\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 06, pp. 1--6.\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-2013/06\hfil Asymptotically periodic solutions] {Asymptotically periodic solutions for differential and difference inclusions in \\ Hilbert spaces} \author[G. Moro\c{s}anu, F. \"Ozpinar \hfil EJDE-2013/06\hfilneg] {Gheorghe Moro\c{s}anu, Figen \"Ozpinar} % in alphabetical order \address{Gheorghe Moro\c{s}anu \newline Department of Mathematics and its Applications\\ Central European University\\ Budapest, Hungary} \email{morosanug@ceu.hu} \address{Figen \"Ozpinar \newline Bolvadin Vocational School\\ Afyon Kocatepe University\\ Afyonkarahisar, Turkey} \email{fozpinar@aku.edu.tr} \thanks{Submitted October 18, 2012. Published January 8, 2013.} \subjclass[2000]{39A10, 39A11, 47H05, 34G25} \keywords{Differential inclusion; difference inclusion; subdifferential; \hfill\break\indent maximal monotone operator; weak convergence; strong convergence} \begin{abstract} Let $H$ be a real Hilbert space and let $A:D(A)\subset H\to H$ be a (possibly set-valued) maximal monotone operator. We investigate the existence of asymptotically periodic solutions to the differential equation (inclusion) $u'(t) + Au(t)\ni f(t) + g(t)$, $t>0$, where $f \in L_{\rm loc}^2(\mathbb{R}_+,H)$ is a $T$-periodic function ($T>0$) and $g \in L^1(\mathbb{R}_+,H)$. Consider also the following difference inclusion (which is a discrete analogue of the above inclusion): $ \Delta u_n + c_n A u_{n+1}\ni f_n + g_n , \ n=0,1, \dots$, where $(c_n)\subset (0,+\infty)$, $(f_n)\subset H$ are $p$-periodic sequences for a positive integer $p$ and $(g_n)\in \ell^{1}(H)$. We investigate the weak or strong convergence of its solutions to $p$-periodic sequences. We show that the previous results due to Baillon, Haraux (1977) and Djafari Rouhani, Khatibzadeh (2012) corresponding to $g\equiv 0$, respectively $g_n=0$, $n=0,1,\dots$, remain valid for $g\in L^1(\mathbb{R}_+,H)$, respectively $(g_n)\in l^1(H)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction}\label{intro} Let $H$ be a real Hilbert space with inner product $(\cdot, \cdot)$ and the induced Hilbertian norm $\Vert \cdot \Vert$. Let $A:D(A)\subset H\to H$ be a (possibly multivalued) maximal monotone operator. Consider the following differential equation (inclusion) \begin{equation} \frac{{d}u}{{d}t}(t) + Au(t)\ni f(t)+ g(t),\quad t>0, \label{Ec} \end{equation} where $f \in L_{\rm loc}^2(\mathbb{R}_+,H)$ is a $T$-periodic function for a given $T>0$ and $g \in L^1(\mathbb{R}_+,H)$. In this paper we investigate the behavior at infinity of solutions to \eqref{Ec}. Consider also the following difference equation (inclusion) (which is the discrete analogue of \eqref{Ec}) \begin{equation} \Delta u_n + c_n A u_{n+1}\ni f_n + g_n,\quad n=0,1,\dots, \label{ODE} \end{equation} where $(c_n)\subset (0,+\infty)$, $(f_n)\subset H$ are $p$-periodic sequences for a positive integer $p$, $(g_n)\in \ell^{1}(H):=\{u=(u_{1},u_{2},\dots):\sum_{n=1}^{\infty } \Vert u_n\Vert <\infty\}$ and $\Delta$ is the difference operator defined as usual, i.e., $\Delta u_n= u_{n+1}-u_n$. We investigate the weak or strong convergence of solutions to $p$-periodic sequences. More precisely, in this article we show that the previous results due to Baillon, Haraux \cite{Haraux77} and