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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 36, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/36\hfil Neutral functional differential equation]
{Neutral functional differential equations of second-order
with infinite delays}

\author[R. Ye, G. Zhang\hfil EJDE-2010/36\hfilneg]
{Runping Ye, Guowei Zhang}  % in alphabetical order

\address{Runping Ye \newline
Education Department of Suqian College,
Jiangsu,  223800, China}
\email{yeziping168@sina.com}

\address{Guowei Zhang \newline
Department of Mathematics, Northeastern University, Shenyang,
110004, China} 
\email{gwzhangneum@sina.com}

\thanks{Submitted November 24, 2009. Published March 9, 2010.}
\thanks{Supported by grant Z2009004 from the
Science and Technology Foundation of Suqian, China.}
\subjclass[2000]{34K30, 34K40, 47D09}
\keywords{Neutral functional differential equations;
mild  solution; \hfill\break\indent
Hausdorff measure of noncompactness; phase space}

\begin{abstract}
 This work shows the existence of mild solutions to
 neutral functional differential equations of second-order
 with infinite delay. The Hausdorff measure of noncompactness and
 fixed point theorem are used, without  assuming compactness
 on the associated family of operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

Differential equations with delays are often
more realistic to describe natural phenomena than those without
delays, and neutral differential equations arise in many areas of
applied mathematics. These two reasons may explain, why they have
received much attention in the previous decades.
Among the published works, we have \cite{SB,BDM, DFL,Her2,HH1,HH2,EHM,Y}
and references therein.
Existence and stability have been studied by Hale \cite{HJK',HJK},
 Travis and Webb \cite{TW1}, and Webb \cite{WE1}.
second-order differential equations and integrodifferential equations
in Banach spaces have been studied in
\cite{KDS,HC} and \cite{EM}, respectively.

In this article, we investigate the existence of mild solutions for
the neutral functional differential equation
\begin{gather}
\frac{d}{dt}(x'(t)+g(t,x_t))=Ax(t)+f(t,x_t), \quad t\in
J=[0,b], \label{e1.1} \\
 x_0=\varphi\in{\mathcal{B}},\quad
x'(0)=z\in X\,.\label{e1.2}
\end{gather}
We also consider the second order problem
\begin{gather}
\frac{d}{dt}(x'(t)+g(t,x_t,x'(t)))=Ax(t)+f(t,x_t,x'(t)),
\quad t\in J=[0,b], \label{e1.3} \\
x_0=\varphi\in\mathcal{B},\quad x'(0)=z\in X,\label{e1.4}
\end{gather}
  where $A$ is the infinitesimal generator of a strongly
continuous cosine family $\{C(t):t\in \mathbb{R}\}$ of bounded
linear operators on a Banach space X. In both cases, the history
$x_t : (-\infty, 0] \to X, x_t (\theta)= x(t+\theta)$,
belongs to some abstract phase space $\mathcal{B}$ defined
axiomatically; $g,f$ are appropriate functions.

In this paper, we prove the existence of mild solution of the initial
value problems \eqref{e1.1}-\eqref{e1.2} and \eqref{e1.3}-\eqref{e1.4}
under the conditions under assumptions on Hausdorff's measure
of noncompactness.

\section{Preliminaries}

  Now we introduce some definitions, notation and
preliminary facts which are used throughout this paper.

We say that a family $\{C(t):t\in \mathbb{R}\}$ of operators in
$B(X)$ is a strongly continuous cosine family if
\begin{itemize}
\item[(i)] $C(0)=I$ (I is the identity operator in $X$);

\item[(ii)] $C(t+s)+C(t-s)=2C(t)C(s)$ for all $s,t\in \mathbb{R}$;

\item[(iii)] The map $t\to C(t)x$ is strongly continuous for each
$x\in X$.
\end{itemize}

The strongly continuous sine family $\{S(t):t\in \mathbb{R}\}$,
associated to the given strongly continuous cosine family
$\{C(t):t\in \mathbb{R}\}$, is defined by
\[
 S(t)x = \int_0^t C(s)x ds, x \in X, t \in
\mathbb{R}.
\]
For more details on strongly continuous cosine and sine families, we
refer the reader to the books by Goldstein \cite{JAG} and
Fattorini \cite{HOF}.

The operator $A$ is the infinitesimal generator of a strongly
continuous cosine function of bounded linear operators,
$(C(t))_{t\in R}$, on $X$ and $S(t)$ is the sine function associated
with $(C(t))_{t\in R}$. We designate by $ N$, $ \widetilde{N}$
certain constants such that $\|C(t)\| \leq N$ and $\|S(t)\| \leq
\widetilde{N} $ for every $t \in J$. We refer the reader to
\cite{HOF} for the necessary concepts about cosine functions. Next
we only mention a few results and notations needed to establish our
results. As usual we denote by $D(A)$ the domain of $ A$ endowed
with the graph norm $\|x\|_A = \|x\| + \|Ax\|$, $x \in D(A)$.

In this work we  employ an axiomatic definition of the phase
space $\mathcal{B}$ which is similar to that introduced by Hale and
Kato \cite{HJK} and it is appropriate to treat retarded differential
equations with infinite delay.

\begin{definition}[\cite{HJK}]\label{dn1} \rm
 Let $\mathcal{B}$  be a linear
space of functions mapping  $(-\infty, 0]$ into $X $ endowed with a
seminorm $\|\cdot\|_{\mathcal{B}}$ and  that
 satisfies the following cinditions:
\begin{itemize}
\item[(A)]
 If $x :(-\infty,\sigma +b]\to X, b>0$,  such that
$x_\sigma \in \mathcal{B}$ and
$x|_{[\sigma,\sigma+b] }\in C([\sigma, \sigma +b] : X)$, then for
every $t \in [\sigma, \sigma +b)$ the following conditions hold:
\begin{itemize}
\item[(i)] $x_t $ is in $\mathcal{B}$,

\item[(ii)] $\|x(t)\|\le H\|x_t\|_{\mathcal{B}}$,

\item[(iii)] $\|x_t\|_{\mathcal{B}}\leq K(t -
\sigma)\sup\{\|x(s)\|: \sigma \leq s\leq t\} +
M(t+\sigma)\|x_{\sigma} \|_{\mathcal{B}}$,
\end{itemize}
 where $H>0$ is a constant; $K,M:[0,\infty)\to[1,\infty)$, $K$
is continuous, $M$ is locally
bounded and $H,K,M $ are independent of $x(\cdot)$.

