\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil A Massera type criterion \hfil EJDE--2002/40} {EJDE--2002/40\hfil Eduardo Hern\'{a}ndez M. \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 40, pp. 1--17. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A Massera type criterion for a partial neutral functional differential equation % \thanks{ {\em Mathematics Subject Classifications:} 35A05, 34G20, 34A09. \hfil\break\indent {\em Key words:} Functional equations, neutral equations, semigroup of linear operators. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted March 06, 2002. Published May 7, 2002.} } \date{} % \author{Eduardo Hern\'{a}ndez M.} \maketitle \begin{abstract} We prove the existence of periodic solutions for partial neutral functional differential equations with delay, using a Massera type criterion. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{prop}[theorem]{Proposition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} By using a Massera type criterion, we prove the existence of a periodic solution for the partial neutral functional differential equation \begin{gather}\label{Ne} \frac{d}{dt}(x(t)+G(t,x_{t})) = Ax(t)+F(t,x_{t}),\quad t>0, \\ \label{Ne1} x_{0}=\varphi \in \mathcal{D}, \end{gather} where $A$ is the infinitesimal generator of a compact analytic semigroup of linear operators, $(T(t))_{t\geq 0}$, on a Banach space $X$. The history $x_{t}$, $x_{t}(\theta)=x(t+\theta)$, belongs to an appropriate phase space $\mathcal{D}$ and $G,F:\mathbb{R}\times \mathcal{D}\to X$ are continuous functions. The article by Massera \cite{Massera} is a pioneer work in the study of the relations between the boundedness of solutions and the existence of periodic solutions. In \cite{Massera}, this relation was explained for a two dimensional periodic ordinary differential equation. Subsequently, several authors considered similar relations, see for example Yoshizawa for a n-dimensional differential equation; Lopes and Hale for n-dimensional ordinary and functional equations with delay and Yong \cite{Yong}, for functional differential equations. Recently Ezzinbi \cite{EZZ}, using a Massera type criterion, showed the existence of a periodic solution for the partial functional differential equation \begin{equation} \label{neba} \begin{gathered} \dot{x}(t) = Ax(t)+F(t,x_{t}), \\ x_0 = \varphi \in \mathcal{C}=C([-r,0]:X), \end{gathered} \end{equation} where $A$ is the infinitesimal generator of a compact semigroup of bounded linear operators on a Banach space. Our purpose in this paper is to establish similar existence results, as those in \cite{EZZ}, for the partial neutral functional differential equation with delay (\ref{Ne})-(\ref{Ne1}). Neutral differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature of neutral functional differential equations with finite delay is the Hale book \cite{HA1} and the references therein. The work in partial neutral functional differential equations with infinite delay was initiated by Hern\'{a}ndez \& Henr\'{\i}quez in \cite{HH1,HH2}. In these papers, it is proved the existence of mild, strong and periodic solutions for a neutral equation \begin{equation}\label{neutral1} \begin{gathered} \frac{d}{dt}(x(t)+G(t,x_{t})) = Ax(t)+F(t,x_{t}), \\ x_{0} = \varphi \in \mathcal{B}. \end{gathered} \end{equation} where $A$ is the infinitesimal generator of an analytic semigroup of linear operators on a Banach space and ${\mathcal{B}}$ is a phase space defined axiomatically. In general, the results were obtained using the semigroup theory and the Sadovskii fixed point Theorem. For the rest of this paper, $X$ will denote a Banach space with norm $\| \cdot \|$. $A:D(A) \to X$ will denote the infinitesimal generator of a compact analytic semigroup, $(T(t))_{t\geq 0}$, of linear operators on $X$. For the theory of $C_{0}$ semigroups, we refer the reader to Pazy \cite{PA}. However, we will review some notation and properties that will be used in this work. It is well known that there exist $\tilde{M}\geq 1$ and $\rho \in \mathbb{R}$ such that $\| T(t) \|\leq \tilde{M}e^{\rho t},$ for every $t\geq 0 $. If $(T(t))_{t\geq 0}$ is a uniformly bounded