\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 31, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/31\hfil Solutions in several types of periodicity] {Solutions in several types of periodicity for partial neutral integro-differential equation} \author[J. C. dos Santos, S. M. Guzzo\hfil EJDE-2013/31\hfilneg] {Jos\'e Paulo C. dos Santos, Sandro M. Guzzo} % in alphabetical order \address{Jos\'e Paulo C. dos Santos \newline Instituto de Ci\^encias Exatas - Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva, 700, 37130-000 Alfenas - MG, Brazil} \email{zepaulo@unifal-mg.edu.br} \address{Sandro M. Guzzo \newline Universidade Estadual do Oeste do Paran\'a - UNIOESTE, Colegiado do curso de Mate\-m\'atica, Rua Universit\'aria, 2069. Caixa Postal 711, 85819-110 Cascavel - PR, Brazil} \email{smguzzo@gmail.com} \thanks{Submitted August 4, 2012. Published January 28, 2013.} \thanks{J. C. dos Santos was supported by grant CEX-APQ-00476-09 from FAPEMIG/Brazil.} \subjclass[2000]{45K05, 34K40, 34K14, 45N05} \keywords{Integro-differential equations; neutral differential equations; \hfill\break\indent asymptotically almost periodic; asymptotic compact almost automorphic; \hfill\break\indent $S$-asymptotically $\omega$-periodic; asymptotically $\omega$-periodic} \begin{abstract} In this article we study the existence of mild solutions in several types of periodicity for partial neutral integro-differential equations with unbounded delays. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article we study the existence of several types of mild solutions for the partial neutral integro-differential equation \begin{gather} \frac{d}{dt} (x(t) + f(t,x_t)) = Ax(t)+ \int_{0}^{t} B(t-s)x(s) ds + g(t,x_t), \label{eqi1} \\ x_0 = \varphi \in \mathcal{B}, \label{eqi2} \end{gather} where $ A:D(A)\subset X\to X$ and $B(t):D(B(t))\subset X\to X$, $t\geq 0$, are closed linear operators; $(X, \| \cdot \|) $ is a Banach space; the history $x_{t}:(-\infty,0]\to X$, $x_{t}(\theta)=x(t+\theta)$, belongs to an abstract phase space ${\mathcal{B}}$ defined axiomatically, and $ f,g:I\times \mathcal{B} \to X $ are appropriated functions. The literature relative to ordinary neutral differential equations is very extensive, thus we suggest the Hale and Lunel book \cite{HA1} concerning this matter. Referring to partial neutral functional differential equations, we cite the pioneer articles Hale \cite{hale2} and Wu \cite{wu3,wu4,wu5} for finite delay equations, Hern\'{a}ndez and Henriquez \cite{HH2,HH1}, Hern\'{a}ndez \cite{HH3} for the unbounded delay, Hern\'{a}ndez and dos Santos \cite{HJ2} and Henr\'{\i}quez et al. \cite{HJ1,Joseh3} and Dos Santos et al. \cite{Dos,Dos1,DosSantos} for partial neutral integro-differential equations with unbounded delay. The existence of almost automorphic, asymptotically almost automorphic, almost periodic, asymptotically almost periodic, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to differential equations is among the most attractive topics in mathematical analysis due to their possible applications in areas such as physics, economics, mathematical biology, engineering, etc. (cf. \cite{Ravi, B, BD,Caiedo,lizama,HH11,TG,DDD,d1,Hui,Dos1,EN, H2008,HJEJ, g2,g3,veech1,zaid1,zaki1}). The concept of asymptotically almost automorphic, was introduced in the literature in the early eighties by N'Gu\'er\'ekata \cite{gaston}. However, the literature concerning $S$-asymptotically $\omega$-periodic functions with values in Banach spaces is recent (cf \cite{Caiedo,Cue1,Cue2,M1,H2008}). The existence of asymptotically almost automorpic, $S$-asymptotically $\omega$-periodic functions and asymptotically $\omega$-periodic for the partial neutral system \eqref{eqi1}-\eqref{eqi2} is an untreated topic in the literature and this fact is the main motivation of the present work. This paper is organized in four sections. In Section \ref{preliminaries} we mention a few results and notations related with resolvent of operators and of several types of periodicity. In Section \ref{ExistenceResults} we study the existence of several types of periodicity mild solutions to the partial neutral system \eqref{eqi1}-\eqref{eqi2}. In Section 4, we discuss the existence and uniqueness of several types of periodicity solution to a concrete partial neutral integro-differential equation with delay, as an illustration to our abstract results. \section{Preliminaries}\label{preliminaries} Let $(Z,\|\cdot\|_{Z})$ and $(W,\|\cdot\|_{W})$ be Banach spaces. We denote by $\mathcal{L}(Z,W)$ the space of bounded linear operators from $Z$ into $W$ endowed with norm of operators, and we write simply $\mathcal{L}(Z)$ when $Z = W$. By $\mathbf{R}(Q)$ we denote the range of a map $Q$ and for a closed linear operator $P: D(P)\subseteq Z \to W$, the notation $[D(P)]$ represents the domain of $P$ endowed with the graph norm, $\|z\|_{1} = \|z\|_{Z} + \|P z\|_{W} $, $z \in D(P)$. In the case $Z=W$, the notation $\rho(P) $ stands for the resolvent set of $P$, and $R(\lambda, P) = (\lambda I - P)^{-1}$ is the resolvent operator of $P$. Furthermore, for appropriate functions $K :[0,\infty)\to Z$ and $S: [0,\infty)\to \mathcal{L}(Z,W)$, the notation $\widehat{K}$ denotes the Laplace transform of $K$, and $S*K$ the convolution between $S$ and $K$, which is defined by $S*K(t)=\int_{0}^{t} S(t-s) K(s) d s$. The notation, $B_{r}(x,Z)$ stands for the closed ball with center at $x$ and radius $r>0$ in $Z$. As usual, $C_0([0, \infty),Z)$ represents the sub-space of $C_b([0, \infty),Z)$ formed by the functions which vanish at infinity and $C_{\omega}([0,\infty),X)$ denote the spaces $C_{\omega}([0,\infty),X)= \{x \in C_{b}([0,\infty),X ): x \,\,\text{is $\omega$-periodic } \}$. If $k:\mathbb{R} \to W$, we denote $\|k \|_{W,\infty}=\sup_{s\in \mathbb{R}}\| k(s)\|_{W}$ or if $k:[0,\infty) \to W$, we denote $\|k \|_{W,\infty}=\sup_{s\in [0,\infty)}\| k(s)\|_{W}$. In this work we will employ an axiomatic definition of the phase