\item[(A1)] For the function $x(\cdot)$ in (A), $x_t$ is a
 $\mathcal{B}$-valued continuous function on $[\sigma,\sigma+b)$.

\item[(B)] The space  $\mathcal{B}$ is complete.
\end{itemize}
\end{definition}

\begin{definition}[\cite{BJ}] \label{dn2} \rm
The Hausdorff's measure of
noncompactness is defined as
$\chi_{Y} (B)=\inf \{r>0,B$ can be covered by  finite number of
balls with radius $r\}$.
for bounded set $B$ in any Banach space $Y$.
\end{definition}

\begin{lemma}[\cite{BJ}]\label{lm1}
Let $Y$ be a real Banach space and
$B,C\subseteq Y$ be bounded, the following properties are satisfied:
\begin{itemize}
\item[(1)] {B} is pre-compact if and only if $\chi_{Y}(B)=0$;

\item[(2)] $\chi_{Y}(B)=\chi_{Y}(\overline{B})=\chi_{Y}(conv B)$, where
$\overline{B}$ and  $conv B$ are the closure and the convex hull of
$B$ respectively;

\item[(3)] $\chi_{Y}(B)\leq \chi_{Y}(C)$ when $B\subseteq C$;

\item[(4)] $\chi_{Y}(B+C)\leq \chi_{Y}(B)+\chi_{Y}(C)$ where
$B+C={\{x+y: x\in B,y\in C}\}$;

\item[(5)] $\chi_{Y}(B\cup C)\leq \max{\{\chi_{Y}(B),\chi_{Y}(C)}\}$;

\item[(6)] $\chi_{Y}(\lambda B)=|\lambda|\chi_{Y}(B)$ for any $\lambda\in
R$;

\item[(7)] If the map $Q:D(Q)\subseteq Y\to Z$ is Lipschitz
continuous with constant $k$, then $\chi_{Z}(QB)\leq k\chi_{Y}(B)$
for  any bounded subset $B\subseteq D(Q)$, where $Z$ is a Banach
space;

\item[(8)] If ${\{W_n}\}_{n=1}^{+\infty}$ is a decreasing sequence of
bounded closed nonempty subsets of $Y$ and $\lim_{n\to
\infty}\chi_{Y}(W_n)=0$, then $\cap_{n=1}^{+\infty}W_n$ is nonempty
and compact in $Y$.
\end{itemize}
\end{lemma}

\begin{definition}[\cite{BJ}]\label{dn3} \rm
 The map $Q:W\subseteq Y\to Y$ is said to be a $\chi_{Y}-contraction$
if $Q$ is bounded
continuous and there exists a positive constant $k<1$ such that
$\chi_{Y}(Q(C))\leq k\chi_{Y}(C))$  for any bounded closed subset
$C\subseteq W$, where $Y$ is a Banach space.
\end{definition}

\begin{lemma}[Darbo-Sadovskii \cite{BJ}]\label{lm3}
If $W\subseteq Y$ is bounded  closed and convex, the  map $Q:W\to W$ is a
$\chi_{Y}-contraction$, then the map $Q$ has at least one fixed
point in $W$.
\end{lemma}

In this paper we denote $\chi$ the Hausdorff's measure of
noncompactness of $X$, $\chi_{C}$ the Hausdorff's measure of
noncompactness of $C([0,b];X)$ and $\chi_{{C^{1}}}$ the Hausdorff's
measure of noncompactness of $C^{1}([0,b];X)$. To discuss the
existence results we need the following auxiliary results.

\begin{lemma}[\cite{BJ}]\label{lm4} \quad
\begin{itemize}
\item[(1)] If $W\subset C([a,b];X)$ is
 bounded, then
$\chi(W(t))\leq\chi_C(W)$,
 for  $t\in[a,b]$,  where
 $W(t)={\{u(t):u\in W}\}\subseteq X$;

\item[(2)] If $W$ is   equicontinuous on $[a,b]$, then
$\chi(W(t))$ is continuous for
 $t\in[a,b]$, and
\[
\chi_{C}(W)=\sup{\{\chi(W(t)),t\in[a,b]}\};
\]

\item[(3)] If $W\subset C([a,b];X)$ is
 bounded and   equicontinuous, then $\chi(W(t))$ is continuous for
 $t\in[a,b]$, and
\[
 \chi(\int_a^t W(s)ds )\leq \int_a^t\chi W(s)ds
\]
for all $t\in [a,b]$,  where $ \int_a^t W(s)ds={\{\int_a^t
x(s)ds:x\in W}\}$.
\end{itemize}
\end{lemma}

The following lemmas are easy to prove.