and analytic semigroup such that $0\in \rho(A)$, then it is possible to define the fractional power $(-A)^\alpha$, for $\alpha\in (0, 1]$, as a closed linear operator on its domain $D(-A)^\alpha$. Furthermore, the subspace $D(-A)^\alpha$ is dense in $X$ and the expression $$\| x\|_{\alpha}= \| (-A)^\alpha x \|, \quad x\in D(-A)^\alpha, $$ defines a norm in $D(-A)^\alpha$. If $X_{\alpha}$ represents the space $D(-A)^\alpha$ endowed with the norm $\| \cdot \|_{\alpha}$, then the following properties are well known (\cite{PA}, pp. 74 ): \begin{lemma}\label{an} If the above conditions hold, then \begin{enumerate} \item If \,$0<\alpha\leq 1$, then $X_{\alpha}$ is a Banach space. \item If \,$0 < \beta <\alpha \leq 1$ then $X_{\alpha}\hookrightarrow X_{\beta}$ and the imbedding is compact whenever the resolvent operator of $A$ is compact. \item For every $0<\alpha \leq 1$ there exists $C_{\alpha}>0$ such that $$ \| (-A)^{\alpha} T(t)\| \leq \frac{C_{\alpha}}{t^{\alpha}}, \quad t>0. $$ \end{enumerate} \end{lemma} In what follows, to avoid unnecessary notation, we suppose that $0 \in \rho (A)$ and that for $0<\vartheta \leq 1$ \begin{eqnarray}\label{des18} \| T(t) \|\leq \tilde{M},\quad t\geq 0, \quad \hbox{and}\quad \|(-A)^{\vartheta} T(t) \| \leq \frac{C_{\vartheta}}{t^{\vartheta}},\quad t>0, \end{eqnarray} for some positive constant $C_{\theta}$. \bigskip \par In this paper, $0< \beta\leq 1$ and $ \omega>0$ are fixed numbers, $G,F:\mathbb{R}\times \mathcal{D}\to X$ are continuous and we use the following conditions \begin{enumerate} \item[${\bf H_{1}}$] The function $G $ is $ X_{\beta}$-valued and $(-A)^{\beta}G$ is continuous. \item[${\bf H_{2}}$] $G(t,\psi)=V(t,\psi)+h(t)$ where $V,\,h$ are $X_{\beta}$-valued; $(-A)^{\beta}V$, $(-A)^{\beta}h$ are continuous; $(-A)^{\beta}V(\cdot,\psi)$, $(-A)^{\beta}h$ are $ \omega$-periodic and $(-A)^{\beta}V(t,\cdot )$ is linear. \item[${\bf H_{3}}$] $G(t,\psi)=V(t,\psi) + G_{1}(t,\psi)$ where the functions $V$, $G_{1}$ are $X_{\beta}$-valued; $(-A)^{\beta}V$, $(-A)^{\beta}G_{1}$ are continuous; $(-A)^{\beta}V(\cdot,\psi)$, $(-A)^{\beta}G_{1}(\cdot,\psi)$ are $ \omega$-periodic and $(-A)^{\beta}V(t,\cdot )$ is linear. \item[${\bf H_{4}}$] $F(t,\psi)=L(t,\psi)+f(t)$ where $L,\, f$ are continuous; $L(\cdot,\psi)$, $f$ are $ \omega$-periodic and $L(t,\cdot )$ is linear. \item[${\bf H_{5}}$] $F(t,\psi)=L(t,\psi)+F_{1}(t, \psi)$ where $L,\, F_{1}$ are continuous; $L(\cdot,\psi)$, $F_{1}(\cdot,\psi)$ are $\omega$-periodic and $L(t,\cdot )$ is linear. \item[${\bf H_{6}}$] For every $R>0$ and all $T>0$, the set of functions $$\{s \to G(s,x_{s}):x\in C([-r, T]:X), \, \sup_{\theta \in [-r,T]}\| x ( \theta)\| \leq R \}$$ is equicontinuous on $[0,T]$. \item[${\bf H_{7}}$] For every $R>0$ and all $T>0$, the set of functions $$\{s \to G(s,x_{s}):x\in C_{b}((-\infty, T]:X), \, \sup_{\theta \in [-\infty,T]}\| x ( \theta)\| \leq R \}$$ is equicontinuous on $[0,T]$. \end{enumerate} This paper has four sections. In section 2, we discuss the existence of a periodic solution for a partial functional neutral differential equation defined on $ \mathbb{R}\times C([-r,0]:X) $. In section 3, by employing the results in section 2, we consider the existence of a periodic solution for a neutral equation with unbounded delay modeled on $ \mathbb{R}\times \mathcal{B} $, where $\mathcal{B}$ is a phase space defined axiomatically as in Hale and Kato \cite{HK1}. The section 4 is reserved for examples. Our results are based on the properties of analytic semigroups and the ideas and techniques in Hern\'{a}ndez $\&$ Henr\'{\i}quez \cite{HH1,HH2} and Ezzinbi \cite{EZZ}. \par Throughout this paper, $x(\cdot, \varphi)$ denotes a solution of (\ref{Ne})- (\ref{Ne1}). In addition, $B_{r}(x:Z)$, $( \,B_{r}[x:Z]\, )$ will be the open $($ the closed $)$ ball in a metric space $Z$ with center at $x$ and radius $r$. For a bounded function $\xi :[a,b]\to [0,\infty)$ and $a\leq t\leq b$ we will employ the notation $\xi_{a,t}$ for \begin{eqnarray}\label{not2} \xi_{a,t} =\sup\{\xi(s): s \in [a,t]\}. \end{eqnarray} If $\mathcal{D}$ is a Banach phase space, the norm in $\mathcal{D}$ will be denoted by $\| \cdot \|_{\mathcal{D}}$. We remark that for the proofs of our results we will use the following results. \begin{theorem}[\cite{HK1}] \label{teo2} Let $Y$ be a Banach space and $ \Gamma:= \Gamma_{1} +y$ where $\Gamma_{1}: Y \to Y$ is a bounded linear operator and $y\in Y $. If there exist $x_{0}\in Y $ such that the set $\{\Gamma^{n} (x_{0}):n\in \mathbb{N}\} $ is relatively compact in $Y$, then $ \Gamma$ has a fixed point in $Y.