Djafari Rouhani, Khatibzadeh \cite{Hadi} related to the equations (inclusions), \begin{equation} \frac{{d}u}{{d}t}(t) + Au(t)\ni f(t),\quad t>0, \label{Ec0} \end{equation} and \begin{equation} \Delta u_n + c_n A u_{n+1}\ni f_n,\quad n=0,1,\dots , \label{ODE0} \end{equation} respectively, remain valid for \eqref{Ec} and \eqref{ODE}, where $g\in L^1(\mathbb{R}_+,H)$ and $(g_n)\in l^1(H)$. \section{Preliminaries}\label{pr} To obtain our main results we recall the following results on the existence of asymptotically periodic solutions of the equations \eqref{Ec0} and \eqref{ODE0}. \begin{lemma}[{\cite{Haraux77}, \cite[p. 169]{Morosanu88}}] \label{prlm1} Assume that $A$ is the subdifferential of a proper, convex, and lower semicontinuous function $\varphi:H\to(-\infty,+\infty]$, $A=\partial \varphi$. Let $f \in L_{\rm loc}^2(\mathbb{R}_+,H)$ be a $T$-periodic function (for a given $T>0$). Then, equation \eqref{Ec0} has a solution bounded on $\mathbb{R}_+$ if and only if it has at least a $T$-periodic solution. In this case all solutions of \eqref{Ec0} are bounded on $\mathbb{R}_+$ and for every solution $u(t)$, $t \geq 0$, there exists a $T$-periodic solution $q$ of \eqref{Ec0} such that \[ u(t)- q(t)\to 0, \quad\text{as } t\to \infty , \] weakly in $H$. Moreover, every two periodic solutions of \eqref{Ec0} differ by an additive constant, and \[ \frac{{d}u_n}{{d}t}\to \frac{{d}q}{{d}t},\quad \text{as } n\to \infty, \] strongly in $L^2(0,T;H)$, where $u_n(t)=u(t+nT)$, $n=1,2,\dots$ \end{lemma} \begin{lemma}[\cite{Hadi}, \cite{Morosanu12}] \label{prlm2} Assume that $A:D(A)\subset H \to H$ is a maximal monotone operator. Let $c_n>0$ and $f_n \in H$ be $p$-periodic sequences; i.e., $c_{n+p}=c_n$, $f_{n+p}=f_n$ $(n=0,1,\dots )$, for a given positive integer $p$. Then \eqref{ODE0} has a bounded solution if and only if it has at least one $p$-periodic solution. In this case all solutions of \eqref{ODE0} are bounded and for every solution $(u_n)$ of \eqref{ODE0} there exists a $p$-periodic solution $(\omega_n)$ of \eqref{ODE0} such that \[ u_n-\omega_n\to 0, \quad \text{weakly in $H$, as $n\to \infty$.} \] Moreover, every two periodic solutions differ by an additive constant vector. \end{lemma} \section{Results on asymptotically periodic solutions}\label{mr} We begin this section with the following result regarding the continuous case, which is an extension of Lemma \ref{prlm1}. \begin{theorem}\label{mrthm1} Assume that $A:D(A)\subset H \to H$ is the subdifferential of a proper, convex, lower semicontinuous function $\varphi:H\to(-\infty,+\infty]$, $A=\partial\varphi$. Let $f\in L_{\rm loc}^2(\mathbb{R}_+,H)$ be a $T$-periodic function ($T>0$) and let $g \in L^1(\mathbb{R}_+,H)$. Then equation \eqref{Ec} has a bounded solution if and only if equation \eqref{Ec0} has at least a $T$-periodic solution. In this case all solutions of \eqref{Ec} are bounded on $\mathbb{R}_+$ and for every solution $u(t)$ of \eqref{Ec} there exists a $T$-periodic solution $\omega(t)$ of \eqref{Ec0} such that \[ u(t) - \omega(t)\to 0, \quad\text{ weakly in $H$, as $t\to \infty$.} \] \end{theorem} \begin{proof} If a solution $u(t)$, $t \geq 0$, of equation \eqref{Ec} is bounded on $\mathbb{R}_+$, then any other solution $\tilde{u}(t)$, $t \geq 0$, of equation \eqref{Ec} is also bounded, because \begin{equation} \Vert u(t) - \tilde{u}(t)\Vert \leq \Vert u(0) - \tilde{u}(0)\Vert.