\begin{lemma}\label{lm5}
If the semigroup $S(t)$ is equicontinuous and
$\eta\in L([0,b];\mathbb{R}^+)$, then the set
${\{\int_0^t S(t-s)u(s)ds, \|u(s)\|\leq \eta(s)
\text{ for a.e. } s\in[0,b]}\}$ is equicontinuous for
$t\in[0,b]$.
\end{lemma}

\begin{lemma}[\cite{GD}]\label{lm6}
Let $W\subset C^{1}(J;X)$ be bounded and $W'$ be equicontinuous, then
\[
\chi_{{C^{1}}}(W)=\max{\{\chi_{{C}}(W),\chi_{{C}}(W')}\}
=\max{\{\max_{t\in J}\chi_{{C}}(W(t)),\max_{t\in J}\chi_{{C}}(W'(t))}\},
\]
where $W'={\{u':u\in W}\}$, $J=[a,b]$.
\end{lemma}

\section{Main results}

 Now we define the mild solution for the initial value
problem\eqref{e1.1}-\eqref{e1.2}.

\begin{definition}\label{dn3.1} \rm
A function $x:(-\infty ,b]\to X$ is a mild solution of the initial
value problem \eqref{e1.1}-\eqref{e1.2}, if
$x_0=\varphi$, $x(\cdot)|_J \in C(J;X)$ and for $t\in J$,
\[
x(t)=C(t)\varphi(0)+S(t)(z+g(0,\varphi))-\int_0^t
C(t-s)g(s,x_s)ds+\int_0^tS(t-s)f(s,x_s)ds.
\]
\end{definition}

 For  \eqref{e1.1}-\eqref{e1.2}, we assume the following
hypotheses:
\begin{itemize}
 \item[(H1f)] $f:J\times{\mathcal{B}}\to X$
satisfies the following two conditions:
\begin{itemize}
\item[(1)] For each $x:(-\infty,b]\to X$, $x_0\in{\mathcal{B}}$
and $x|_J \in C(J;X)$, the function $t\to {f(t,x_t)}$ is
strongly measurable  and $f(t,\cdot)$ is continuous for a.e.
$t\in J$;

\item[(2)] There exist an integrable  function $\alpha:J\to
[0,+\infty)$ and a monotone continuous nondecreasing function
$\Omega:[0,+\infty)\to(0,+\infty)$, such that
$\|f(t,v)\|\leq\alpha(t)\Omega(\|v\|_{\mathcal{B}})$,
for all $t\in J, v\in \mathcal{B}$;

\item[(3)] There exists an integrable  function
$\eta:J\to [0,+\infty)$, such that
\[
\chi(S(s)f(t,D))\leq \eta(t)\sup_{-\infty \leq \theta \leq
0}\chi(D(\theta))\quad\text{for a.e. }s,t\in J,
\]
 where $D(\theta)={\{v(\theta):v\in D}\}$.
\end{itemize}

\item[(H1g)] The function $g(\cdot)$ is continuous and
$g(t,\cdot)$ satisfies a Lipschitz condition; that is, there exists
a positive constant $L_g$, such that
\[
\|g(t,v_1)-g(t,v_2)\|\leq L_g\|v_1 - v_2\|_{\mathcal{B}},
\quad (t,v_i)\in J \times {\mathcal{B}}, \; i=1,2.
\]

\item[(H1)] \begin{itemize}
\item[(1)] $K_b(NbL_g+\widetilde{N}\int_0^b\alpha(s)ds\limsup
_{\tau\to\infty}\frac{\Omega(\tau)}{\tau})<1$

\item[(2)] $ K_bNL_gb +\int_{0}^{b}\eta(s)ds<1$.

\end{itemize}
\end{itemize}

In this section, $y : (-\infty, b]\to X $ is the function
defined by $y_0 =\varphi$ and $y(t) =
C(t)\varphi(0)+S(t)(z+g(0,\varphi))$ on $J$. Clearly,
$\|y_t\|_{\mathcal{B}}\leq K_b\|y\|_b
+M_b\|\varphi\|_{\mathcal{B}}$, where
\[
K_b=\sup_{0\leq t\leq b}K(t),\quad
M_b=\sup_{0\leq t\leq b}M(t),\|y\|_b=\sup_{0\leq
t\leq b}\|y(t)\|.
\]

 Now we are in position to estate our main results.

\begin{theorem}\label{tm1}
If the hypotheses {\rm (H1f), (H1g), (H1)} are satisfied,
then the initial value problem \eqref{e1.1}-\eqref{e1.2} has
at least one mild solution.
\end{theorem}

\begin{proof}
 Let $S(b)$ be the space
$S(b)={\{x:(-\infty,b]}\to X \mid   x_0=0,\;x|_J\in C(J;X)\}$
endowed with supremum norm $\|\cdot\|_b$ . Let $\Gamma :
S(b)\to S(b)$ be the map defined by
\begin{equation}
(\Gamma x)(t)= \begin{cases}
0, & t\in (-\infty,0],\\
-\int_0^t C(t-s)g(s,x_s+y_s)ds\\
+\int_0^tS(t-s)f(s,x_s+y_s)ds,
& t\in J.
\end{cases}
 \label{e3.1}
\end{equation}

 It is easy to see that $\|x_t+y_t\|_{\mathcal{B}}\leq
K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}+K_b\|x\|_t$,
 where
$\|x\|_t=\sup_{0\leq s\leq t}\|x(s)\|$. Thus, $\Gamma$ is
well defined and with values in $S(b)$. In addition, from the axioms
of phase space, the Lebesgue dominated convergence theorem and the
conditions (H1f)  (H1g),  we can show that $\Gamma$ is
continuous.