$ \end{theorem} \begin{theorem}[\cite{BoKa}] \label{teoK} Let $X$ be a Banach space and $M$ be a nonempty convex subset of $X$. If \, $ \Gamma: M \to 2^{X}$ is a multivalued map such that \begin{itemize} \item[{\bf (i)}] For every $x\in M$, the set $ \Gamma (x)$ is nonempty, convex and closed, \item[{\bf (ii)}] The set $\Gamma (M) =\bigcup_{x\in M} \Gamma x$ is relatively compact, \item[{\bf (iii)}] $\Gamma$ is upper semi-continuous, \end{itemize} then $\Gamma$ has a fixed point in $M$. \end{theorem} \section{A periodic solution for a partial neutral\\ differential equation with bounded delay}\label{sec1} In this section, we prove the existence of a periodic solution of the initial value problem \begin{gather}\label{ne} \frac{d}{dt}(x(t)+G(t,x_{t})) = Ax(t)+F(t,x_{t}),\\ \label{ne1} x_{0} = \varphi \in\mathcal{C}=C([-r,0]:X). \end{gather} \paragraph{Definition} A function $x:[-r,T]\to X$ is a mild solution of the abstract Cauchy problem (\ref{ne})-(\ref{ne1}) if: $x_{0}=\varphi$; the restriction of $x(\cdot)$ to the interval $[0,T]$ is continuous; for each $0\leq t < T$ the function $AT(t-s)G(s,x_{s})$, $s\in [0,t)$, is integrable and \begin{eqnarray}\label{milsol} x(t)&=&T(t)(\varphi(0)+G(0,\varphi))-G(t,x_{t}) -\int_{0}^{t}AT(t-s)G(s,x_{s})ds \nonumber \\ &&+ \int_{0}^{t}T(t-s)F(s,x_{s})ds, \quad t\in [0,T]. \end{eqnarray} The existence of mild solutions for the abstract Cauchy problem (\ref{ne})-(\ref{ne1}) follows from \cite[theorems 2.1, 2.2]{HH2}, for this reason, we choose to omit the proof of the next two results. \begin{theorem}\label{edo1} Let $\varphi\in\mathcal{C}$, $T>0$ and assume that the following conditions hold: \begin{enumerate} \item[{\bf (a)}] There exist constants $ \beta \in (0,1)$ and $L \geq 0$ such that the function $ G $ is $ X_{\beta}$-valued, $ L\| (- A)^{-\beta}\|< 1$ and \begin{equation} \| (-A)^{\beta} G(t,\psi_{1} )-(-A)^{\beta} G(s, \psi_{2})\| \leq L(|t-s| + \| \psi_{1} - \psi_{2} \|_{\mathcal{C}}), \label{G1} \end{equation} for every $0\leq s,t \leq T$ and $ \psi_{1},\psi_{2}\in \mathcal{C}$. \item[{\bf (b)}] The function $F$ is continuous and takes bounded sets into the bounded sets. \end{enumerate} Then there exists a mild solution $x(\cdot,\varphi) $ of the abstract Cauchy problem (\ref{ne})-(\ref{ne1}) defined on $ [-r,b]$, for some $00$. Assume that condition ${\bf{(a)}}$ of the previous Theorem holds and that there exists $N>0$ such that \begin{equation} \| F(t,\varphi ) - F(t, \psi) \| \leq N \|\varphi - \psi\|_{\mathcal{C}} , \label{G3} \end{equation} for all $0\leq t \leq T $ and every $\varphi , \psi \in \mathcal{C}$. Then there exists a unique mild solution $x(\cdot, \varphi )$ of (\ref{ne})-(\ref{ne1}) defined on $[-r,b]$ for some $0 < b \leq T $. Moreover, $ b$ can be chosen as $\min\{ T, b_{0}\}$, where $b_{0}$ is a positive constant independent of $\varphi $. \end{theorem} To prove the main result of this section, it is fundamental the next result. \begin{theorem}\label{teo1} Let $T>r$ and assume that assumption ${\bf{H_{1}, H_{6}}}$ hold. Suppose, furthermore, that the following conditions hold. \begin{itemize} \item[{\bf{(a)}}] For every $\varphi \in \mathcal{C}$ the set $$\mathrm{X}(\varphi)=\{\,x\in C([-r,T]:X): x \hbox{ is solution of {(\ref{ne})-(\ref{ne1})}}\}$$ is nonempty. \item[{\bf{(b)}}] For every $R>0$, the set $$\{ (-A)^{\beta}G(s,x_{s}),\,F(s,x_{s})\,:s\in [0,T],\, x\in \mathrm{X}(\varphi)\hbox{ and }\| \varphi\|_{\mathcal{C}} \leq R\},$$ is bounded. \end{itemize} Then the multivalued map $ \Upsilon:\mathcal{C}\to 2^{\mathcal{C}} $; $ \varphi \to \mathrm{X}_{T}(\varphi)=\{x_{T}:x\in \mathrm{X}(\varphi)\}$ is compact, that is, for every $R>0 $ the set $\mathcal{U}_{R,T}=\bigcup_{\|\varphi\|_{\mathcal{C}}\leq