space $\mathcal{B}$ similar at those in \cite{HMN}. More precisely, $\mathcal{B}$ will denote a vector space of functions defined from $(-\infty,0]$ into $X$ endowed with a semi-norm denoted by $\|\cdot\|_{\mathcal{B}}$ and such that the following axioms hold: \begin{itemize} \item[(A1)] If $x:(-\infty,\sigma+b)\to X$ with $b>0$ is continuous on $[\sigma,\sigma +b)$ and $x_{\sigma}\in \mathcal{B}$, then for each $t\in [\sigma,\sigma+b)$ 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, and $K,M:[0,\infty) \mapsto [1,\infty)$ are functions such that $K(\cdot)$ and $M(\cdot)$ are respectively continuous and locally bounded, and $H,K,M$ are independent of $x(\cdot)$. \item[(A2)] If $x(\cdot)$ is a function as in (A1), then $x_{t}$ is a $\mathcal{B}$-valued continuous function on $[\sigma,\sigma+b)$. \item[(B1)] The space $\mathcal{B}$ is complete. \item[(C1)] If $(\varphi^n )_{n\in\mathbb{N}}$ is a sequence in $C_{b}((-\infty,0],X) $ formed by functions with compact support such that { $\varphi^{n}\to \varphi$} uniformly on compact, then $\varphi \in \mathcal{B}$ and $\|\varphi^n - \varphi\|_{\mathcal{B}} \to 0$ as $n \to \infty$. \end{itemize} \begin{definition} \rm Let $S(t):\mathcal{B} \to \mathcal{B}$ be the $C_{0}$-semigroup defined by $S(t)\varphi (\theta)= \varphi(0)$ on $[-t,0]$ and $ S(t)\varphi (\theta)=\varphi( t + \theta )$ on $(-\infty, -t]$. The phase space $\mathcal{B}$ is called a fading memory if $\| S(t)\varphi\|_{\mathcal{B}}\to 0$ as $t \to \infty$ for each $\varphi \in \mathcal{B}$ with $\varphi (0)=0$. \end{definition} \begin{remark} \label{rem2} \rm In this work we assume there exists positive $\mathfrak{K}$ such that $$ \max\{K(t), M(t)\}\leq \mathfrak{K} $$ for each $t\geq 0$. Observe that this condition is verified, for example, if $\mathcal{B}$ is a fading memory, see \cite[Proposition 7.1.5]{HMN}. \end{remark} \begin{example} \label{example1} \rm The phase space $ C_{r} \times L^{p}(\rho,X)$. Let $r \geq 0$, $1 \leq p < \infty$ and let $\rho:(-\infty,-r] \to \mathbb{R}$ be a nonnegative measurable function which satisfies the conditions (g-5), (g-6) in the terminology of \cite{HMN}. Briefly, this means that $\rho$ is locally integrable and there exists a non-negative, locally bounded function $\gamma$ on $(- \infty, 0]$ such that $\rho(\xi+\theta) \leq \gamma(\xi) \rho(\theta)$, 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. The space $C_{r} \times L^{p}(\rho,X)$ consists of all classes of functions $\varphi: (- \infty , 0] \to X $ such that $ \varphi $ is continuous on $[- r,0]$, Lebesgue-measurable, and $\rho \|\varphi\|^{p} $ is Lebesgue integrable on $ (- \infty , -r )$. The seminorm in $ C_{r}\times L^{p}(\rho,X)$ is defined by $$ \| \varphi \|_{\mathcal{B}} := \sup \{ \|\varphi(\theta)\| : -r \leq \theta \leq 0 \} +\Big( \int_{- \infty }^{-r} \rho(\theta ) \|\varphi(\theta)\|^{p} d\theta \Big)^{1/p}. $$ The space $\mathcal{B} = C_{r} \times L^{p}(\rho;X) $ satisfies axioms (A1), (A2), (B1). Moreover, when $ r=0$ and $p=2$, we can take $H = 1$, $M(t) = \gamma(-t)^{1/2}$ and ${ K(t) = 1 + (\int_{-t}^{0} \rho(\theta) \,d \theta )^{1/2}}$, for $t \geq 0$ and $$ \mathfrak{K} = \Big(\sup_{s\leq 0} |\gamma(s)^{1/2}|+ \Big(1+ (\int_{-\infty}^0 \rho(\theta) d\theta)^{1/2}\Big)\Big). $$ See \cite[Theorem 1.3.8]{HMN} for details. \end{example} For better comprehension of the subject we shall introduce the following definitions, hypothesis and results. Throughout the rest of the paper we always assume that the abstract integro-differential problem \begin{gather} \frac{dx(t)}{dt} = Ax(t)+ \int_{0}^{t} B(t-s)x(s) \, ds, \label{eqa1} \\ x(0) = x \in X. \label{eqa2} \end{gather} \begin{definition} \label{D3} \rm A one-parameter family of bounded linear operators $(\mathcal{R}(t))_{t\geq 0} $ on $X$ is called a resolvent operator of \eqref{eqa1}-\eqref{eqa2} if the following conditions are satisifed. \begin{itemize} \item[(a)] Function $\mathcal{R}(\cdot): [0, \infty) \to \mathcal{L}(X)$ is strongly continuous and $\mathcal{R}(0)x=x$ for all $x\in X$. \item[(b)] For $x \in D(A) $, $\mathcal{R}(\cdot)x \in C([0,\infty), [D(A)]) \cap C^{1}([0,\infty),X)$, and \begin{gather}\label{eqrp1} \frac{d \mathcal{R}(t)x}{dt} = A \mathcal{R}(t)x + \int_{0}^{t} B(t-s) \mathcal{R} (s) x d s, \\ \label{eqrp2} \frac{d \mathcal{R}(t)x}{dt} = \mathcal{R}(t) A x + \int_{0}^{t} \mathcal{R}(t-s) B(s)x d s, \end{gather} for every $t\geq 0 $, \item[(c)] There exists constants $M>0,\delta$ such that $\|\mathcal{R}(t)\| \leq M e^{\delta t}$ for every $ t\geq 0$. \end{itemize} \end{definition} \begin{definition} \label{def3} \rm A resolvent operator $(\mathcal{R}(t))_{t\geq 0}$ of \eqref{eqa1}-\eqref{eqa2} is called exponentially stable if there exists positive constants $M, \beta$ such that $\| \mathcal{R}(t) \| \leq M e^{- \beta t}$. \end{definition} In this work we assume that the following conditions are satisfied: \begin{itemize} \item[(H1)] Operator $A : D(A)\subseteq X \to X $ is the infinitesimal generator of an analytic semigroup $(T(t))_{t\geq 0}$ on $X$, and there are constants $M_{0} >0, \omega \in \mathbb{R}$ and $\vartheta \in (\pi/2,\pi) $ such that $ \rho(A) \supseteq \Lambda_{ \omega, \vartheta } = \{ \lambda \in \mathbb{C}: \lambda \neq \omega, \, |\arg(\lambda - \omega)| < \vartheta \} $ and $\| R(\lambda,A) \| \leq \dfrac{M_{0}}{|\lambda - \omega|} $ for all $\lambda \in \Lambda_{\omega, \vartheta}$. \item[(H2) ] For all $t\geq 0$, $B(t):D(B(t)) \subseteq X \to X $ is a closed linear operator, $D(A) \subseteq D(B(t)) $ and $B(\cdot)x $ is strongly measurable on $(0,\infty) $ for each $x \in D(A)$. There exists $b(\cdot) \in L^{1}([0,\infty)) $ such that $\widehat{b}(\lambda)$ exists for $\operatorname{Re}(\lambda) > 0$ and $\| B(t) x \| \leq b(t) \|x\|_{1} $ for all $t>0 $ and $x \in D(A)$. Moreover, the operator valued function $\widehat{B} : \Lambda_{\omega,\pi/2} \to \mathcal{L}([D(A)],X)$ has an analytical extension (still denoted by $\widehat{B}$) to $\Lambda_{\omega, \vartheta}$ such that $\| \widehat{B}(\lambda) x \| \leq \|\widehat{B}(\lambda)\| \, \|x\|_{1}$ for all $x \in D(A)$, and $\|\widehat{B}(\lambda)\| = O(\frac{1}{|\lambda|})$ as $| \lambda | \to \infty$. \item[(H3)] There exists a subspace $D \subseteq D(A) $ dense in $[D(A)] $ and positive constants $C_{i}$, $i=1,2$, such that $A(D) \subseteq D(A) $, $\widehat{B}(\lambda)(D) \subseteq D(A)$, $\|A \widehat{B}(\lambda) x \| \leq C_{1} \|x\|$ for every $x\in D$ and all $ \lambda \in \Lambda_{\omega, \vartheta}$. \end{itemize} For $r>0$, $ \theta \in (\frac{\pi}{2}, \vartheta )$ and $w \in \mathbb{R}$, set $$ \Lambda_{r, \omega, \theta}= \{ \lambda \in \mathbb{C}: \lambda \neq \omega, |\lambda| >r, \,|\arg(\lambda-\omega)| < \theta \} , $$ and $\omega+\Gamma^{i}_{r,\theta }$, $i=1,2,3$, the paths \begin{gather*} \omega+\Gamma^{1}_{r,\theta } =\{\omega + t e^{i\theta}: t \geq r \}, \\ \omega+\Gamma^{2}_{r,\theta } =\{\omega + re^{i\xi}: -\theta \leq \xi \leq \theta \}, \\ \omega+\Gamma^{3}_{r,\theta } =\{\omega + t e^{-i\theta}: t \geq r \}, \end{gather*} with $\omega+\Gamma_{r,\theta }=\bigcup_{i=1}^{3}\omega+\Gamma^{i}_{r,\theta }$ oriented counterclockwise. In addition, $\Psi(G)$ is the set $$ \Psi(G) = \{ \lambda \in \mathbb{C}: G(\lambda):=(\lambda I- A - \widehat{B}(\lambda) )^{-1} \in \mathcal{L}(X)\}. $$ The next results establish that the operator family $(\mathcal{R}(t))_{t\geq 0} $ defined by \begin{equation}\label{defnresolv1} \mathcal{R}(t) = \begin{cases} \frac{1}{2\pi i } \int_{\omega+\Gamma_{r,\theta}} e^{\lambda t} G(\lambda) d\lambda, & t>0, \\ I, & t=0. \end{cases} \end{equation} is an exponentially stable resolvent operator for \eqref{eqa1}-\eqref{eqa2}. \begin{theorem}[{\cite[Corollary 3.1]{Dos1}}] \label{constnresolvente} Suppose that conditions {\rm (H1)--(H3)} are satisfied. Then, the function $\mathcal{R}(\cdot) $ is a resolvent operator for system \eqref{eqa1}-\eqref{eqa2}. If $ \omega+r < 0$, the function $\mathcal{R}(\cdot)$ is an exponentially stable resolvent operator for system \eqref{eqa1}-\eqref{eqa2}. \end{theorem} In the next result we denote by $(-A)^{\vartheta}$ the fractional power of the operator $(-A)$, (see \cite{PA} for details). \begin{theorem}[{\cite[Corollary 3.2]{Dos1}}] \label{estpr1} Suppose that conditions {\rm (H1)--(H3)} are satisfied. Then there exists a positive number $ C$ such that \begin{equation} \label{despderr} \| (-A)^{\vartheta}\mathcal{R}(t) \| \leq \begin{cases} C e^{(r+\omega) t}, & t\geq1, \\ Ce^{(r+\omega)t}t^{-\vartheta}, & t \in (0,1), \end{cases} \end{equation} for all $ \vartheta \in (0,1)$. If $ \omega +r< 0$ and $\vartheta \in(0,1)$, then there exists $\phi \in L^1([0,\infty))$ such that \begin{equation} \label{estnec} \|(-A)^{\vartheta}\mathcal{R}(t)\| \leq \phi (t). \end{equation} \end{theorem} In the remaining of this section we discuss the existence of solutions to \begin{gather} \label{eqanhp1} \frac{dx(t)}{dt} = Ax(t)+ \int_{0}^{t} B(t-s)x(s) \ d s + f(t), \quad t \in [0,a], \\ \label{eqanhp2} x(0) = z \in X, \end{gather} where $f\in L^{1}([0,a], X) $. In the sequel, $\mathcal{R}(\cdot)$ is the operator function defined by \eqref{defnresolv1}. We begin by introducing the following concept of classical solution. \begin{definition} \label{D1} \rm A function $x : [0,b] \to X $, $00} \big[\frac{ r}{2 \mathfrak{K}}- L_f(2\mathfrak{K}r )r \mu - \frac{M}{\beta}L_g(2\mathfrak{K}r )r\big] \\ &\geq \frac{1}{2 \mathfrak{K}} (M \|\varphi\|_{\mathcal{B}} + M \|f(0,\varphi)\| + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y \mu + \frac{M}{\beta}\sup_{t \in [0,\infty)}\|g(t,0)\| ), \end{align*} where $\mu =( \|i_c\|_{\mathcal{L}(Y,X)} + \|\phi\|_{L^1} + \frac{M}{\beta} \|b\|_{L^1} )$. \end{itemize} Motivated by the theory of resolvent operator, we introduce the following concept of mild solution for \eqref{eqi1}-\eqref{eqi2}. \begin{definition} \rm A function $u:(-\infty, b] \to X$, $0 0$ there exists a relatively dense subset of $\mathbb{R}$, denoted by $\mathcal{H}( \varepsilon, f,Z)$, such that $$ \| f(t+ \xi ) - f(t) \|_{Z} < \varepsilon, \quad t \in\mathbb{R},\, \xi \in \mathcal{H}( \varepsilon, f,Z).$$ \end{definition} \begin{definition} \rm A function $f\in C([0, \infty),Z) $ is asymptotically almost periodic (a.a.p.) if there exists an almost periodic function $g(\cdot)$ and $w \in C_0([0, \infty),Z)$ such that $f(\cdot) = g(\cdot) + w(\cdot)$. \end{definition} In this paper, $AP(Z)$ and $AAP(Z)$ are the spaces \begin{gather*} AP(Z)=\{f\in C(\mathbb{R},Z): f \text{ is a.p. }\}, \\ AAP(Z)= \{f\in C([0,\infty ),Z): f \text{ is a.a.p. }\}, \end{gather*} endowed with the norm of the uniform convergence. We know from the result in \cite{z5} that $AP(Z)$ and $AAP(Z)$ are Banach spaces. \begin{definition} \label{def4} \rm A function $u\in C_{b}([0, \infty),X) $ is said $S$-asymptotically $\omega$-periodic if \[ \lim_{t \to \infty} (u(t+\omega )-u(t) )=0. \] \end{definition} In the rest of this paper, the notation $SAP_{\omega}(X)$ stands for the space $$ SAP_{\omega}(X) = \{f\in C_b(\mathbb{R},X): f \text{ is $S$-asymptotically $\omega$-periodic }\}, $$ endowed with the norm of the uniform convergence. It is clear that $SAP_{\omega}(X)$ is a Banach space. \begin{definition} \label{def7}\rm A continuous function $f:[0,\infty)\times Z \to W$ is said uniformly $S$-asymptotically $\omega$-periodic on bounded sets if $f(\cdot, x)$ is bounded for each $x \in Z$, and for every $\varepsilon >0$ and for all bounded set $K \subseteq Z $, there exists $L(K,\varepsilon) \geq 0$ such that $ \| f(t,x)-f(t+\omega ,x)\|_{W}\leq \varepsilon$ for every $t\geq L(K,\varepsilon) $ and all $x\in K$. \end{definition} \begin{definition} \label{def8} \rm A continuous function $f:[0,\infty) \times Z\to W$ is said asymptotically uniformly continuous on bounded sets, if for every $\varepsilon>0$ and for all bounded set $K\subseteq Z $ there exist constants $L(K,\varepsilon) \geq 0$ and $\delta = \delta(K,\varepsilon) > 0$ such that $\|f(t,x) - f(t,y)\|_{W} \leq \varepsilon$ for all $t \geq L(K,\varepsilon)$ and every $x,y \in K$ with $\| x-y \|_{Z} \leq \delta$. \end{definition} \begin{lemma}[{\cite[Lemma 4.1]{M1}}] \label{composition} Assume that $f:[0,\infty)\times Z\to W$ is a function uniformly $S$-asymptotically $\omega$-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. Let $u\in SAP_{\omega}(Z)$, then the function $\theta: \mathbb{R} \to W$ defined by $\theta(t) = f(t,u(t))$ is $S$-asymptotically $\omega$-periodic. \end{lemma} By using a similar procedure to the proof of the \cite[Lemma 3.5]{H2008}, we prove the next result. \begin{lemma} \label{lema4H} Suppose that condition {\rm (P1)(b)} holds and $ f\in SAP_{\omega}(X)$. Let $F:[0,\infty)\to X$ be the function defined by $$ F(t):=\int_{0}^{t} \mathcal{R}(t-s) f(s)d s. $$ Then $F \in SAP_{\omega}(X)$. \end{lemma} \begin{lemma}[{\cite[Lemma 2.10]{H2008}}] \label{lem8H} Assume that $\mathcal{B}$ is a fading memory space and $u \in C(\mathbb{R},X)$ is such that $u_0 \in \mathcal{B}$ and $u|_{[0,\infty)} \in SAP_{\omega}(X)$, then $ t \mapsto u_t \in SAP_{\omega}(\mathcal{B})$. \end{lemma} \begin{definition} \label{def6}\rm A function $u\in C_{b}([0, \infty),X) $ is called asymptotically $\omega$-periodic if there exists an $\omega$-periodic function $v$ and $w \in C_{0}([0, \infty), X)$ such that $u = v + w$. \end{definition} \begin{remark} \rm In \cite{H2008} the authors have shown that the set of the asymptotically $\omega$-periodic functions is properly contained in $SAP_{\omega}(W)$. \end{remark} \begin{lemma}[{\cite[Remark 3.13]{H2008}}] \label{caiedo0} If $u\in C_{b}([0,\infty), X)$ is a function such that $\lim_{t\to \infty}(u(t +n \omega)-u(t))=0$, uniformly for $n \in \mathbb{N}$, then $u(\cdot)$ is asymptotically $\omega$-periodic. \end{lemma} In the rest of this paper, $S_{\omega}(X)$ stands for the space $$ S_{\omega}(X) = \{f\in C_b([0,\infty),X): \lim_{t\to \infty}f(t +n \omega)-f(t)=0, \text{ uniformly for } n \in \mathbb{N} \}, $$ endowed with the norm of the uniform convergence. \begin{lemma}[{\cite[Lemma 2.3]{Caiedo}}] \label{caiedo1} Let $f : [0,\infty)\times Z \to W$ be asymptotically uniformly continuous on bounded sets. Suppose that for all bounded subset $K \subset Z$, the set $\{f(t,z)\geq 0, z \in K \}$ is bounded and $ \lim_{t\to \infty} \| f(t+n\omega, z) - f(t,z) \| = 0$, uniformly for $z \in K$ and $n \in \mathbb{N}$. If $ u \in S_{\omega}(Z)$, then $ f(\cdot, u(\cdot)) \in S_{\omega}(W)$. \end{lemma} \begin{lemma}\cite[Lemma 3.7]{Caiedo} \label{caiedo2} Suppose that condition {\rm (P1)(b)} holds and $f \in S_{\omega}(X)$. If $F$ is the function defined by $F(t):=\int_{0}^t \mathcal{R}(t-s) f(s)ds$, $t\geq 0$, then $F \in S_{\omega}(X)$. \end{lemma} We now introduce some notion of asymptotically almost automorphic. \begin{definition} \rm A function $f\in C( \mathbb{R} , X)$ is said to be almost automorphic if for every sequence of real numbers $(s'_{n})_{n \in \mathbb{N}}$, there exists a subsequence $(s_{n})_{n \in \mathbb{N}} \subset (s'_{n})_{n \in \mathbb{N}}$ such that $$ g(t):= \lim_{n \to \infty} f(t+s_{n}) $$ is well defined for each $t \in \mathbb{R}$, and $$ f(t) = \lim_{n \to \infty} g(t-s_{n}) $$ for all $t \in \mathbb{R}$. \end{definition} It is well known that the range of an almost automorphic function is relatively compact on $X$, and hence it is bounded. Moreover, the space of all almost automorphic functions, denoted by $AA(X)$, endowed with the norm of the uniform convergence is a Banach space \cite{g2}. \begin{definition} \rm A function $f\in C( [0, \infty) , Z)$ is said to be asymptotically almost automorphic if it can be written as $f=g+h$ where $g \in AA(Z)$ and $ h \in C_0([0,\infty),Z)$. Denote by $AAA(Z)$ the set of all such functions. \end{definition} \begin{definition} \label{def1} \rm A function $f\in C( \mathbb{R}, Z)$ is said to be compact almost automorphic if for every sequence of real numbers $(\sigma_{n})_{n \in \mathbb{N}}$ there exists a subsequence $(s_{n})_{n \in \mathbb{N}} \subset (\sigma_{n})_{n \in \mathbb{N}}$ such that \begin{gather*} g(t): = \lim_{n \to \infty} f(t+s_{n}),\\ f(t)= \lim_{n \to \infty} g(t-s_{n}) \end{gather*} uniformly on compact subsets of $\mathbb{R}$. The collection of those functions will be denoted by $AA_c(Z)$. \end{definition} \begin{definition} \label{def2} \rm A function $f\in C( \mathbb{R} \times Z, W)$ is said to be compact almost automorphic in $t \in \mathbb{R}$, if for every sequence of real numbers $(\sigma_{n})_{n \in \mathbb{N}}$ there exists a subsequence $(s_{n})_{n \in \mathbb{N}} \subset (\sigma_{n})_{n \in \mathbb{N}}$ such that \begin{gather*} g(t,z):= \lim_{n \to \infty} f(t+s_{n}, z), \\ f(t,z) = \lim_{n \to \infty} g(t-s_{n},z), \end{gather*} where the limits are uniform on compact subset of $\mathbb{R}$, for each $z \in Z$. The space of such functions will be denoted by $AA_c(Z, W)$. \end{definition} \begin{definition} \rm A continuous function $f\in C( [0, \infty), Z)$ is said to be compact asymptotically almost automorphic if it can be written as $f=g+h$ where $g \in AA_c(Z)$ and $ h \in C_0(\mathbb{R}^{+},Z)$. Denote by $AAA_c(Z)$ the set of all such functions. \end{definition} \begin{definition} \rm Let $K\subset Z$ and $I \subset \mathbb{R}$. Let $C_{K}(I\times Z, W)$ denote the collection of functions $f:I\times Z\to W$ such that $f(t, \cdot)$ is uniformly continuous on $K$ for every $t \in I \subseteq \mathbb{R}$. \end{definition} \begin{definition} \rm A function $f\in C([0, \infty) \times Z,W)$ is said to be compact asymptotically almost automorphic if it can be written as $f=g+h, $ where $g\in AA_c(Z,W)$ and $h \in C_0([0, \infty)\times Z, W) $. Denote by $AAA_c(Z,W)$ the set of all such functions. \end{definition} \begin{lemma}[{\cite[Lemma 3.3]{Jose3}}] \label{lemaJ1} Let $u \in AAA_c(Z)$ and $f\in AAA_c(Z, W)\cap C_{R}(\mathbb{R}\times Z,W)$, where $R=\overline{\{ u(t): t \in \mathbb{R} \}}$. Then the function $\Phi: \mathbb{R} \to W$ defined by $\Phi(t) = f (t , u(t)) \in AAA_c(W)$. \end{lemma} \begin{lemma}[{\cite[Lemma 3.4]{Jose3}}] \label{lemaJ2} %\label{Jose3} Suppose that condition (P1)-$(b) $ holds and $f \in AAA_c(X)$. If $F$ is the function defined by $$ F(t):=\int_{0}^t \mathcal{R}(t-s) f(s)ds, \quad t\geq 0, $$ then $F \in AAA_c(X)$. \end{lemma} \begin{lemma}[{\cite[Lemma 3.5]{Jose3}}] \label{lemaJ3} If $u\in AA_c(X)$, then the function $s \mapsto u_{s}$ belongs to $AA_c(\mathcal{B})$. Moreover, if $\mathcal{B}$ is a fading memory space and $u \in C(\mathbb{R},X)$ is such that $u_0 \in \mathcal{B}$ and $u|_{[0,\infty)} \in AAA_c(X)$, then $ t \mapsto u_t \in AAA_c(\mathcal{B})$. \end{lemma} \section{Several types of periodicity of mild solutions} \label{ExistenceResults} In this section we establish the existence of several type of periodicity for solutions to partial neutral integro-differential equations system \eqref{eqi1}-\eqref{eqi2}. For that, we need to introduce a few preliminaries and important results. Following, we consider the problem of the existence of compact asymptotically almost automorphic solutions. In the following, we let $\mathcal{A}(Z)$ stands for one of the spaces $AAA_{c}(Z)$, $SAP_{\omega}(Z)$ or $S_{\omega}(Z)$. \begin{lemma} \label{apide1} Assume the condition {\rm(P1)} is fulfilled. Let $u \in \mathcal{A}(Y)$ and $G(\cdot ):[0,\infty )\to X$ be the function defined by $$ G(t)=\int_{0}^t \mathcal{R}(t-s) \int_0^s B(s-\tau)u(\tau) \ d\tau ds,\quad t\geq 0. $$ Then $ G(\cdot) \in \mathcal{A}(X)$. \end{lemma} \begin{proof} First we consider the $AAA_{c}(Y)$ case. By Lemma \ref{lemaJ2} is sufficient to prove that $H(t)=\int_0^t B(t-s) u(s) ds \in AAA_c(Y)$. Suppose $u=k+h$ where $k \in AA_c(Y)$ and $ h \in C_0([0,\infty),Y)$. Then \begin{align*} H(t)&= \int_{-\infty}^t B(t-s) k(s)ds - \int_{-\infty}^0 B(t-s) k(s)ds + \int_{0}^t B(t-s) h(s)ds \\ &= w(t) + q(t), \end{align*} where \begin{gather*} w(t) = \int_{-\infty}^t B(t-s) k(s)ds, \\ q(t) = \int_{0}^t B(t-s) h(s)ds - \int_{-\infty}^0 B(t-s) k(s)ds. \end{gather*} For a given sequence $(\sigma_{n})_{n \in \mathbb{N}}$ of real numbers, fix a subsequence $(s_{n})_{n \in \mathbb{N}}$, and a continuous functions $v\in C_b(\mathbb{R}, Y)$ such that $k(t+s_{n})$ converges to $v(t)$ in $Y$, and $v(t-s_{n})$ converges to $k(t)$ in $Y$, uniformly on compact sets of $\mathbb{R}$. From the Bochner's criterion related to integrable functions and the estimate \begin{eqnarray}\label{est1} \|B(t-s)k(s)\| = \| B(t-s)\|_{\mathcal{L}({Y,X})}\|k(s) \|_{{Y}} \leq b(t-s) \|k(s) \|_{{Y}} \label{des3} \end{eqnarray} it follows that the function $s \mapsto B(t -s ) k(s) $ is integrable over $(-\infty, t)$ for each $t\in \mathbb{R}$. Furthermore, since $$ w(t+s_{n}) = \int^t_{-\infty} B(t-s) k(s+s_{n})ds, \quad t\in \mathbb{R}, \; n \in \mathbb{N}, $$ using the estimate \eqref{est1} and the Lebesgue Dominated Convergence Theorem, it follows that $w(t+s_{n})$ converges to ${z(t)= \int^t_{-\infty}B(t-s) v(s) ds}$ for each $t\in \mathbb{R}$. The remaining task consists of showing that the convergence is uniform on all compact subsets of $\mathbb{R}$ and that $ q (\cdot) \in C_0([0,\infty), X)$. Let $K \subset \mathbb{R}$ be an arbitrary compact and let $\varepsilon >0$. Since $ h \in C_0([0,\infty), Y)$ and $ k(\cdot) \in AA_c(Y)$, there exists a constant $L$ and $N_{\varepsilon}$ such that $K \subset [\frac{-L}{2},\frac{L}{2}]$ with \begin{gather*} \int_{\frac{L}{2}}^{\infty} b(s) ds < \varepsilon, \\ \| k(s+s_{n})-v(s)\|_{Y} \leq \varepsilon, \quad n \geq N_{\varepsilon}, \; s \in [-L,L], \\ \| h(s) \|_{Y} \leq \varepsilon, \quad s \geq L. \end{gather*} For each $t\in K$, one has \begin{align*} &\|w(t+s_{n})-z(t)\| \\ & \leq \int^t_{-\infty} \|B(t-s)\|_{\mathcal{L}(Y,X)} \|k(s+s_{n})-v(s)\|_{Y} ds \\ & \leq \int^{-L}_{-\infty} b(t-s)\| k(s+s_{n})-v(s)\|_{Y} ds + \int^t_{-L} b(t-s) \| k(s+s_{n})-v(s)\|_{Y} ds \\ & \leq 2\|k\|_{Y,\infty} \int^{\infty}_{t+L} b(s) ds + \varepsilon\,\int^{\infty}_{0} b(s) ds \\ & \leq 2\|k\|_{Y,\infty} \int^{\infty}_{\frac{L}{2}} b(s) ds + \varepsilon\,\int^{\infty}_{0} b(s) ds \\ & \leq \varepsilon\Big(2\| k \|_{Y,\infty} +\int^{\infty}_{0} b(s) ds\Big), \end{align*} which proves that the convergence is uniform on $K$, from the fact that the last estimate is independent of $t\in K$. Proceeding as previously, one can similarly prove that $z(t-s_{n})$ converges to $w$ uniformly on all compact subsets of $\mathbb{R}$. Next, let us show that $q(\cdot) \in C_0([0,\infty), X)$. For all $t \geq 2L$ we obtain \begin{align*} \|q(t) \| & \leq \int_{-\infty}^0 \|B(t-s)\|_{\mathcal{L}(Y,X)} \|k(s)\|_{Y} ds + \int_0^t \|B(t-s)\|_{\mathcal{L}(Y,X)} \|h(s)\|_{Y}ds \\ & \leq \int_{-\infty}^0 b(t-s)\| k(s)\|_{Y} ds + \int_{t/2}^t b(t-s)\| h(s)\|_{Y} ds + \int_0^{t/2} b(t-s)\| h(s)\|_{Y} ds \\ & \leq \int_{\frac{L}{2}}^{\infty} b(s) ds\| k\|_{Y,\infty} + \varepsilon \int_{t/2}^t b(s) ds + \int_{\frac{L}{2}}^{\infty} b(s) ds\| h \|_{Y,\infty} \\ & \leq \varepsilon ( \|k\|_{Y,\infty} + \int^{\infty}_{0} b(s) ds + \| h \|_{Y,\infty} ). \end{align*} Now we consider the $SAP_{\omega}(Y)$ case. From Lemma \ref{lema4H} is sufficient to prove that $$ H(t) = \int_{0}^{t} B(t-s) u(s) ds $$ is $SAP_{\omega}(X)$. For all $t \geq 0$, \begin{align*} \|H(t)\| & \leq \int_{0}^{t} \|B(t-s) \|_{\mathcal{L}(Y,X)} \|u(s)\|_Y d\tau \\ & \leq \int_{0}^{t} b(t-s) \|u(s)\|_Y ds \\ & \leq \|u\|_{Y,\infty} \int_0^{\infty}b(s) ds. \end{align*} This shows that $H \in C_{b}([0,\infty), X)$. Furthermore, for $\omega \geq 0$, we have for $t \geq L > 0$, \begin{align*} &\| H(t+\omega) - H(t) \| \\ & = \| \int_{0}^{t+\omega} B(t+\omega-s) u(s) ds - \int_{0}^{t} B(t-s) u(s) ds \| \\ & \leq \int_{0}^{\omega} b(t+\omega-s) \|u(s)\|_Y ds + \| \int_{0}^{t} B(t-s) u(s+\omega) ds - \int_{0}^{t} B(t-s) u(s) ds \| \\ & \leq \|u\|_{Y,\infty} \int_{0}^{\omega} b(t+\omega-s) ds + \int_{0}^{t} \|B(t-s)( u(s+\omega) - u(s) )\|ds \\ & \leq \|u\|_{Y,\infty} \int_{0}^{\omega} b(t+\omega-s) ds + \int_{0}^{L} b(t-s) \|u(s+\omega) - u(s)\|_Y ds \\ & \quad + \int_{L}^{t} b(t-s) \|u(s+\omega) - u(s)\|_Y ds. \end{align*} For all $\varepsilon >0$, we choose $L$ sufficiently large such that $\|u(s+\omega)-u(s)\|_Y < \varepsilon$ for all $s \geq L$ and $\int_{L}^{\infty} b(s) ds < \varepsilon$. Hence, for $t \geq 2L$ we obtain \begin{align*} \| H(t+\omega) - H(t) \| & \leq \|u\|_{Y,\infty} \int_{t}^{t+\omega} b(s) ds + 2\|u\|_{Y,\infty} \int_{t-L}^{t} b(s) ds + \varepsilon \int_{0}^{t-L} b(s) ds\\ & \leq \|u\|_{Y,\infty} \varepsilon + 2\|u\|_{Y,\infty} \varepsilon + \varepsilon \int_{0}^{t-L} b(s) ds\\ & \leq \varepsilon \Big( 3\|u\|_{Y,\infty} + \int_{0}^{\infty} b(s) ds \Big). \end{align*} Finally, let us prove the $S_{\omega}(Y)$ case. From the Lemma \ref{caiedo2} is sufficient prove that $\lim_{t\to \infty} H(t + n\omega) - H(t) = 0$, uniformly in $n \in \mathbb{N}$, where $H(t)=\int_0^t B(t-s) u(s) ds$. For all $\varepsilon >0$, we choose $L$ sufficiently large such that $\|u(s+n \omega)-u(s)\|_Y < \varepsilon$ for all $s \geq L$ and $\int_{L}^{\infty} b(s) ds < \varepsilon$. Hence, for $t \geq 2L$ we obtain \begin{align*} &\| H(t+ n\omega) - H(t) \|\\ & \leq \| \int_0^{t+ n\omega} B(t+ n\omega-s) u(s) d s - \int_0^{t} B(t-s) u(s) ds \| \\ & \leq \|u\|_{Y,\infty} \int_{0}^{n\omega} b(t + n\omega - s) ds + \int_{0}^{L} b(t-s) \|u(s+n \omega)-u(s)\|_Y ds \\ & + \quad \int_{L}^{t} b(t- s) \|u(s+n \omega)-u(s)\|_Y ds \\ & \leq \|u\|_{Y,\infty} \int_{t}^{t +n\omega} b(s) ds + 2\|u\|_{Y,\infty} \int_{t-L}^{t} b(s) ds + \varepsilon \int_{0}^{\infty} b(s) ds\\ & \leq \varepsilon ( 3 \|u\|_{Y,\infty} + \int_{0}^{\infty} b(s) ds ) . \end{align*} This completes the proof. \end{proof} \begin{lemma} \label{apide2} Let condition {\rm (P1)(c)} hold and $u$ be a function in $\mathcal{A}(Y)$. If $I:[0,\infty )\to X$ is the function defined by $ I(t) = \int_{0}^t A\mathcal{R}(t-s)u(s)ds$, then $ I(\cdot)\in \mathcal{A}(X)$. \end{lemma} \begin{proof} All the $AAA_{c}(Y)$, $SAP_{\omega}(Y)$ and $S_{\omega}(Y)$ cases require small modifications in the proof of Lemma \ref{apide1}. \end{proof} \begin{theorem} \label{p1} Let $f \in AAA_c([0,\infty) \times\mathcal{B},Y) $ and $g \in AAA_c([0,\infty) \times\mathcal{B}, X)$. Assume that $\mathcal{B}$ is a fading memory space and {\rm (P1), (P2), (PF), (PG)} hold. Then there exists $ \varepsilon >0$ such that for each $\varphi \in B_{\varepsilon}(0,\mathcal{B})$ there exists a unique mild solution $u(\cdot,\varphi) \in AAA_c(X)$ of \eqref{eqi1}-\eqref{eqi2}. \end{theorem} \begin{proof} By the hypothesis there exists a constant $r>0$ such that \begin{align*} & [ r- L_f(2\mathfrak{K}r) 2 \mathfrak{K}r \mu - \frac{M}{\beta}L_g(2\mathfrak{K}r )2 \mathfrak{K}r] \\ &\geq M \|\varphi\|_{\mathcal{B}} + M \|f(0,\varphi)\| + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y \mu + \frac{M}{\beta}\sup_{t \in [0,\infty)}\|g(t,0)\|, \end{align*} where $ \mathfrak{K}$ is the constant introduced in Remark \ref{rem2}. We affirm that the assertion holds for $ \varepsilon \leq r$. Let $\varphi \in B_{\varepsilon}(0,\mathcal{B})$ and the space $$ \mathfrak{D}= \{ x \in AAA_c(X): x(0) = \varphi(0), \|x(t)\| \leq r,\, t\geq 0 \} $$ endowed with the metric $d(u,v)=\|u-v\|_{\infty} $, we define the operator $ \Gamma: \mathfrak{D}\to C([0,\infty);X )$ by \begin{align*} \Gamma u(t)&=\mathcal{R}(t)(\varphi(0)+f(0,\varphi))-f(t,\widetilde{u}_{t}) -\int_{0}^{t}A\mathcal{R}(t-s)f(s,\widetilde{u}_{s})ds \\ &\quad - \int_{0}^{t}\mathcal{R}(t-s) \int_{0}^{s}B(s- \xi) f(\xi, \widetilde{u}_{\xi}) d\xi ds+ \int_{0}^{t}\mathcal{R}(t-s)g(s,\widetilde{u}_{s})ds, \quad t\geq 0 \end{align*} where $ \widetilde{u}:\mathbb{R} \to X$ is the function defined by the relation $\widetilde{u}_{0}= \varphi$ and $\widetilde{u} = u$ on $[0, \infty)$. From the hypothesis (P1) (PF) and (PG) we obtain that $\Gamma u$ is well defined and that $\Gamma u \in C([0, \infty);X)$. Moreover, from Lemma \ref{lemaJ3}, we have that function $s \mapsto \widetilde{u}_s \in AAA_c(\mathcal{B})$. By Lemma \ref{lemaJ1}, we conclude that $s \mapsto f(s,\widetilde{u}_s) \in AAA_c([0,\infty),Y) $ and $s \mapsto g(s,\widetilde{u}_s) \in AAA_c([0,\infty),X)$. From Lemmas \ref{lemaJ2}, \ref{apide1}, \ref{apide2} and $\lim_{t \to \infty} \| \mathcal{R}(t)(\varphi(0)+f(0,\varphi)) \| =0$, we obtain that $ \Gamma u\in AAA_c(X)$. Next, we prove that $\Gamma(\cdot)$ is a contraction from $\mathfrak{D}$ into $\mathfrak{D}$. If $u\in \mathfrak{D}$ and $t\geq 0$, we obtain \begin{align*} %\label{iii} &\|\Gamma u(t)\|\\ &\leq \| \mathcal{R}(t)(\varphi(0)+ f(0,\varphi)) \| + \|i_c\|_{\mathcal{L}(Y,X)}(\|f(t,\widetilde{u}_{t}) - f(t,0)\|_Y + \|f(t,0)\|_Y) \\ &\quad + \int_{0}^{t}\|A\mathcal{R}(t-s)(f(s,\widetilde{u}_{s}) - f(s,0))\| ds + \int_{0}^{t} \|A\mathcal{R}(t-s)f(s,0)\| ds \\ &\quad + \int_{0}^{t}\|\mathcal{R}(t-s) \int_{0}^{s}B(s- \xi) (f(\xi, \widetilde{u}_{\xi}) - f(\xi,0)) d\xi \| ds \\ &\quad + \int_{0}^{t}\|\mathcal{R}(t-s) \int_{0}^{s}B(s- \xi)f(\xi,0) d\xi \| ds \\ &\quad + \int_{0}^{t}\|\mathcal{R}(t-s)(g(s,\widetilde{u}_{s}) - g(s,0))\| ds + \int_{0}^{t}\| \mathcal{R}(t-s) g(s,0) \| ds \\ &\leq M \|\varphi\|_{\mathcal{B}} + M \|f(0,\varphi)\| + \|i_c\|_{\mathcal{L}(Y,X)} ( L_f(\| \widetilde{u}_{t} \|_{\mathcal{B}})\|\widetilde{u}_{t}\|_{\mathcal{B}} + \sup_{t \in [0,\infty)} \|f(t,0)\|_Y ) \\ &\quad + \int_{0}^{t} \phi(t-s)L_f(\|\widetilde{u}_{s}\|_{\mathcal{B}}) \|\widetilde{u}_{s}\|_{\mathcal{B}} ds + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y \int_{0}^{t} \phi(s) ds \\ &\quad + \int_{0}^{t}Me^{-\beta(t-s)} \int_{0}^{s}b(s-\xi)L_f (\|\widetilde{u}_{\xi}\|_{\mathcal{B}}) \|\widetilde{u}_{\xi}\|_{\mathcal{B}} d\xi ds \\ &\quad + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y \int_{0}^{t} Me^{-\beta(t-s)} \int_{0}^{s}b(s-\xi) d\xi ds \\ &\quad + \int_{0}^{t} Me^{-\beta(t-s)} L_g(\|\widetilde{u}_{s}\|_{\mathcal{B}}) \|\widetilde{u}_{s}\|_{\mathcal{B}} ds + \sup_{t \in [0,\infty)}\|g(t,0)\| \int_{0}^{t}M e^{-\beta(t-s)} ds \\ &\leq M \|\varphi\|_{\mathcal{B}} + M \|f(0,\varphi)\| \\ &\quad + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y ( \|i_c\|_{\mathcal{L}(Y,X)} + \int_0^{\infty} \phi(s) ds + \frac{M}{\beta} \int_0^{\infty} b(s) ds ) \\ &\quad + \frac{M}{\beta}\sup_{t \in [0,\infty)}\|g(t,0)\| \\ &\quad + L_f(\|\widetilde{u}_{t}\|_{\mathcal{B}}) ( \|i_c\|_{\mathcal{L}(Y,X)} + \int_0^{\infty} \phi(s) ds + \frac{M}{\beta} \int_0^{\infty} b(s) ds ) \|\widetilde{u}_{t}\|_{\mathcal{B}} \\ &\quad + \frac{M}{\beta}L_g(\|\widetilde{u}_{t}\|_{\mathcal{B}}) \|\widetilde{u}_{t}\|_{\mathcal{B}} \\ &\leq M \|\varphi\|_{\mathcal{B}} + M \|f(0,\varphi)\| \\ &\quad + \sup_{t \in [0,\infty)}\|f(t,0)\|_Y( \|i_c\|_{\mathcal{L}(Y,X)} + \|\phi\|_{L^1} + \frac{M}{\beta} \|b\|_{L^1}) \\ &\quad + \frac{M}{\beta}\sup_{t \in [0,\infty)}\|g(t,0)\| + L_f(2\mathfrak{K}r) ( \|i_c\|_{\mathcal{L}(Y,X)} + \|\phi\|_{L^1} + \frac{M}{\beta} \|b\|_{L^1} )2\mathfrak{K}r \\ &\quad + \frac{M}{\beta}L_g(2\mathfrak{K}r)2 \mathfrak{K}r \leq r \end{align*} where the inequality $\|\widetilde{u}_t\| \leq 2\mathfrak{K}r $ has been used and $i_c:Y \to X$ represents the continuous inclusion of $Y$ on $X$. Thus, $ \Gamma(\mathfrak{D}) \subset \mathfrak{D}$. On the other hand, for $ u,v \in \mathfrak{D} $ we see that \begin{align*} &\| \Gamma u(t)-\Gamma v(t) \| \\ &\leq \|i_c\|_{\mathcal{L}(Y,X)} \| f(t,\widetilde{u}_t) - f(t, \widetilde{v}_t ) \|_Y \\ & \quad + \int_{0}^{t} \|A\mathcal{R}(t-s)\|_{\mathcal{L}(Y,X)} \|f(s,\widetilde{u}_s) - f(s, \widetilde{v}_s )\|_{Y} ds \\ & \quad + \int_{0}^{t} \| \mathcal{R}(t-s) \| ( \int_{0}^{s} \|B(s-\xi)\|_{\mathcal{L}(Y,X)}\| f(\xi,\widetilde{u}_\xi) - f(\xi,\widetilde{v}_\xi ) \|_{Y} d\xi ) ds \\ & \quad + \int_{0}^{t}\| \mathcal{R}(t-s) \| \| g(s,\widetilde{u}_s) - g(s,\widetilde{v}_s ) \| ds \\ &\leq \Big(L_{f}(2\mathfrak{K}r) \mathfrak{K} \mu + L_{g}(2\mathfrak{K}r) \mathfrak{K}\frac{M}{\beta} \Big) \|u-v\|_{\infty} \\ &\leq \Big(L_{f}(2\mathfrak{K}r)2\mathfrak{K} \mu + L_{g}(2\mathfrak{K}r)2\mathfrak{K}\frac{M}{\beta} \Big) \|u-v\|_{\infty}, \end{align*} we observe that $r- L_f(2\mathfrak{K}r) 2\mathfrak{K}r \mu - \frac{M}{\beta}L_g(2\mathfrak{K}r )2 \mathfrak{K}r >0$, this implies that \[ L_f(2\mathfrak{K}r)2 \mathfrak{K} \mu + \frac{M}{\beta}L_g(2\mathfrak{K}r )2 \mathfrak{K}<1, \] which shows that $ \Gamma(\cdot)$ is a contraction from $\mathfrak{D}$ into $\mathfrak{D}$. The assertion is now a consequence of the contraction mapping principle. The proof is complete. \end{proof} \begin{remark} \rm A similar result was obtained by Dos Santos et al. \cite{Dos1} for the existence of asymptotically almost periodic solutions for the system \eqref{eqi1}-\eqref{eqi2}. \end{remark} \begin{proposition} \label{p2} Let $f :[0,\infty) \times\mathcal{B}\to Y $ and $g:[0,\infty) \times\mathcal{B} \to X$ be uniformly $S$-asymptotically $\omega$-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. Assume that $\mathcal{B}$ is a fading memory space and {\rm (P1), (P2), (PF), (PG)} hold. Then there exists $ \varepsilon >0$ such that for each $\varphi \in B_{\varepsilon}(0,\mathcal{B})$ there exists a unique mild solution $u(\cdot,\varphi ) \in SAP_{\omega}(X)$ of \eqref{eqi1}-\eqref{eqi2} on $[0,\infty)$. \end{proposition} \begin{proof} Let the space $$ \mathfrak{D}_{\omega}= \{ x \in SAP_{\omega}(X): x(0) = \varphi(0), \| x(t) \| \leq r, \, t\geq 0 \} $$ endowed with the metric $d(u,v)=\| u-v \|_{\infty} $, we define the operator $ \Gamma: \mathfrak{D}_{\omega}\to C([0,\infty);X )$ by \begin{align*} \Gamma u(t) &=\mathcal{R}(t)(\varphi(0)+f(0,\varphi))-f(t,\widetilde{u}_{t}) -\int_{0}^{t}A\mathcal{R}(t-s)f(s,\widetilde{u}_{s})ds \\ &\quad - \int_{0}^{t}\mathcal{R}(t-s) \int_{0}^{s}B(s- \xi)f(\xi, \widetilde{u}_{\xi}) d\xi ds+ \int_{0}^{t}\mathcal{R}(t-s)g(s,\widetilde{u}_{s})ds, \quad t\geq 0, \end{align*} where $ \widetilde{u}:\mathbb{R} \to X$ is the function defined by the relation $\widetilde{u}_{0}= \varphi$ and $\widetilde{u} = u$ on $[0, \infty)$. From the hypothesis (P1), (PF) and (PG) we obtain that $\Gamma u$ is well defined and that $\Gamma u \in C([0, \infty);X)$. Moreover, from Lemma \ref{lem8H}, we have that function $s \mapsto \widetilde{u}_s \in SAP_{\omega}(\mathcal{B})$. By Lemma \ref{composition}, we conclude that $s \mapsto f(s,\widetilde{u}_s) \in SAP_{\omega}([0,\infty),Y) $ and $s \mapsto g(s,\widetilde{u}_s) \in SAP_{\omega}([0,\infty),X)$. From Lemmas \ref{lema4H}, \ref{apide1} and \ref{apide2} it follows that $ \Gamma u\in SAP_{\omega}(X)$. Using the same argument of Theorem \ref{p1} proof, we obtain that $\Gamma (\mathfrak{D}_{\omega}) \subset \mathfrak{D}_{\omega}$ and $\Gamma$ is a contraction. This completes the proof. \end{proof} \begin{proposition} \label{p3} Let $f :[0,\infty) \times\mathcal{B}\to Y $ and $g:[0,\infty) \times\mathcal{B} \to X$ be asymptotically uniformly continuous on bounded subset $K \subset \mathcal{B}$, and $\lim_{t\to \infty} \| f(t + n\omega, \psi) - f(t, \psi) \|_Y = 0$, $\lim_{t\to \infty} \| g(t + n\omega, \psi) - g(t, \psi) \| = 0 $ uniformly for $\psi \in K $ and $n \in \mathbb{N}$. Assume that $\mathcal{B}$ is a fading memory space and (P1), (P2), (PF) and (PG) hold. Then there exists $ \varepsilon >0$ such that for each $\varphi \in B_{\varepsilon}(0,\mathcal{B})$ there exists a unique asymptotically $\omega$-periodic mild solution $u(\cdot,\varphi)$ of \eqref{eqi1}-\eqref{eqi2} on $[0,\infty)$. \end{proposition} \begin{proof} We define the space $$ \mathfrak{D}_0= \{ x \in S_{\omega}(X): x(0) = \varphi(0), \| x(t) \| \leq r,\, t\geq 0 \} $$ endowed with the metric $d(u,v)=\| u-v \|_{\infty}$. It is easy see that $\mathfrak{D}_0$ is a closed subspace of $S_{\omega}$. We define the operator $ \Gamma:\mathfrak{D}_0 \to C([0,\infty);X )$ by \begin{align*} \Gamma