\noindent {\em Step 1.} There exists $k> 0$ such that
$\Gamma(B_k)\subset B_k$, where $B_k={\{x\in S(b):\|x\|_b\leq
k}\}$. In fact, if we assume that the assertion is false, then for
$k > 0$ there exist $x_k\in B_k$ and $t_k\in I$ such that $k
<\|\Gamma x_k(t_k)\|$. This yields
\begin{align*}
 k &<\|\Gamma x_k(t_k)\|\\
&\leq N\int_0^{t_k}(L_g\|x_{ks}+y_s\|_{\mathcal{B}}+\|g(s,0)\|)ds
+\widetilde{N}
\int_0^{t_k}\alpha(s)\Omega(\|x_{ks}+y_s\|_{\mathcal{B}})ds \\
&\leq N\int_0^bL_g(K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}
 +K_bk+\|g(s,0)\|)ds\\
&\quad
+\widetilde{N}\int_0^b\alpha(s)ds\Omega(K_b\|y\|_b
+M_b\|\varphi\|_{\mathcal{B}}+K_bk)
\end{align*}
 which implies
\begin{align*}
1&<K_bNbL_g+\widetilde{N}\int_0^b\alpha(s)ds\limsup_{k\to\infty}
\frac{\Omega(K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}+K_bk)}{k}\\
& \leq
K_b(NbL_g+\widetilde{N}\int_0^b\alpha(s)ds\limsup_{\tau\to\infty}
\frac{\Omega(\tau)}{\tau})<1,
\end{align*}
which is  a contradiction.

\noindent {\em Step 2.} Next, we show that $\Gamma$ is
$\chi-contraction$.
To clarify this, we decompose $\Gamma$ in the form
$\Gamma={\Gamma}_1+{\Gamma}_2$, for $t\geq 0$, where
\begin{gather*}
 {\Gamma}_1x(t)=-\int_0^t C(t-s)g(s,x_s+y_s)ds , \\
 {\Gamma}_2x(t)=\int_0^t S(t-s)f(s,x_{s}+y_{s})ds.
\end{gather*}
 First, we show the ${\Gamma}_1$ is Lipschitz continuous.
For  arbitrary $ x_1,x_2\in B_k$, from Definition \ref{dn1} and
hypotheses, we obtain
\begin{align*}
\|{\Gamma}_1x_1(t)-{\Gamma}_1x_2(t)\|
& \leq \|\int_0^tC(t-s)(g(s,x_{1s}+y_s)-g(s,x_{2s}+y_s))ds\|\\
& \leq NL_gb\|x_{1t}-x_{2t}\|_{\mathcal{B}}
 \leq K_bNL_gb\|x_{1t}-x_{2t}\|_{b};
\end{align*}
that is,
$\|{\Gamma}_1x_1(t)-{\Gamma}_1x_2(t)\|_b \leq
K_bNL_gb\|x_{1t}-x_{2t}\|_{b}$;
hence, ${\Gamma}_1$ is Lipschitz continuous, with
Lipschitz constant $L'=K_bNL_gb $.

Next, taking $W\subset \Gamma( B_k)$. Obviously,  $S(t)$ is
equicontinuous. From Lemma \ref{lm5}, $W$ is equicontinuous. As
$\chi_{C}(W)=\sup{\{\chi(W(t)),t\in J}\}$, we have
\begin{align*}
\chi({\Gamma}_2W(t))
&=\chi(\int_0^t S(t-s)f(s,W_{s}+y_{s})ds)\\
& \leq \int_0^t \eta
(s)\sup_{-\infty<\theta\leq 0} \chi(W(s+\theta)+y(s+\eta))ds\\
& \leq \int_0^t \eta (s)\sup_{0\leq\tau\leq s} \chi W(\tau)ds\\
&\leq \chi_{C}(W)\int_0^t \eta (s)ds,
\end{align*}
 for each bounded set $W\in C(J;X)$. Since
\begin{align*}
\chi_{C}(\Gamma W)& =\chi_{C}(\Gamma_1 W +\Gamma_2 W)\\
&\leq \chi_{C}(\Gamma _1 W)+\chi_{C}(\Gamma_2 W) \\
& \leq (L'+\int_0^t \eta
(s)ds))\chi_{C}(W) \leq \chi_{C}(W),
\end{align*}
the function  $\Gamma$ is $\chi$-contraction. In view of Lemma
\ref{lm3},  Darbo-Sadovskii fixed point theorem, we conclude
that $\Gamma$ has at least one fixed point in $W$. Let $x$ be a
fixed of $\Gamma$ on $S(b)$, then $z=x+y$ is a mild solution of
\eqref{e1.1}-\eqref{e1.2}. So we deduce the existence of a
mild solution of \eqref{e1.1}-\eqref{e1.2}.
\end{proof}

For  \eqref{e1.3}-\eqref{e1.4}, it is possible to establish similar
results as those given in the first part of this section.
Furthermore, we denote by ${C}^{1}$ the space of  smooth functions
in the sense above described endowed with the norm $\|u\|_1
=\|u\|+\|u'\|$.

Now we define the mild solution for the initial value
problem \eqref{e1.3}-\eqref{e1.4}.

\begin{definition}\label{dn3.2} \rm
A function $x:(-\infty ,b]\to X$
is a mild solution of the initial value problem \eqref{e1.3}-\eqref{e1.4}
if $x_0=\varphi$, $x(\cdot)|_J \in C^{1}(J;X)$ and for $t\in J$,
\begin{align*}
x(t)&=C(t)\varphi(0)+S(t)(z+g(0,\varphi,z))-\int_0^t
C(t-s)g(s,x_s,x'(t))ds\\
&\quad +\int_0^tS(t-s)f(s,x_s,x'(t))ds.
\end{align*}
\end{definition}

For  \eqref{e1.3}-\eqref{e1.4}, we assume  the following hypotheses:
\begin{itemize}

\item[(H2f)] $f:J\times{\mathcal{B}}\times X\to X$
satisfies the following conditions:
\begin{itemize}
\item[(1)] For each $x:(-\infty,b]\to X$, $x_0=\varphi\in{\mathcal{B}}$
and $x|_J \in C^{1}$, the function $t\to {f(t,x_t,x'(t))}$ is
strongly measurable  and $f(t,\cdot,\cdot)$
is continuous for a.e.  $t\in J$;