R} \mathrm{X}_{T}(\varphi)$ is relatively compact in $\mathcal{C}$. \end{theorem} \paragraph{Proof:} Let $R>0$ and $\mathcal{U}_{R}=\bigcup_{\|\varphi\|_{\mathcal{C}}\leq R} \mathrm{X}(\varphi) $. From {\bf{(b)}}, we fix $N>0$ such that $\|(-A)^{\beta}G(s,x_{s})\| \leq N$ and $\| F(s,x_{s})\|\leq N$ for every $x\in\mathcal{U}_{R} $ and every $s\in [0,T]$. In order to use the Ascoli Theorem, we divide the proof in two steps. \noindent{\bf Step 1} The set $\mathcal{U}_{R}(t)= \{x(t): x\in \mathcal{U}_{R}\}$ is relatively compact for $t\in (0,T]$. Let $0<\epsilon 0$. The equicontinuity in $t_{0}>0$ is proved in similar form, we omit details. Thus, $\mathcal{U}_{R}$ is equicontinuous on $(0,T]$. From the steps $1$ and $2$, it follow that $ \{x|_{[\mu, T]}:x\in\mathcal{U}_{R}\}$ is relatively compact in $C([\mu, T]: X)$ for every $\mu>0$, which in turn implies that $\mathcal{U}_{R,T}$ is relatively compact in $C([-r, 0]:X)$. This completes the proof. \begin{corollary}\label{cor1} Assume that the hypothesis in Theorems \ref{teoexi2} and \ref{teo1} are fulfilled. Then the map $\varphi\to x_{T}(\cdot, \varphi)$ is a completely continuous function. \end{corollary} \paragraph{Proof:} The assertion follows from (\ref{dess1}), the Lebesgue dominated convergence Theorem and Theorem \ref{teo1}. We omit details. \paragraph{Definition} A function $ x:\mathbb{R}\to X$ is an $\omega$-periodic solution of equation (\ref{ne}) if: $x(\cdot)$ is a mild solution of (\ref{ne}) and $x(t + \omega) = x(t)$ for every $t \in \mathbb{R}$. \smallskip Using the ideas and techniques in \cite{HH1}, it is possible to establish sufficient conditions for the existence of global solutions of (\ref{ne}). In what follows, we always assume that the mild solutions are defined on $[0,\infty)$. \begin{theorem}\label{exis1} Let conditions ${\bf{H_{2}}},\,{\bf{H_{4}}}$ and ${\bf{H_{6}}}$ be satisfied. If the equation (\ref{ne}) has a bounded mild solution, then there exists an $\omega$-periodic solution of (\ref{ne}). \end{theorem} \paragraph{Proof:} For a mild solution $x(\cdot )= x(\cdot, \varphi)$, we introduce the decomposition $x(\cdot)=v(\cdot)+z(\cdot)$ where $v(\cdot)$ is the mild solution of \begin{gather*} \frac{d}{dt}(u(t)+V(t,u_{t})) = Au(t)+L(t,u_{t}), \\ u_{0} = \varphi, \end{gather*} and $z(\cdot)$ is the mild solution of \begin{gather*} \frac{d}{dt}(u(t)+V(t,u_{t})+h(t)) = Au(t)+L(t,u_{t})+f(t),\\ u_{0} = 0. \end{gather*} Let $y:[-r,\infty)\to X$ be a bounded mild solution of (\ref{ne}) and $\Gamma : \mathcal{C}\to \mathcal{C}$ be the map $\Gamma (\varphi):=\Gamma_{1}(\varphi)+z_{\omega}:=v_{\omega}+z_{\omega}$. Since $\Gamma_{1} $ is a bounded linear operator and $ \bigcup_{n\geq 1}\Gamma^{n} (y_{0})=\{y_{n\omega}:n\in \mathbb{N}\} $ is relatively compact in $\mathcal{C}$, see Theorem \ref{teo1}, it follows from Theorem \ref{teo2} that $\Gamma$ has a fixed point in $\mathcal{C}$. This fixed point give a periodic solution. The proof is complete. In what follows $CP$ is the space $CP=\{u:\mathbb{R}\to X: \mbox{$u$ is $\omega$-periodic}\}$ endowed with the uniform convergence topology. Now we prove the main result of this work. \begin{theorem}\label{mainteo} Let conditions ${\bf{H_{3}}}, {\bf{H_{5}}}$ and ${\bf{H_{6}}}$ be satisfied and assume that the following conditions are fulfilled. \begin{itemize} \item[${\bf (a)}$] The functions $ (-A)^{\beta}V,\, (-A)^{\beta}G_{1},\, L$ and $F_{1}$ takes bounded sets into the bounded sets. \item[${\bf (b)}$] There is $\rho>0 $ such that for every $v\in B_{\rho}[0,CP]$ the neutral equation $$ \frac{d}{dt}(x(t)+V(t,x_{t})+G_{1}(t, v_{t})) = Ax(t)+L(t,x_{t})+F_{1}(t, v_{t}), $$ has an $\omega$-periodic solution $u(\cdot, v )\in B_{\rho}[0,CP]$. \end{itemize} Then the equation (\ref{ne}) has an $\omega$-periodic solution. \end{theorem} \paragraph{Proof:} On $B_{\rho}=B_{\rho}[0,CP]$, we define the multivalued map $ \Gamma:B_{\rho}\to 2^{B_{\rho}} $ by: $x\in \Gamma (v)$ if, and only if, \begin{eqnarray*} x(t) &=& T(t-s)(x(s)+V(s,x_{s})+G_{1}(s,v_{s})) -V(t,x_{t})-G_{1}(t,v_{t})\nonumber\\ && -\int_{s}^{t}AT(t-\tau) (V(\tau ,x_{\tau})+G_{1}(\tau,v_{\tau}))d\tau \nonumber\\&&+ \int_{s}^{t}T(t-\tau)( L(\tau,x_{\tau})+F_{1}(\tau, v_{\tau}))d\tau, \quad t>s.