad (x,t)\in[a,b]\times J, \\
w(a,t)=w(b,t)=0,\quad t\in J,\\
w(x,0)=\sum_{k=1}^{p}\arctan\frac{1}{2k^2}w(x,k),\quad
x\in[a,b],
 \end{gathered} \label{e4.1}
\end{equation}
where $\kappa>0$ is the coefficient of heat conductivity, $J=[0,K]$,
$g:[a,b]\times J\times\mathbb{R}\to \mathbb{R}$ is
continuous.

Let $H=L^2(a,b)$ with the norm $\|\cdot\|_2$. Define
an operator $A$ in Hilbert space $H$ by
\begin{equation}
D(A)=H^2(a,b)\cap H_0^1(a,b),\quad
Au=-\kappa\frac{\partial^2}{\partial x^2}u,\label{e4.2}
\end{equation}
where
$H^2(a,b)=W^{2,2}(a,b)$, $H_0^1(a,b)=W_0^{1,2}(a,b)$.
From \cite{h1,p1},
we know that $A$ is a positive definite self-adjoint operator on $H$
and $-A$ is the infinitesimal generator of an analytic, compact
semigroup $T(t)(t\geq0)$. Moreover, $A$ has discrete spectrum with
eigenvalues $\lambda_n={\kappa n^2\pi^2}/{(b-a)^2}$, $n\in
\mathbb{N}$, associated normalized eigenvectors
$v_n(x)=\sqrt{2/z}\sin {n\pi x}/{(b-a)}$,
$z=\sqrt{b-a+(\sin2n\pi a-\sin2n\pi b)/(2n\pi)}$, the set
 $\{v_n: n\in \mathbb{N}\}$ is an
orthonormal basis of $H$ and
\begin{equation}
T(t)u=\sum_{n=1}^{\infty}e^{-\frac{\kappa n^2\pi^2t}{(b-a)^2}}(u,v_n)v_n,\quad
\| T(t)\|\leq e^{-\frac{\kappa\pi^2t}{(b-a)^2}}, \quad \forall
 t\geq0.\label{e4.3}
\end{equation}
Let $u(t)=w(\cdot,t)$, $f(t,u(t))=g(\cdot,t,w(\cdot,t))$,
$c_k=\arctan\frac{1}{2k^2}$, $t_k=k$, $k=1,2,\dots,p$, then
\eqref{e4.1} can be rewritten into the abstract form of problem
\eqref{e1.5}--\eqref{e1.6}.

\begin{theorem} \label{thm4.1}
 Assume that the nonlinear term $g$ satisfies the following conditions:
\begin{itemize}
\item[(G1)] there exist positive constants $\eta$ and $M$ with
$\eta<\frac{\kappa\pi^2}{(b-a)^2(\pi+4)}
\big(4-\pi e^{-\frac{\kappa\pi^2}{(b-a)^2}}\big)$ such that
$$
| g(x,t,w)|\leq \eta | w|+M,\quad x\in [a,b],\;t\in J,\;w\in \mathbb{R};
$$

\item[(G2)] there exists a function
$c:\mathbb{R}^+\to\mathbb{R}^+$ such that
$$
| g(x,t,\xi)-g(y,s,\eta)|\leq c(\rho)\big(| x-y|^\mu
 +|t-s|^{\mu/2}+| \xi-\eta|\big),
$$ 
for any $\rho>0$, $\mu\in(0,1)$ and $(x,t,\xi)$, $(y,s,\eta)\in[a,b]\times
J\times[-\rho,\rho]$.
\end{itemize}
Then  \eqref{e4.1} has at least one classical solution 
$u\in C^{2+\mu,1+\mu/2}([a,b]\times J)$.
\end{theorem}

\begin{proof}  Since 
$$
\sum_{k=1}^{p}|c_k|\leq\sum_{k=1}^{\infty}\arctan\frac{1}{2k^2}
=\pi/4<e^{\frac{\kappa\pi^2}{(b-a)^2}},
$$
condition (P0) holds. From  (G1), we 
see that the condition (P1) is satisfied. Hence by 
Theorem \ref{thm3.1},  problem \eqref{e4.1} has a strong solution 
$u\in C(J,H_0^1(a,b))\cap
L^2(J,H^2(a,b))\cap W^{1,2}(J,L^2(a,b))$ 
in the $L^2(a,b)$ sense.
Since the nonlinear term $g$ satisfies (G2), by
using a similar regularization method in \cite[Lemma 4.2]{a1},
 we can prove that $u\in C^{2+\mu,1+\mu/2}([a,b]\times J)$ is
a classical solution of  \eqref{e4.1}. 
\end{proof}

Similarly, from Theorem \ref{thm3.2} we obtain the following result.

\begin{theorem} \label{thm4.2}
 Assume that the nonlinear term $g$ satisfies  {\rm (G2)} and 
\begin{itemize}
\item[(G3)] there exists a positive constant
$$
\eta<\frac{\kappa\pi^2}{(b-a)^2(\pi+4)}
\big(4-\pi e^{-\frac{\kappa\pi^2}{(b-a)^2}}\big)
$$
 such that
$$
| g(x,t,w)-g(x,t,v)|\leq \eta| w-v|,\quad x\in [a,b],\;t\in J,\;w,v\in \mathbb{R}.
$$ 
\end{itemize}
Then \eqref{e4.1} has a unique classical solution $\widehat{u}\in
C^{2+\mu,1+\mu/2}([a,b]\times J)$.
\end{theorem}

\subsection*{Acknowledgments}
This research supported by grants 11261053 from the  NNSF of China,
and  1208RJZA129 from the NSF of Gansu Province.

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\end{document}