\item[(2)] There exist an integrable  function $\alpha:J\to
[0,+\infty)$ and a monotone continuous nondecreasing function
$\Omega:[0,+\infty)\to(0,+\infty)$, such that
\[
 \|f(t,v,w)\|\leq\alpha(t)\Omega(\|v\|_{\mathcal{B}}+\|w\|),\quad
  t\in J,\;  (v,w)\in{\mathcal{B}}\times X;
\]
\item[(3)] There exist  integrable  functions $\eta_i:J\to
[0,+\infty),\ i=1,2$, such that
\begin{gather*}
 \chi(S(s)f(t,D_1,D_2))\leq \eta_1(t)\sup_{-\infty \leq \theta \leq
0}\chi(D_1(\theta)), \\
 \chi(C(s)f(t,D_1,D_2))\leq \eta_2(t)\sup_{-\infty \leq \theta \leq
0}\chi(D_2(\theta))\quad\text{for a.e.  }s,t\in J,
\end{gather*}
 where $D_i(\theta)={\{D_i(\theta):v\in D}\}$, $i=1,2$.
\end{itemize}

\item[(H2g)] There exists a positive constant $L_g$ such that
\[
\|g(t,v_1,w_1)-g(t,v_2,w_2)\|\leq L_g(\|v_1 -
v_2\|_{\mathcal{B}}+\|w_1-w_2\|),\]
$(t,v_i,w_i)\in J \times {\mathcal{B}}\times X$, $i=1,2$.

\item[(H2)]
\begin{itemize}
\item[(1)]
$(K_b+1)(L_g(Nb+1+\|A\|\widetilde{N}b)+(N+\widetilde{N})
\int_0^b\alpha(s)ds\limsup_{\tau\to\infty}\frac{\Omega(\tau)}{\tau})<1;$

 \item[(2)] $L_g(Nb+1+\|A\|\widetilde{N}b)(K_b+1)+\max{\{\int_0^b
\eta_1 (s)ds,\int_0^b \eta_2 (s)ds}\}<1$.
\end{itemize}
\end{itemize}

In this section, $y : (-\infty, b]\to X $ is the function
defined by $y_0 =\varphi$ and
$y(t) = C(t)\varphi(0)+S(t)(z+g(0,\varphi,z))$ on $J$. Clearly,
$\|y_t\|_{\mathcal{B}}\leq K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}$,
where $K_b=\sup_{0\leq t\leq b}K(t) $,
$M_b=\sup_{0\leq t\leq b}M(t),\|y\|_b=\sup_{0\leq t\leq b}\|y(t)\|$.

\begin{theorem}\label{tm3}
If the hypotheses {\rm (H2f) (H2g), (H2)} are satisfied,
then the initial value problem \eqref{e1.3}-\eqref{e1.4} has
at least one mild solution.
\end{theorem}

\begin{proof}
Let $S^{1}(b)$ be the space
\[
S^{1}(b)={\{x:(-\infty,b]}\to X:  x_0=0,\; x|_J\in
C^{1}(J;X),\; x'(0)=-g(0,\varphi,z)\}
\]
endowed with supremum norm
 $\|\cdot\|_{1b}$. Let $\Gamma : S^{1}(b)\to S^{1}(b)$ be
the map defined by
\begin{equation}
(\Gamma x)(t)= \begin{cases}
0, &t\in (-\infty,0],\\
-\int_0^t C(t-s)g(s,x_s+y_s,x'(s)+y'(s))ds\\
+\int_0^tS(t-s)f(s,x_s+y_s,x'(s)+y'(s))ds,
&t\in J,
\end{cases}
 \label{e3.4}
\end{equation}
where $y_0 =\varphi$ and $y(t) = C(t)\varphi(0)+S(t)(z+g(0,\varphi,z))$
on $J$. It is easy to see that
\[
\|x_t+y_t\|_{\mathcal{B}}\leq K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}
+K_b\|x\|_t,
\]
where
$\|x\|_t=\sup_{0\leq s\leq t}\|x(s)\|$.
  Thus, $\Gamma$ is well defined and with values in $S^{1}(b)$, and
\begin{align*}
 (\Gamma x)'(t)&=-g(t,x_t+y_t,x'(t)+y'(t))-\int_0^t
AS(t-s)g(s,x_s+y_s,x'(s)+y'(s))ds\\
&\quad +\int_0^tC(t-s)f(s,x_s+y_s,x'(s)+y'(s))ds,\quad
t\in J.
\end{align*}
 In addition,
from the axioms of phase space, the Lebesgue dominated convergence
theorem and the conditions (H2f), (H2g),  we can show that
$\Gamma$ and $\Gamma'$ are continuous.