\nonumber\end{eqnarray*} Next we prove that $\Gamma$ verifies the conditions ${\bf{(i)}}$-${\bf{(iii)}}$ of Theorem \ref{teoK}. Clearly assumption ${\bf{ ( i)}}$ holds. The condition ${\bf{ (ii)}}$ follows using the steps in the proof of Theorem \ref{teo1}. In relation to ${{\bf (iii)}}$, we observe that from ${{\bf (ii)}}$ is sufficient to show that $\Gamma$ is closed. If $(v^{n})_{n\in\mathbb{N}}$ and $\,(x^{n})_{n\in\mathbb{N}}$ are convergent sequences in $CP$ to points $v, x $ then \begin{eqnarray*} (-A)^{\beta}(V(\tau,x^{n}_{\tau})+G_{1}(\tau,v_{\tau}^{n})) &\to& (-A)^{\beta}(V(\tau,x_{\tau})+G_{1}(\tau,v_{\tau})),\\ T(t)(L( \tau,x^{n}_{\tau})+F_{1}(\tau,v^{n}_{\tau})) &\to & T(t)(L(\tau,x_{\tau})+F_{1}(\tau,v_{\tau})), \end{eqnarray*} for $\tau\in \mathbb{R}$ and $t\geq s$. From the Lebesgue dominated convergence Theorem, assumption ${\bf (a)}$ and the estimate $$ \| AT(t-s)(V(\tau,x^{n}_{\tau})+ G_{1}(\tau,v_{\tau}^{n}))\|\leq C_{1-\beta}\frac{\|(-A)^{\beta}(V(\tau,x^{n}_{\tau})+ G_{1}(\tau,v_{\tau}^{n}))\|}{(t-s)^{1-\beta}} ,\nonumber$$ we conclude that \begin{eqnarray*} x(t) &=& T(t-s)(x(s)+V(s,x_{s})+G_{1}(s,v_{s})) -V(t,x_{t})-G_{1}(t,v_{t})\nonumber\\ && -\int_{s}^{t}AT(t-\tau) (V(\tau ,x_{\tau})+G_{1}(\tau,v_{\tau}))d\tau \nonumber\\&&+ \int_{s}^{t}T(t-\tau)( L(\tau,x_{\tau})+F_{1}(\tau,v_{\tau}))d\tau, \quad t>s, \nonumber\end{eqnarray*} which proves that $x\in\Gamma v$. Thus, $\Gamma$ is closed and consequently upper semi-continuous. From Theorem \ref{teo2} the operator $\Gamma$ has a fixed point. This fixed point is an $\omega$-periodic solution of (\ref{ne}). The proof is finished. \section{A periodic solution for a partial neutral\\ differential equation with unbounded delay}\label{sec3} In this section, we discuss the existence of an $\omega$-periodic solution for a partial functional neutral differential equation with unbounded delay modeled in the form \begin{gather} \label{eud1} \frac{d}{dt}(x(t)+G(t,x_{t})) = Ax(t)+F(t,x_{t}), \\ x_{\sigma} = \varphi \in \mathcal{B}, \label{eud2} \end{gather} where the history $x_{t}:(-\infty,0]\to X$, $x_{t}(\theta)=x(t+\theta)$, belongs to some abstract phase space $\mathcal{B}$ defined axiomatically and $F, G:\mathbb{R}\times \mathcal{B}\to X$ are appropriate continuous functions. For the rest of this paper, $\mathcal{B}$ will be an abstract phase space defined axiomatically as in Hale and Kato \cite{HK}. To establish the axioms of the space $\mathcal{B}$, we follow the terminology used in \cite{HMN}, and thus, $\mathcal{B}$ will be a linear space of functions mapping $(-\infty,0]$ into $X$, endowed with a semi-norm $\| \cdot \| _{\mathcal{B}}$. We will assume that $\mathcal{B}$ satisfies the following axioms: \begin{enumerate} \item[$(\mathbf{A})$] If $x:(-\infty,\sigma+a)\to X$, $a>0$, is continuous on $[\sigma,\sigma +a)$ and $x_{\sigma}\in \mathcal{B}$, then for every $t\in [\sigma,\sigma+a)$, the following conditions hold: \begin{itemize} \item[i)] $ x_{t}$ is in $\mathcal{B}$. \item[ii)] $\| x(t)\| \leq H \|x_{t}\|_{\mathcal{B}}$. \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 [0,\infty)$, $K$ is continuous, $M$ is locally bounded and $H,K,M$ are independent of $x(\cdot)$. \item[$(\mathbf{A1})$] For the function $x(\cdot)$ in $(\mathbf{A})$, $x_{t}$ is a $\mathcal{B}$-valued continuous function on $[\sigma,\sigma+a)$. \item[$(\mathbf{B})$] The space $\mathcal{B}$ is complete. \item [$(\mathbf{C\,2})$] If a uniformly bounded sequence $(\varphi^n )_n $ in $C_{00}$ converges to a function $\varphi$ in the compact-open topology, then $\varphi \in \mathcal{B} $ and $\|\varphi^n - \varphi\|_{\mathcal{B}} \to 0\;\mbox{as}\;n \to~\infty$. \end{enumerate} \begin{example} \label{example1} \rm We consider the phase space $\mathcal{B}:= C_{r} \times L^{p}(g;X)$, $r \geq 0$, $ 1 \leq p < \infty $, see \cite{HMN}, which consists of all classes of functions $ \varphi : (- \infty , 0]\to X \; $ such that $ \varphi $ is continuous on $[- r,0], $ Lebesgue-measurable and $ g | \varphi(\cdot)|^{p} $ is Lebesgue integrable on $ (- \infty , -r ),$ where $ g: (- \infty , -r) \to \mathbb{R} $ is a positive Lebesgue integrable function. The seminorm in $\|\cdot\|_{\mathcal{B}} $ is defined by $$ \| \varphi \|_{\mathcal{B}} : = \sup \{ \| \varphi (\theta ) \| : -r\leq \theta \leq 0 \}\; +\left( \int_{- \infty }^{-r} g(\theta ) \| \varphi (\theta ) \|^{p} d \theta \right)^{1/p}. $$ We will assume that $ g $ satisfies conditions (g-6) and (g-7) in the terminology of \cite{HMN}. This means that $\;g\;$ is integrable on $(- \infty, -r)$ and that there exists a non-negative and locally bounded function $\gamma$ on $\;(- \infty, 0]\;$ such that $$ g(\xi \: + \: \theta) \leq \gamma(\xi) \: g(\theta) , $$ for all $ \xi \leq 0$ and $ \theta \in (- \infty , -r) \setminus N_{\xi }$, where $\; N_{\xi} \subseteq (- \infty, -r)\;$ is a set with Lebesgue measure zero. In this case, $ \mathcal{B} $ is a phase space which satisfies axioms $\mathbf{(A)}$, $\mathbf{(A1)}$, $\mathbf{(B)}$ and $\mathbf{(C2)}$\,( see \cite{HMN}, Theorem 1.3.8). \end{example} \paragraph{Definition} A function $x:(-\infty,T]\to X$ is a mild solution of the abstract Cauchy problem (\ref{eud1})-(\ref{eud2}) if: $x_{0}=\varphi$; the restriction of $x(\cdot)$ to the interval $[0,T]$ is continuous; for each $0\leq t < T$ the function $AT(t-s)G(s,x_{s})$, $s\in [0,t)$, is integrable and \begin{equation} \label{milsol1} \begin{aligned} x(t)=&T(t)(\varphi(0)+G(0,\varphi))-G(t,x_{t}) -\int_{0}^{t}AT(t-s)G(s,x_{s})ds \\ &+ \int_{0}^{t}T(t-s)F(s,x_{s})ds, \quad t\in [0,T]. \end{aligned} \end{equation} \begin{lemma} Let Assumptions ${\bf{H_{1}, H_{7}}}$ be satisfied. If $x:(-\infty,T]\to X$ is a bounded mild solution of (\ref{eud1}), then the set $ U=\{ x_{n\omega}:n\in \mathbb{N}\}$ is relatively compact in $\mathcal{B}$. \end{lemma} \paragraph{Proof:} Let $(x_{n_{k}\omega})_{k\in \mathbb{N}}$ be a sequence in $U$. For $n>0$, we define the set $U(n,r)=\{{x_{n_{k}\omega}}_{\left|_{[-r,0]}\right.}:n_{k}\geq n \}$. Using the same arguments in the proof of Theorem \ref{teo1}, it follows that $U(n,r)$ is relatively compact in $C([-r,,0];X)$ when $n\omega>r$. Now we can choose a subsequence of $(x_{n_{k}\omega})_{k\in \mathbb{N}}$; which is indicated by the same index, that converges uniformly on compact subsets of $(-\infty,0] $ to some function $x\in C_{b}((-\infty,0]:X)$. Since $\mathcal{B}$ verifies axiom $\mathbf{C\,2}$, it follow that $x\in \mathcal{B}$ and that $x_{n_{k}\omega}\to x $ in $\mathcal{B}$. Thus, $U$ is relatively compact in $\mathcal{B}$. \paragraph{Definition} A function $ x:\mathbb{R}\to X$ is an $\omega$-periodic solution of equation (\ref{eud1}) if: $x(\cdot)$ is a mild solution of (\ref{eud1}) and $x(t + \omega) = x(t)$ for every $t \in \mathbb{R}$. \smallskip The proofs of the following results are similar to the proofs of Theorems \ref{exis1} and \ref{mainteo}. We only remark that the continuity of $\varphi\to x_{\omega}(\cdot, \varphi )$ is discussed in \cite{HH1}. \begin{theorem}\label{exis2} Let conditions ${\bf{H_{2}}},\,{\bf{H_{4}}}$ and ${\bf{H_{7}}}$ be satisfied. If the equation (\ref{eud1}) has a bounded mild solution then there exists an $\omega$-periodic solution of (\ref{eud1}). \end{theorem} \begin{theorem}\label{mainteo2} Let conditions ${\bf{H_{3}}}, {\bf{H_{5}}}$ and ${\bf{H_{7}}}$ be satisfied and assume that the following conditions are fulfilled. \begin{itemize} \item[${\bf (a)}$] The functions $ (-A)^{\beta}V,\, (-A)^{\beta}G_{1},\, L$ and $F_{1}$ takes bounded sets into the bounded sets. \item[${\bf (b)}$] There is $\rho>0 $ such that for every $v\in B_{\rho}[0,CP]$ the neutral equation \begin{eqnarray*} \frac{d}{dt}(x(t)+V(t,x_{t})+G_{1}(t, v_{t})) &=& Ax(t)+L(t,x_{t})+F_{1}(t, v_{t}), \nonumber\end{eqnarray*} has an $\omega$-periodic solution $u(\cdot, v )\in B_{\rho}[0,CP]$. \end{itemize} Then the neutral equation (\ref{eud1}) has an $\omega$-periodic solution. \end{theorem} \section{Applications}\label{sec4} In this section, we illustrate some of the results in this work. Let $X=L^{2}([0,\pi])$ and $A $ and $Ax=x''$ with domain $$ D(A) := \{f(\cdot) \in L^{2}([0, \pi]) : f''(\cdot) \in L^{2}([0, \pi]), \;f(0) = f(\pi) = 0 \}. $$ It's well known that $A$ is the infinitesimal generator