u(t)&=\mathcal{R}(t)(\varphi(0)+f(0,\varphi))-f(t,\widetilde{u}_{t}) -\int_{0}^{t}A\mathcal{R}(t-s)f(s,\widetilde{u}_{s})ds \\ &\quad - \int_{0}^{t}\mathcal{R}(t-s) \int_{0}^{s} B(s- \xi)f(\xi, \widetilde{u}_{\xi}) d\xi ds+ \int_{0}^{t}\mathcal{R}(t-s)g(s,\widetilde{u}_{s})ds, \quad t\geq 0, \end{align*} where $ \widetilde{u}:\mathbb{R} \to X$ is the function defined by the relation $\widetilde{u}_{0}= \varphi$ and $\widetilde{u} = u$ on $[0, \infty)$. We observe that $\mathcal{R}(\cdot)(\varphi(0)+f(0,\varphi)) \in C_b([0,\infty),X))$ and $$ \lim_{t \to \infty}(\mathcal{R}(t+n\omega) - R(t))(\varphi(0)+f(0,\varphi))=0, $$ uniformly in $n \in \mathbb{N}$. Moreover, from \cite[Lemma 3.16]{Mi} and Lemma \ref{caiedo1}, we obtain that $\lim_{t \to \infty} \| f(t+n\omega, \widetilde{u}_{t + n\omega}) - f(t, \widetilde{u}_{t}) \|_Y = 0$ and $\lim_{t \to \infty} \| g(t+n\omega, \widetilde{u}_{t+ n\omega}) - g(t, \widetilde{u}_{t}) \| = 0$, uniformly in $n \in \mathbb{N}$. By Lemmas \ref{caiedo2}, \ref{apide1} and \ref{apide2} we have that $$ \lim_{t \to \infty} \Gamma x (t+n\omega) -\Gamma x(t)=0, $$ uniformly in $n \in \mathbb{N}$. From Lemma \ref{caiedo0} and using the same argument of the Theorem \ref{p1} proof we conclude that $u=\Gamma u \in \mathfrak{D}_0$ and $u$ is asymptotically $\omega$-periodic. The proof is ended. \end{proof} \section{Applications}\label{Examples} In this section we study the existence of several type of asymptotically periodicity solutions of the partial neutral integro-differential system \begin{gather} \begin{aligned} &\frac{\partial}{\partial t} \Big[ u(t, \xi) + \int_{-\infty}^{t} \int_{0}^{\pi} b(s-t,\eta,\xi) u(s,\eta)d \eta ds\Big] \\ & = (\frac{\partial^{2}} {\partial \xi^{2}} + \nu) \Big[u(t, \xi) + \int_0^t e^{- \gamma(t-s)} u(s,\xi) ds\Big] + \int_{-\infty}^{t} a_0(s-t) u(s, \xi)ds, \end{aligned} \label{eqexemcseg1} \\ u(t, 0) = u(t, \pi) = 0, \quad u(\theta, \xi) = \varphi(\theta, \xi), \label{eqexemcseg2} \end{gather} for $(t,\xi)\in [0,a] \times [0,\pi]$, $\theta\leq 0, \nu <0$ and $\gamma>0$. Moreover, we have identified $\varphi(\theta)(\xi) =\varphi(\theta, \xi)$. To represent this system in the abstract form \eqref{eqi1}-\eqref{eqi2}, we choose the spaces $X=L^{2}([0,\pi]) $ and $\mathcal{B} = C_{0} \times L^{2}(\rho,X)$, see Example \ref{example1} for details. We also consider the operators $A,B(t): D(A) \subseteq X \to X$, $t\geq 0 $, given by $ A x= x'' + \nu x$, $ B(t)x = e^{-\gamma t} Ax $ for $x\in D(A) = \{x \in X : x'' \in X,\, x(0) = x(\pi) = 0 \}$. Moreover, $A$ has discrete spectrum, the eigenvalues are $- n^{2}+\nu$, $n \in \mathbb{N}$, with corresponding eigenvectors $z_{n} (\xi) = (\frac{2}{\pi})^{1/2} \sin (n \xi)$, the set of functions $\{z_{n} : n \in \mathbb{N} \}$ is an orthonormal basis of $X$ and $T(t)x = \sum_{n=1}^{\infty} e^{-(n^{2}- \nu)t} \langle x, z_{n} \rangle z_{n}$ for $x\in X$. For $\alpha \in (0,1)$, from \cite{PA} we can define the fractional power $(-A)^{\alpha}: D((-A)^{\alpha}) \subset X \to X$ of $A$ is given by $(-A)^{\alpha}x= \sum_{n=1}^{\infty} (n^2-\nu)^{\alpha} \langle x, z_{n} \rangle z_n$, where $D((-A)^{\alpha}) =\{ x \in X: (-A)^{\alpha}x \in X \}$. In the next Theorem we consider $Y=D((-A)^{1/2})$. We observe that $\rho(A) \supset \{ \lambda \in \mathbb{C}: \operatorname{Re}(\lambda) \geq \nu\}$ and $\|\lambda R(\lambda,A)\| \leq M_1$ for $ \operatorname{Re}(\lambda)\geq \nu$, from \cite[Proposition 2.2.11]{Alessandra2} we obtain that $A$ is a sectorial operator satisfying $\|R(\lambda,A) \| \leq \frac{M}{|\lambda -\nu|}, M>0$, therefore (H1) is satisfied. Moreover, it is easy to see that conditions (H2)--(H3) are satisfied with $b(t)= e^{-\gamma t}$, and $D=C_{0}^{\infty}([0,\pi])$ the space of infinitely differentiable functions that vanishes at $\xi =0$ and $\xi = \pi$. Under the above conditions we can represent the system \begin{gather} \frac{\partial u(t,\xi)}{\partial t}= \Big(\frac{\partial^{2}} {\partial \xi^{2}} + \nu\Big) \Big[u(t, \xi) + \int_0^t e^{- \gamma(t-s)} u(s,\xi) ds\Big], \label{sit1} \\ u(t,\pi) = u(t,0) = 0, \label{sit2} \end{gather} in the abstract for \begin{gather*} \frac{d x(t)}{dt} = A x(t) + \int_{0}^{t} B(t-s)x(s) ds, \\ x(0) = z \in X. \end{gather*} We define the functions $f, g : \mathcal{B}\to X$ by \begin{gather*} f(\psi)(\xi) = \int_{-\infty}^{0}\int_{0}^{\pi} b(s,\eta,\xi) \psi (s,\eta)d\eta ds, \\ g(\psi)(\xi) = \int_{-\infty}^{0} a_0(s) \psi(s, \xi)ds, \end{gather*} where \begin{itemize} \item[(i)] The function $a_{0}:\mathbb{R}\to \mathbb{R}$ is continuous and $L_g := (\int_{-\infty}^{0} \frac{(a_0(s))^{2}}{\rho (s)}ds )^{\frac{1}{2}} < \infty$. \item[(ii)] The functions $ b(\cdot)$, $ \frac{\partial b(s, \eta, \xi)} {\partial \xi} $ are measurable, $b(s, \eta, \pi) = b(s,\eta,\, 0) = 0$ for all $(s,\eta)$ and \[ L_f:= \max \Big\{\Big(\int_{0}^{\pi} \int_{-\infty}^{0} \int_{0}^{\pi} \rho^{-1}(\theta)\Big(\frac{\partial^{i}}{\partial\xi ^{i}} b(\theta, \eta, \xi)\Big)^2 d\eta d\theta d\xi \Big)^{1/2}:i=0,1 \Big\}<\infty. \] \end{itemize} Moreover, $f, g $ are bounded linear operators, $\|f\|_{\mathcal{L}(\mathcal{B},X)} \leq L_f$, $\|g\|_{\mathcal{L}(\mathcal{B},X)} \leq L_g$ and a straightforward estimation using $\bf (ii)$ shows that $f(I\times \mathcal{B})\subset D((-A)^{\frac{1}{2}})$ and $$ \|(-A)^{\frac{1}{2}}f(t,\cdot)\|_{\mathcal{L}(\mathcal{B},X)} \leq L_f $$ for all $t\in I$. { This allows us to rewrite } the system \eqref{eqexemcseg1}-\eqref{eqexemcseg2} in the abstract form \eqref{eqi1}-\eqref{eqi2} with $u_0=\varphi \in \mathcal{B}$. \begin{theorem} \label{thm1} Assume that the previous conditions are verified. Let $2 \max\{M(K+1+ \gamma ),\gamma\}$. If $ \frac{1}{2 \mathfrak{K}} \geq L_f\mu + \frac{M}{|r + \nu|}L_g$, where $\mu = ( \| (-A)^{-\frac{1}{2}}\| + M(2 + \dfrac{e^{r + \nu}}{|r + \nu|} + \dfrac{1}{|r + \nu| \gamma} ) )$, then there exists $R>0$ such that if $\| \varphi \|_{\mathcal{B}} M \|\varphi\|_{\mathcal{B}} (1+L_f). $$ Now, for $\|\varphi\|_{\mathcal{B}}