\label{thm1.1} \end{equation} If a solution $u(t)$ of \eqref{Ec} is bounded, then any solution $v(t)$ of \eqref{Ec0} is bounded and conversely, because \[ \Vert u(t)-v(t)\Vert \leq \Vert u(0)-v(0)\Vert + \int_0^t \Vert g(s)\Vert{d}s \leq \Vert u(0)-v(0)\Vert + \int_0^{\infty} \Vert g(s)\Vert{d}s < \infty, \] for $t\ge 0$. Thus, the first part of the theorem follows by Lemma \ref{prlm1}. To prove the second part, we define $g_m: \mathbb{R}_+ \to H$ as follows: \[ g_{m}(t)= \begin{cases} g(t) &\text{for a.e. } t \in (0, m) \\ 0 &\text{if } t\ge m, \end{cases} \] where $m=1,2, \dots$. Let $u(t)$, $t\ge 0$, be an arbitrary bounded solution of \eqref{Ec}. For each $m=1,2,\dots$ denote by $u_m(t)$, $t\ge 0$, the solution of the Cauchy problem \begin{gather} \frac{{d}u_m(t)}{{d}t} + A(u_m(t))\ni f(t)+ g_m(t),\quad t>0, \label{Ecm} \\ u_{m}(0)= u(0). \label{ICcm} \end{gather} Since $u_m(t)$, $t\ge m$, is a solution of equation \eqref{Ec0}, it follows by Lemma \ref{prlm1} that there is a $T$-periodic solution $q_m(t)$ of \eqref{Ec0}, such that \begin{equation} u_m(t) - q_m(t)\to 0, \quad \text{weakly in $H$, as $t\to \infty$.} \label{thm1.2} \end{equation} In fact, since any two periodic solutions of \eqref{Ec0} differ by an additive constant (cf. Lemma \ref{prlm1}), it follows that \[ q_m(t) = q(t) + c_m, \quad m=1,2, \dots, \] for a fixed periodic solution $q(t)$ of \eqref{Ec0}, where $(c_m)$ is a sequence in $H$. Thus, \eqref{thm1.2} becomes \begin{equation} u_m(t) - q(t) \to c_m \quad \text{as } t\to \infty, \label{99} \end{equation} weakly in $H$. Moreover, \begin{equation} \frac{{d}q(t)}{{d}t} + A(q(t) + c_m)\ni f(t). \label{thm1.3} \end{equation} On the other hand, it is easy to see that, for all $m < r$, we have \[ \Vert [u_m(t) - q(t)] - [u_r(t) - q(t)] \Vert = \Vert u_m(t) - u_r(t)\Vert \le \Vert u(0)-u(0)\Vert + \int_m^r \Vert g(t) \Vert \, dt. \] Therefore, taking the limit as $t\to \infty$, it follows (see \eqref{99}), \begin{equation} \Vert c_m - c_r \Vert \le \int_m^r \Vert g(t) \Vert \, dt, \end{equation} which shows that $(c_m)$ is a convergent sequence; i.e., there exists a point $a\in H$, such that \begin{equation} \Vert c_m - a\Vert \to 0, \quad \text{as } m\to \infty. \label{9999} \end{equation} Since $A$ is maximal monotone (hence demiclosed), we can pass to the limit in \eqref{thm1.3}, as $m \to \infty$, to deduce that $\omega (t) := q(t) + a$ is a solution of \eqref{Ec0} (which is $T$-periodic). Note also that \begin{equation} \Vert u(t) - u_m(t) \Vert \le \int_m^t \Vert g(s) \Vert \, ds \le \int_m^{\infty} \Vert g(s) \Vert \, ds, \ t\ge m. \label{99999} \end{equation} To conclude, we use the decomposition \[ \begin{split} u(t) - \omega(t) &= [u(t)-u_m(t)] + [u_m(t)-q_m(t)] + [q_m(t) - \omega(t)] \\ &= [u(t)-u_m(t)] + [u_m(t) - q(t) - c_m] + [(q(t) + c_m) -(q(t) +a)], \end{split} \] which shows that $u(t) - \omega(t)$ converges weakly to zero, as $t \to \infty$ (cf. \eqref{99}, \eqref{9999}, \eqref{99999}). In other words, $u(t)$ is asymptotically periodic with respect to the weak topology of $H$. \end{proof} It is well known that, even in the case $g\equiv 0$, the above result (Theorem \ref{mrthm1}) is not valid for a general maximal monotone operator $A$, so we cannot expect more in