of a $C_{0}$ semigroup, $(T(t))_{t\geq 0}$, on $X$, which is compact, analytic and self-adjoint. Moreover, $A$ has discrete spectrum, the eigenvalues are $- n^{2},\;n \in \mathbb{N},$ with corresponding normalized eigenvectors $z_{n} (\xi) := (2/\pi)^{1/2} \sin (n \xi)$ and the following properties hold: \begin{description} \item[(a)] $\{z_{n} : n \in \mathbb{N} \}$ is an orthonormal basis of $X$. \item[(b)] If $f \in D(A)$ then $A(f) = - \sum_{n=1}^{\infty}n^{2} \langle f,z_{n}\rangle z_{n}$. \item[(c)] For $f \in X$, $(-A)^{-\frac{1}{2}}f = \sum_{n=1}^{\infty} \frac{1}{n} \langle f, z_{n}\rangle z_{n}$. In particular, $\|(-A)^{-1/2} \| =1$. \item[(d)] The operator $(-A)^{1/2}$ is given as $(-A)^{1/2} f = \sum_{n=1}^{\infty} n \langle f, z_{n}\rangle z_{n}$ on the space $D((-A)^{1/2}) = \{f \in X: \sum_{n=1}^{\infty} n\langle f, z_{n}\rangle z_{n} \in X \}$. \item[(e)] For every $f \in X$, $T(t)f = \sum_{n=1}^{\infty} e^{-n^{2}t} \langle f, z_{n}\rangle z_{n}$. Moreover, it follows from this expression that $ \| T(t) \|\leq e^{-t}$, $t\geq 0$, and that $\|(-A)^{1/2} T(t) \| \leq \frac{1}{\sqrt{2}}e^{-t/2}t^{-1/2}$, for $t>0$. \end{description} \subsection*{A neutral equation with bounded Delay}\label{exem1} Considering the example in Ezzimby \cite{EZZ}, in this section we study the neutral equation \begin{gather} \label{H2} \begin{aligned} \frac{d}{d t}&[u(t,\xi) +\int_{-r}^{0}\int_{0}^{\pi} a_{0}(t) b(s,\eta,\xi) u(t+s,\eta)d \eta ds]\\ &=\frac{\partial^{2}} {\partial \xi^{2}} u(t,\xi) + a_{1}(t)x(t-r,\xi) +a_{2}(t)p(t,x(t-r,\xi))+q(t,\xi), \end{aligned}\\ u(t, 0)=u(t, \pi) = 0, \quad t \geq 0, \label{H3} \\ u(\tau, \xi)=\varphi(\tau, \xi),\quad \tau \in [-r,0],\;\; 0 \leq \xi \leq \pi, \label{H4} \end{gather} where \begin{enumerate} \item[{\bf{(i)}}] The function $b(\cdot)$ is measurable and $$\sup_{t\in [-r, \infty )} \int_{0}^{\pi} \int_{0}^{\pi} b^{2}(t ,\eta,\xi ) d\eta d\xi < \infty. $$ \item[{\bf{(ii)}}] The function $ {\displaystyle \frac{\partial} {\partial \zeta}\, b(\tau, \eta, \zeta)}$ is measurable; $\;b(\tau, \eta,\pi) = 0;$ $b(\tau, \eta,\, 0) =0$ and $ N_{1}r< 1$ where $$ N_{1} := \int_{0}^{\pi} \int_{-r}^{0} \int_{0}^{\pi} \big( \frac{\partial}{\partial \zeta}b(\tau, \eta, \zeta) \big)^{2} d\eta d\tau d\zeta .$$ \item[{\bf{(iii)}}] The functions $p, q:\mathbb{R}^{2}\to \mathbb{R}$ are continuous and $\omega$-periodic in the first variable. \item[{\bf{(iv)}}] The substitution operators $g:\mathbb{R}\times X \to X$, $f:\mathbb{R}\to X$ defined by $g(t,x)(\xi)=p(t,x(\xi))$ and $f(t)(\xi)=q(t,\xi)$ are $ \omega$-periodic, continuous and there exists $k>0$ such that $ \| g(t,x) \|\leq k \| x\|, \quad (t,x)\in \mathbb{R}\times X$. \item[{\bf{(v)}}] The functions $ a_{1}, a_{2} :\mathbb{R}\to\mathbb{R}$ are continuous, $\omega$-periodic and there exists a constant $ l$ such that $-1 +| a_{1}(t)| +| a_{2}(t)| k\leq -{l}$, $ t\geq 0$. \item[{\bf{(vi)}}] The function $a_{0}:\mathbb{R}\to\mathbb{R}$ is continuous, nondecreasing, $\omega$-periodic and $0\leq a_{0} (t) \leq ( 1- e^{- t})$, for $t\geq 0$. \end{enumerate} On the space $\mathbb{R}\times {\mathcal{C}}$, we define the maps \begin{gather*} G(t,\psi)(\xi) := a_{0}(t)V(t,\psi), \\ V(t,\psi)(\xi) := \int_{-r}^{0}\int_{0}^{\pi} b(s,\eta,\xi) \psi (s,\eta)d\eta ds, \label{H29} \\ %\label{H30} L(t,\psi)(\xi):= a_{1}(t)\psi(-r)(\xi),\\ F_{1}(t,\psi)(\xi) :=a_{2}(t)g(t,\psi(-r))(\xi)+f(t)(\xi). \end{gather*} With the previous notation, the initial-boundary value problem (\ref{H2})-(\ref{H4}) can be written as the abstract Cauchy problem \begin{gather}\label{equ1} \frac{d}{dt} (x(t) + G(t, x_{t})) = Ax(t)+L(t,x_{t})+F_{1}(t,x_{t}),\quad t \geq 0 \\ \label{H27} x_{0} = \varphi, \quad \varphi \in C([-r,0]:X)=:\mathcal{C}. \end{gather} A straightforward estimation using {\bf{(i)}}-{\bf{(iv)}} shows that the functions $G, \,L $ and $F_{1}$ are continuous. Moreover, from {\bf{(d)}} and {\bf{(ii)}}, it follow that $G$ is a bounded linear operator with values in $X_{1/2}$ and that $\| (- A)^{1/2}G(t,\cdot )\| \leq a_{0}(t) (N_{1}r)^{1/2} $ for every $t\in \mathbb{R}$. Next we prove that $G$ satisfies ${\bf{H_{6}}}$. Considering the condition $ {\bf {( vi)}}$, we only proof that $V$ verifies ${\bf{ H_{6}}}$. Let $R >0$ and $x\in C([-r,T];X)$ such that $\| x\|_{-r,T} \leq R $. For $t>0$ we get \begin{eqnarray*} \lefteqn { V(t+h, x_{t+h})(\xi)-V(t, x_{t})(\xi) }\\ &=& \int_{-r+t+h}^{-r+t}\int_{0}^{\pi} b(\theta-t-h,\eta,\xi)x(\theta,\eta ) d\eta d\theta \\ &&+\int_{-r+t}^{t}\int_{0}^{\pi} ( b(\theta-t-h,\eta,\xi)- b(\theta-t,\eta,\xi) ) x(\theta,\eta ) d\eta d\theta\nonumber\\ &&+\int_{t}^{t+h}\int_{0}^{\pi} b(\theta-t-h,\eta,\xi) x(\theta,\eta )d\eta d\theta \end{eqnarray*} and hence \begin{eqnarray*} | \frac{d}{dt}V(t,x_{t})|^{2}_{X} &\leq& 4\| x(-r+t)\|^{2} \int_{0}^{\pi}\int_{0}^{\pi} b^{2}(-r,\eta,\xi)^{2}d\eta d\xi \\ && +\, 4\| x\|_{\infty}^{2}r\int_{0}^{\pi} \int_{-r+t}^{t} \int_{0}^{\pi}( \frac{\partial b}{\partial \theta}(\theta-t,\eta,\xi))^{2}d\eta d\theta d\xi \\ && +\, 4\| x\|_{\infty}^{2}\int_{0}^{\pi}\int_{0}^{\pi} b^{2}(0,\eta,\xi ) d\eta d\xi. \end{eqnarray*} Thus, \begin{equation} \label{dgds} \| \frac{d}{dt} V(t,x_{t})\| ^{2} \leq C(\frac{\partial b}{\partial\theta},b), \end{equation} where $ C(\frac{\partial b}{\partial\theta},b)>0$ is independent of $t> 0$ and $x$ with $\| x\|_{-r,T} \leq R $. This implies that the set $$\mathcal{U}=\{s \to V (s,x_{s}):x\in C([-r, T];X), \, \sup_{\theta \in [-r,T]}\| x ( \theta)\| \leq R \}$$ is equicontinuous from the right side at $t>0$. The equicontinuity of $\mathcal{U}$ on $ \mathbb{R}$ is proved in similar form. Thus, $ V $ verifies condition ${\bf{H_{6}}}$. \begin{prop} Assume that the above conditions hold and that \begin{equation}\label{ineq} l > (N_{1}r)^{1/2}\rho ( 1 +\frac{1}{\sqrt{2}} ( 2e^{\frac{-1}{2}} +2 )), \end{equation} where $\rho=1+\| f\|_{\infty}/l$. Then there exists an $\omega$-periodic solution of (\ref{H2})-(\ref{H4}). \end{prop} \paragraph{Proof:} Let $v\in B_{\rho}(0,CP)$ and $ \varphi\in B_{\rho}(0,\mathcal{C})$. From Theorem \ref{edo1} and the condition $\| (- A)^{1/2}G(t,\cdot )\| \leq (N_{1}r)^{1/2}<1 $, we know that there exist a local mild solution, $x(\cdot, \varphi)$, of \begin{equation}\label{maisuma} \begin{gathered} \frac{d}{dt} (y(t) + G(t,y_{t}))=Ay(t)+L(t,y_{t})+F_{1}(t,v_{t}),\quad t \geq 0, \\ y_{0}=\varphi. \end{gathered} \end{equation} We claim that $x(\cdot, \varphi)$ is bounded by $ \rho$ on $[0, a_{\varphi})$, where $[0,a_{\varphi})$ is the maximal interval of definition of $x(\cdot, \varphi)$. Assume that the claim is false and let $t_{0}=\inf\{t>0:\| x(t)\| > \rho \}.$ Clearly, $ x( t_{0})=\rho $. If $t_{0}> 1$, by employing the estimates in Ezzinbi \cite{EZZ}, pp. $ 227$, we have that \begin{eqnarray} \| x(t_{0})\| &\leq& \rho - l(1- e^{-t_{0}})+ a_{0}(t_{0})\| V( t_{0},x_{ t_{0}})\| \nonumber\\ && + a_{0}(t_{0})\int_{0}^{ t_{0}} \|AT(t_{0}-s)V(s,x_{s})\| ds \nonumber\\ & \leq & \rho - l (1- e^{-t_{0}}) + a_{0}(t_{0})(N_{1}r)^{1/2}\rho \nonumber\\ && + a_{0}(t_{0})(\frac{N_{1}r}{2})^{1/2}\int_{0}^{t_{0}-1} e^{-\frac{(t_{0}-s)}{2}}\| x_{s} \|_{\mathcal{C}}ds \nonumber\\&& + a_{0}(t_{0})\,(\frac{N_{1}r}{2})^{1/2} \int_{t_{0}-1}^{t_{0}} \frac{e^{-\frac{(t_{0}-s)}{2}}}{(t_{0}-s)^{1/2}}\| x_{s} \|_{\mathcal{C}}ds \nonumber\\& \leq& \rho - l( 1- e^{-t_{0}}) +a_{0}(t_{0})(N_{1}r)^{1/2}\rho +\,a_{0}(t_{0})(\frac{N_{1}r}{2})^{1/2}\rho(2 e^{-\frac{1}{2}} +2), \nonumber\end{eqnarray} thus, \begin{equation}\label{des1} \| x(t_{0})\| \leq \rho - \big( l-(N_{1}r)^{1/2}\rho ( 1 + \frac{2}{\sqrt{2}} ( e^{-\frac{1}{2}} +1 )\big) ( 1- e^{-t_{0}}). \end{equation} Similarly, if $t_{0}\in (0,1]$ \begin{equation}\label{des2} \| x(t_{0})\| \leq \rho - \big( l - (N_{1}r )^{1/2}\rho ( 1+ \frac{2}{\sqrt{2}} )\big) ( 1-e^{-t_{0}}) . \end{equation} From (\ref{ineq}), (\ref{des1}) and (\ref{des2}), it follows that $\| x(t_{0})\| <\rho $, which is a contradiction. Now we prove that $a_{\varphi}=\infty $. Assume that $a_{\varphi}<\infty $ and let $N_{2}$ be the number $N_{2}=\rho ( | a_{1}|_{\infty}+| a_{2}|_{\infty}k )+ \| f\|_{\infty}$. For $\epsilon>0$, we fix $0<\delta <\frac{a_{\varphi}}{2}$ such that $$\| T(s)x-T(s')x\| <\epsilon,\quad x\in T(\epsilon)B_{2\rho+N_{2}}[0,X],$$ when $ | s-s'| <\delta$ and $s,s'\in [0,a_{\varphi}]$. Let $\frac{a_{\varphi}}{2}