\noindent {\em Step 1.} There exists $k> 0$ such that
$\Gamma(B_k)\subset B_k:={\{x\in S^{1}(b):\|x\|_{1b}\leq k}\}$. In
fact, if we assume that the assertion are false, then for $k> 0$
there exist $x_k\in B_k$ and $t_k\in J$ such that $k <\|\Gamma
x_k(t_k)\|_1$. This yields
\begin{align*}
k &<\|\Gamma x_k(t_k)\|_1\\
&=\|\Gamma x_k(t_k)\|+\|(\Gamma x_k)'(t_k)\| \\
&\leq N\int_0^{t_k}(L_g(\|x_{ks}+y_s\|_{\mathcal{B}}
+\|x_k'(s)+y'(t)\|)+\|g(s,0,0)\|)ds\\
&\quad+\int_0^{t_k}\widetilde{N}\alpha(s)\Omega(\|x_{ks}
 +y_s\|_{\mathcal{B}}+\|x_k'(s)+y'(t)\|)ds\\
&\quad +L_g(\|x_{kt_k}+y_{t_k}\|_{\mathcal{B}}+\|x_k'(t_k)+y'(t_k)\|)
+\|g(t_k,0,0)\|\\
&\quad +\int_0^{t_k}\|A\|\widetilde{N}(L_g(\|x_{ks}
+y_s\|_{\mathcal{B}}+\|x_k'(s)+y'(s)\|)+\|g(s,0,0)\|)ds\\
&\quad +\int_0^{t_k}N\alpha(s)\Omega(\|x_{ks}+y_s\|_{\mathcal{B}}
+\|x_k'(s)+y'(s)\|)ds \\
&\leq bNL_g(K_bk+K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}+k
+\|y'\|_b)+N\int_0^b\|g(s,0,0)\|ds\\
&\quad +\widetilde{N}\int_0^b\alpha(s)ds
\Omega(K_bk+K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}+k+\|y'\|_b)\\
&\quad +L_g(K_bk+K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}
 +k+\|y'\|_b)+\|g(t_k,0,0)\|\\
&\quad +b\|A\|\widetilde{N}L_g(K_bk+K_b\|y\|_b
 +M_b\|\varphi\|_{\mathcal{B}}+k+\|y'\|_b)
 +\|A\|\widetilde{N}\int_0^b\|g(s,0,0)\|ds\\
&\quad +N\int_0^b\alpha(s)ds\Omega(K_bk+K_b\|y\|_b
 +M_b\|\varphi\|_{\mathcal{B}}+k+\|y'\|_b),
\end{align*}
 which implies
\begin{align*}
1&<L_g(K_b+1)(Nb+1+\|A\|\widetilde{Nb})\\
&+(N +\widetilde{N})
\int_0^b\alpha(s)ds\limsup_{k\to\infty}
\frac{\Omega(K_bk+K_b\|y\|_b+M_b\|\varphi\|_{\mathcal{B}}
 +k+\|y'\|_b)}{k})\\
&\leq (K_b+1)(L_g(Nb+1+\|A\|\widetilde{N}b)+(N+\widetilde{N})
\int_0^b\alpha(s)ds\limsup_{\tau\to\infty}\frac{\Omega(\tau)}{\tau})<1,
\end{align*}
which is a contradiction.

\noindent {\em Step 2.} Next we show that $\Gamma$ is
$\chi-contraction$. To clarify this, we decompose $\Gamma$ in the
form $\Gamma={\Gamma}_1+{\Gamma}_2$, for $t\geq 0$, where
\begin{gather*}
 {\Gamma}_1x(t)=-\int_0^t
C(t-s)g(s,x_s+y_s,x'(s)+y'(s))ds ,\\
{\Gamma}_2x(t)=\int_0^t
S(t-s)f(s,x_{s}+y_{s},x'(s)+y'(s))ds.
\end{gather*}
 First, we show the ${\Gamma}_1$ is Lipschitz continuous.
 For arbitrary  $x_1,x_2\in B_k$, from Definition \ref{dn1} and
hypotheses conditions, we obtain
\begin{align*}
&\|{\Gamma}_1x_1(t)-{\Gamma}_1x_2(t)\|_1\\
&\leq \|{\Gamma}_1x_1(t)-{\Gamma}_1x_2(t)\|+\|({\Gamma}_1x_1)'(t)
-({\Gamma}_1x_2)'(t)\|\\
& \leq \|\int_0^tC(t-s)(g(s,x_{1s}+y_s,x_1'(s)+y'(s))
 -g(s,x_{2s}+y_s,x_2'(s)+y'(s)))ds\|\\
&\quad +\|g(t,x_{1t}+y_t,x_1'(t)+y'(t))-g(t,x_{2t}+y_t,x_2'(t)+y'(t))\|\\
&\quad +\|\int_0^tAS(t-s)(g(s,x_{1s}+y_s,x_1'(s)+y'(s))-
g(s,x_{2s}+y_s,x_2'(s)+y'(s)))ds\| \\
&\leq N\int_0^tL_g(\|x_{1s}-x_{2s}\|_{\mathcal{B}}+\|x_1'(s)
-x_2'(s)\|)ds \\
&\quad + L_g(\|x_{1t}-x_{2t}\|_{\mathcal{B}}+\|x_1'(t)-x_2'(t)\|)\\
&\quad +\|A\|
\widetilde{N}\int_0^tL_g(\|x_{1s}-x_{2s}\|_{\mathcal{B}}
+\|x_1'(s)-x_2'(s)\|)ds \\
&\leq N\int_0^tL_g(K(s)\sup_{0\leq\tau\leq
s}\|x_1(\tau)-x_2(\tau)\|+\|x_1'(s)-x_2'(s)\|)ds \\
&\quad + L_g(K(t)\sup_{0\leq\tau\leq
t}\|x_1(\tau)-x_2(\tau)\|+\|x_1'(t)-x_2'(t)\|) \\
&\quad +\|A\|
\widetilde{N}\int_0^tL_g(K(s)\sup_{0\leq\tau\leq
s}\|x_1(\tau)-x_2(\tau)\|+\|x_1'(s)-x_2'(s)\|)ds\\
&\leq L_g(Nb+1+\|A\|\widetilde{N}b)\sup_{0\leq\tau\leq
t}(K(t)\|x_1(\tau)-x_2(\tau)\|+\|x_1'(t)-x_2'(t)\|)\\
& \leq L_g(Nb+1+\|A\|\widetilde{N}b)(K_b+1)\|x_1-x_2\|_{1};
\end{align*}
 that is,
\[
\|{\Gamma}_1x_1(t)-{\Gamma}_1x_2(t)\|_{1b} \leq
L_g(Nb+1+\|A\|\widetilde{N}b)(K_b+1)\|x_1-x_2\|_{1b}.
\]
 Hence, ${\Gamma}_1$ is  Lipschitz continuous with
Lipschitz constant $L'=L_g(Nb+1+\|A\|\widetilde{N}b)(K_b+1) $.