our case. \begin{theorem}\label{mrthm2} \rm Assume that $A:D(A)\subset H \to H$ is a maximal monotone operator. Let $(g_n)\in \ell^{1}(H)$ and let $c_n>0$, $f_n \in H$ be $p$-periodic sequences, i.e., $c_{n+p}=c_n$, $f_{n+p}=f_n$ $(n=0,1,\dots )$, for a given positive integer $p$. Then equation \eqref{ODE} has a bounded solution if and only if equation \eqref{ODE0} has at least one $p$-periodic solution. In this case all solutions of \eqref{ODE} are bounded and for every solution $(u_n)$ of \eqref{ODE} there exists a $p$-periodic solution $(\omega_n)$ of \eqref{ODE0} such that \[ u_n-\omega_n\to 0, \quad\text{weakly in $H$, as $n\to \infty$.} \] \end{theorem} \begin{proof} Consider the initial condition \begin{equation} u_{0}= x, \label{IC} \end{equation} for a given $x \in H$. We can rewrite equation \eqref{ODE} in the form: \[ u_{n+1}-u_n + c_n A u_{n+1}\ni f_n+g_n. \] The solution of the problem \eqref{ODE}-\eqref{IC} is calculated successively from \[ u_{n+1}=\big(I+ c_n A\big)^{-1}\big( u_n+f_n+g_n\big),\quad n=0,1, \dots, \] in a unique manner, which will give a unique solution $(u_n)_{n\geq0}$. If a solution $(u_n)$ of \eqref{ODE} is bounded, then any other solution $(\tilde{u}_n)$ of \eqref{ODE} is bounded, because \begin{equation} \Vert u_n-\tilde{u}_n\Vert \leq \Vert u_{0}-\tilde{u}_{0} \Vert \quad \text{for } n=0,1, \dots \label{thm2.1} \end{equation} If a solution $(u_n)$ of \eqref{ODE} is bounded, then any solution $(v_n)$ of \eqref{ODE0} is bounded and conversely, because \[ \Vert u_n-v_n\Vert \leq \Vert u_0-v_0\Vert + \sum_{k=0}^{n-1} \Vert g_k\Vert \leq \Vert u_0-v_0\Vert + \sum_{k=0}^{\infty} \Vert g_k\Vert < \infty . \] According to Lemma \ref{prlm2} the first part of the theorem is proved. For the second part we define $(g_{n,m})_{n,m\ge 0}$ as follows: \[ g_{n,m}= \begin{cases} g_n &\text{if } n < m, \\ 0 &\text{if } n\ge m. \end{cases} \] Let $(z_n)$ be an arbitrary solution of \eqref{ODE} (which is bounded). For each $m=0,1,\dots$ denote by $(z_{n,m})_{n\ge 0}$ the (unique) solution of the problem \begin{gather} z_{n+1,m}-z_{n,m} + c_n A z_{n+1,m} \ni f_n+g_{n,m} \label{ODEm} \\ z_{0,m}= z_{0}. \label{ICm} \end{gather} Note that $(z_{n,m})_{n\ge m}$ is a solution of equation \eqref{ODE0}. By Lemma \ref{prlm2} there is a $p$-periodic (with respect to $n$) solution ($\omega_{n,m}$) of \eqref{ODE0} such that \begin{equation} z_{n,m} - \omega_{n,m}\to 0, \quad \text{weakly in $H$, as $n\to \infty$.} \label{thm2.2} \end{equation} For each $m\ge 0$ we have \begin{gather*} \omega_{1,m}-\omega_{0,m}+ c_{0} A \omega_{1,m} \ni f_0, \\ \omega_{2,m}-\omega_{1,m}+ c_{1} A \omega_{2,m} \ni f_1, \\ \dots \\ \omega_{p,m}-\omega_{p-1,m}+ c_{p-1} A \omega_{p,m} \ni f_{p-1}, \end{gather*} where $\omega_{p,m} = \omega_{0,m}$. Since any two periodic solutions of \eqref{ODE0} differ by an additive constant, we can write \begin{equation} \omega_{t,m} = \zeta_{t} + a_m \quad t \in \{0,1, \dots, p-1\}, \label{thm2.3} \end{equation} where $(\zeta_t)$ is a an arbitrary but fixed periodic solution of \eqref{ODE0}, and ${(a_m)}_{m\ge 0}$ is a sequence in $H$. Thus \begin{equation} \begin{gathered} \zeta_{1}-\zeta_{0}+ c_{0} A (\zeta_{1}+a_m) \ni f_0, \\ \zeta_{2}-\zeta_{1}+ c_{1} A (\zeta_{2}+a_m) \ni f_1, \\ \dots \\ \zeta_{p}-\zeta_{p-1}+ c_{p-1} A (\zeta_{p}+a_m) \ni f_{p-1}, \\ \end{gathered}\label{thm2.4} \end{equation} for all $m\ge 0$, where $\zeta_p = \zeta_{0}$. Also we can rewrite \eqref{thm2.2} as \begin{equation} z_{kp+t,m} \to \zeta_t + a_m, \quad \text{weakly in $H$, as $k\to \infty$}, \label{100} \end{equation} for $m \ge 0$ and $t\in \{ 0,1, \dots, p-1\} $. On the other hand, for $0\le m < r$, we have (cf. \eqref{ODEm}, \eqref{ICm}) $$ \Vert z_{kp+t,m} - z_{kp+t,r} \Vert \le \sum_{j=m}^{r-1}\Vert g_j\Vert. $$ According to \eqref{100} this implies \begin{equation} \Vert a_m - a_r\Vert \le \sum_{j=m}^{r-1}\Vert g_j\Vert \le \sum_{j=m}^{\infty}\Vert g_j\Vert, \end{equation} for all $0\le m< r$, so there exists an $a\in H$ such that \begin{equation} \Vert a_m - a \Vert \to 0, \quad \text{as } m\to \infty. \label{1000} \end{equation} Since $A$ is maximal monotone (hence demiclosed), we can pass to the limit in \eqref{thm2.4} as $m\to \infty$ to obtain \begin{gather*} \zeta_{1}-\zeta_{0}+ c_{0} A (\zeta_{1}+a) \ni f_0, \\ \zeta_{2}-\zeta_{1}+ c_{1} A (\zeta_{2}+a) \ni f_1, \\ \dots \\ \zeta_{p}-\zeta_{p-1}+ c_{p-1} A (\zeta_{p}+a) \ni f_{p-1}, \\ \end{gather*} where $\zeta_p = \zeta_0$. So $\omega_n : = \zeta_n + a$ is a $p$-periodic solution of equation \eqref{ODE0}. We can also see that \begin{equation} \Vert z_n - z_{n,m} \Vert \le \Vert z_0 - z_{0,m}\Vert + \sum_{j=m}^{n-1} \Vert g_j\Vert \le \sum_{j=m}^{\infty} \Vert g_j \Vert . \label{10000} \end{equation} Finally, for all natural $n$, we have $n=kp+t$, $t\in \{ 0,1,\dots,p-1 \}$, and \begin{align*} z_n-\omega_n &= [z_n-z_{n,m}] + [z_{n,m} - \omega_{t,m}] +[\omega_{t,m} - \omega_n]\\ &= [z_n-z_{n,m}] + [z_{kp+t,m} - \zeta_t -a_m] + [\zeta_t + a_m - \zeta_t - a], \end{align*} thus the conclusion of the theorem follows by \eqref{100}, \eqref{1000} and \eqref{10000}. \end{proof} If in addition $A$ is strongly monotone, then we can easily extend Theorem 2 in \cite{Morosanu12}, as follows. \begin{theorem} \label{mrthm3} Assume that $A:D(A)\subset H \to H$ is a maximal monotone operator, that is also strongly monotone; i.e., there is a constant $b>0$, such that \[ (x_1-x_2, y_1-y_2)\ge b{\Vert x_1-x_2\Vert}^2, \quad \forall x_i\in D(A), \; y_i\in Ax_i, \; i=1,2. \] Let $c_n>0$ and $f_n \in H$ be $p$-periodic sequences for a given positive integer $p$ and $(g_n)\in \ell^{1}(H)$. Then equation \eqref{ODE0} has a unique $p$-periodic solution $(\omega_n)$ and for every solution $(u_n)$ of \eqref{ODE} we have \[ u_n-\omega_n\to 0, \quad\text{strongly in $H$, as } n\to \infty. \] \end{theorem} The proof relies on arguments similar to the one above. \begin{thebibliography}{00} \bibitem{Haraux77} J. B.~Baillon, A.~Haraux; \emph{Comportement \'{a} l'infini pour les \'equations d'\'evolution avec forcing p\'eriodique}, Archive Rat. Mech. Anal., 67(1977), 101-109. \bibitem{Hadi} B.~Djafari Rouhani, H.~Khatibzadeh; \emph{Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing}, J. Difference Eqns Appl., DOI:10.1080/10236198.2012.658049. \bibitem{Morosanu88} G.~Moro\c{s}anu; \emph{Nonlinear evolution equations and applications.} D.Reidel, Dordrecht-Boston-Lancaster-Tokyo, 1988. \bibitem{Morosanu12} G.~Moro\c{s}anu and F.~\"{O}zp\i nar; \emph{Periodic forcing for some difference equations in Hilbert spaces}, Bull. Belgian Math. Soc. (Simon Stevin), to appear. \end{thebibliography} \end{document}