Next, taking $W\subset \Gamma( B_k)$. Obviously, $S(t)$ is
equicontinuous. From Lemma \ref{lm5}, $W$ is equicontinuous and
$\chi_{{C}}(W)=\sup{\{\chi(W(t)),t\in[a,b]}\}$, we have
\begin{align*}
&\chi_{{C^{1}}}({\Gamma}_2W(t))\\
&=\chi_{{C^{1}}}(\int_0^t S(t-s)f(s,W_{s}+y_{s},W'(s)+y'(s))ds)
\\
&=\max\big\{\max_{t\in J}\chi_{C}(\int_0^t S(t-s)f(s,W_{s}+y_{s},
W'(s)+y'(s)))\big\}ds),
\\
&\quad{\max_{t\in J}\chi_{C}(\int_0^t
C(t-s)f(s,W_{s}+y_{s},W'(s)+y'(s))ds)}\}
\\
&\leq \max{\{\max_{t\in J}\int_0^t{\eta_1}(s)\sup _{-\infty<\theta\leq 0}\chi_{C}(
W(s+\theta)+y(s+\theta))}ds),\\
&\quad{\ \max _{t\in J} \int_0^t {\eta
_2}(s)\sup _{-\infty<\theta\leq 0}\chi_{C}(
W'(s+\theta)+y'(s+\theta))ds)}\}\\
&\leq \max{\{\int_0^t \eta_1
(s)\sup_{0\leq\tau\leq s} \chi_{C} (W(\tau))ds,\int_0^t
\eta_2 (s)\sup_{0\leq\tau\leq s} \chi_{C}
(W'(\tau))ds}\} \\
&\leq \max{\{\int_0^b \eta_1
(s)ds,\int_0^b \eta_2 (s)ds}\}\max{\{\sup_{0\leq\tau\leq b}
\chi_{C} (W(\tau)),\sup_{0\leq\tau\leq b} \chi_{C}
(W'(\tau))}\}\\
&\leq \max{\{\int_0^b \eta_1
(s)ds,\int_0^b \eta_2 (s)ds}\}\max{\{ \chi_{C} (W), \chi_{C}
(W')}\}\\
&\leq \max{\{\int_0^b \eta_1
(s)ds,\int_0^b \eta_2 (s)ds}\}\chi_{C{^{1}}} (W),
\end{align*}
 for each bounded set $W\in C^{1}(J;X)$.
Since
\begin{align*}
 \chi_{{C^{1}}}(\Gamma W)
  &=\chi_{{C^{1}}}(\Gamma_1 W
 +\Gamma_2 W) \leq \chi_{{C^{1}}}(\Gamma _1 W)+\chi_{{C^{1}}}(\Gamma_2 W) \\
 &\leq
 (L_g(Nb+1+\|A\|\widetilde{N}b)+\max{\{\int_0^b \eta_1 (s)ds,\int_0^b
 \eta_2 (s)ds}\})\chi_{{C^{1}}}(W) .
\end{align*}
The function $\Gamma$ is $\chi$-contraction. In view of Lemma
\ref{lm3}, we conclude that $\Gamma$ has at least one fixed point in
$W$. Let $x$ be a fixed of $\Gamma$ on $S^{1}(b)$, then $z=x+y$ is a
mild solution of \eqref{e1.3}-\eqref{e1.4}. So we deduce
the existence of a mild
solution of \eqref{e1.3}-\eqref{e1.4}.
\end{proof}

\section{Examples}

\subsection{The phase space $C_r \times L^2(h,X)$}
Let $h(\cdot) : (-\infty,-r]\to R$ be a positive Lebesgue integrable
function and
${\mathcal{B}} := C_r \times L^2(h;X)$, $r \geq 0$, be the space
formed of all classes of functions $\varphi : (-\infty, 0]
\to X$ such that $\varphi|_{[-r,0]} \in C([-r, 0],X)$,
$\phi(\cdot)$ is Lebesgue-measurable on $(-\infty,-r]$ and
$h|\varphi|^2$ is Lebesgue integrable on $(-\infty,-r]$. The
seminorm in $\| \cdot \|_{\mathcal{B}}$ is defined by
\[
\|\varphi\|_{\mathcal{B}} := \sup
_{\theta\in[-r,0]} |\varphi(\theta) | +(
\int_{-\infty}^{-r}h(\theta)|\varphi(\theta)|^2d\theta)^{1/2}.
\]
 Assume that $h(\cdot)$ satisfies \cite[conditions (g-6) and
(g-7)]{YH}, function $G$  is locally
bounded on $(-\infty,0]$. Proceeding as in the proof of
\cite[Theorem 1.3.8]{YH} it follows that ${\mathcal{B}}$ is a phase
space which satisfies the axioms (A) and (B). Moreover, when $r = 0$
this space coincides with $X \times L^2(h,X)$ and the parameter
$H = 1$, as in \cite[Theorem 1.3.8]{YH}; $M(t) = G(-t)^{1/2}$ and $
K(t) = 1+(\ \int_{-t}^{0} h(\tau ) d\tau )^{1/2}$,
for $t \geq 0$ (see \cite{YH}).

 Let $X = L^2([0,\pi])$ and let $A$ be the operator  $Af
= f''$ with domain
\[
 D(A) := {\{f \in L^2([0,\pi]): f'' \in
L^2([0,\pi]), f(0) = f (\pi) = 0 }\} .
\]
 It is well known that $A$ is the infinitesimal generator
of a $C_0$-semigroup and of a strongly continuous cosine function on
$X$, which will be denoted by   $(C(t))_{t\in R}$. Moreover, $A$ has
discrete spectrum, the eigenvalues are $-n^2$, $n \in N$, with
corresponding normalized eigenvectors
$z_n(\xi ) := (\frac{ 2}{\pi} )^{1/2} \sin(n\xi)$ and the following
properties hold:
\begin{itemize}
\item[(a)] ${\{z_n: n \in \textbf{N}}\}$ is an orthonormal basis of
$X$.

\item[(b)] For $f \in X$, $(-A)^{-1/2}f = \sum_{
n=1}^{\infty}\frac{1}{n} \langle f, z_n\rangle z_n$ and
$\|(-A)^{-1/2}\|=1$.

\item[(c)] For $f \in X$, $ C(t)f = \sum_{
n=1}^{\infty} \cos(nt)\langle f, z_n\rangle z_n$. Moreover, it
follow from this expression that $ S(t)\varphi
=\sum_{ n=1}^{\infty}\frac {\sin(nt)}{ n}\langle \varphi,
z_n\rangle z_n$, that $S(t)$ is compact for $t
> 0$ and that $\|C(t)\| = 1$ and $\|S(t)\|=1$ for every $t \in R$.

\item[(d)] If $\Phi$ denotes the group of translations on $X$ defined by
$\Phi(t)x(\xi) = \widetilde{x}(\xi + t)$, where $\widetilde{x}$ is
the extension of $x$ with period $2\pi$, then $C(t) =\frac{1}{2}
(\Phi(t) + \Phi(-t))$; $A = B^2$ where $B$ is the infinitesimal
generator of the group $\Phi$ and $E = {\{x \in H^1(0,\pi): x(0) =
x(\pi) = 0}$, see \cite{HOF} for details.
\end{itemize}

In the next applications, $\mathcal{B}$ will be the phase space
$X\times L^2(h,X)$.

\subsection{A second order neutral equation}
 Now we discuss
the existence of solutions for the second order neutral differential
equation
\begin{gather}
\begin{aligned}
&\frac{\partial}{\partial t}(
\frac{\partial u(t, \xi)}{
\partial t} + \int_{-\infty}^t\int_0^{\pi} b(t-s,\eta, \xi)u(s,\eta)
d\eta ds )\\
&=\frac{{\partial}^ 2u(t, \xi )}{ \partial
{\xi}^2 }+ \int_{-\infty}^{t}F(t, t-s,\xi,u(s, \xi)) ds, \quad
t \in [0, a], \xi \in [0,\pi],
\end{aligned} \label{e4.1}\\
  u(t, 0) = u(t,\pi) = 0, \quad t\in[0, a],
\label{e4.2}
\\
 u(\tau, \xi ) =\varphi(\tau,\xi),\quad  \tau \leq 0,\; 0 \leq
\xi \leq \pi, \label{e4.3}
\end{gather}
 where $\varphi \in X \times L^2(h;X)$, and
\begin{itemize}
\item[(a)] The functions $b(s, \eta, \xi)$, $\frac{\partial b(s,\eta,\xi
)}{\partial \xi}$ are measurable, $b(s, \eta,\pi) = b(s, \eta, 0) =
0$ and
\[ L_g := \max
{\{(\int_{0}^{\pi}\int_{-\infty}^{0}\int_{0}^{\pi}\frac {1}{
h(s)}(\frac{{\partial}^i b(s, \eta, \xi)}{\partial \xi^i})^2 d\eta
ds d\xi)^{1/2} : i = 0, 1}\} < \infty;
\]

\item[(b)] The function $F :\mathbb{R}^4
\to \mathbb{R}$ is continuous and there is continuous
function $\mu:\mathbb{R}^2\to \mathbb{R}$ such that
\[
\int_{-\infty}^0\frac{\mu(t,s)^2}{h(s)}ds<\infty
\]
and
$|F(t, s,\xi,x)|\leq \mu(t, s)|x|$,  $(t,s,\xi,x)\in \mathbb{R}^4;$
\end{itemize}

Assuming that conditions (a),(b) are satisfied,  problem
\eqref{e4.1}-\eqref{e4.3} can be modelled as the abstract  Cauchy problem
\eqref{e1.1}-\eqref{e1.2} by defining
\begin{gather}
 g(t,\psi)(\xi ) :=
\int_{-\infty}^{0}\int_0^{\pi} b(s, \nu, \xi)\psi(s, \nu) d\nu ds,
 \label{e4.4} \\
 f (t,\psi)(\xi ) := \int_{-\infty}^{0} F
( t, s, \xi,\psi(s, \xi) ) ds.  \label{e4.5}
\end{gather}
 Moreover, $\|f (t,\psi)\| \leq  d(t)\|\psi\|_{\mathcal{B}}$ for
every $t \in[0,a]$, where
$  d(t) := (\int_{-\infty}^0\frac{ \mu(t,
s)^2}{ h(s)} ds)^{1/2}$ is a  Lebesgue integrable function.

The next result is a consequence of Theorem \ref{tm1}.

\begin{proposition} \label{prop4.1}
 Let the previous conditions be
satisfied. If
\[
( 1+ (\int_{-a}^{0}h(\tau ) d\tau )^{1/2})( aL_g +  \int_0^a d(t) dt ) <
1,
\]
 then there exists a mild solution of \eqref{e4.1}-\eqref{e4.3}.
\end{proposition}

\subsection*{ Acknowledgements}
The authors express their gratitude to the anonymous referee for
his or her valuable comments and suggestions.



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\end{document}