\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 103, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/103\hfil Periodic trajectories] {Periodic trajectories for evolution equations in Banach spaces} \author[M. D. Voisei\hfil EJDE-2005/103\hfilneg] {Mircea D. Voisei} \address{Mircea D. Voisei \hfill\break Department of Mathematics, MAGC 3.734, The University of Texas - Pan American, Edinburg, TX 78539, USA} \email{mvoisei@utpa.edu} \date{} \thanks{Submitted August 3, 2005. Published September 28, 2005.} \subjclass[2000]{47J35, 34C25, 35K55} \keywords{Periodic solution, evolution equation of parabolic type} \begin{abstract} The existence of periodic solutions for the evolution equation $$y'(t)+Ay(t)\ni F(t,y(t))$$ is investigated under considerably simple assumptions on $A$ and $F$. Here $X$ is a Banach space, the operator $A$ is $m$-accretive, $-A$ generates a compact semigroup, and $F$ is a Carath\'{e}odory mapping. Two examples concerning nonlinear parabolic equations are presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Consider the nonlinear evolution equation $$\label{e1.1} y'(t)+Ay(t)\ni F(t,y(t)), \quad 0\leq t\leq T,$$ where $(X,\|\cdot \|)$ is a Banach space, $A:D(A)\subset X\to 2^{X}$ is $m$-accretive such that $-A$ generates a compact semigroup, $F:[0,T]\times \overline{D(A)}\to X$ is a Carath\'{e}odory mapping, i.e., $F(\cdot ,x):[0,T]\to X$ is strongly measurable for every $x\in \overline{D(A)}$ and $F(t,\cdot ):\overline{D(A)}\to X$ is continuous for almost every $t\in [0,T]$. The aim of this note is to investigate the existence of a mild periodic solution $y\in C([0,T];X)$, $y(0)=y(T)$ for \eqref{e1.1} in general Banach spaces. Our main result is the following. \begin{theorem} \label{thm1.1} Assume that $A$ is $m$-accretive, $-A$ generates a compact semigroup, $F$ is Carath\'{e}odory with $$\label{e1.2} \int_{0}^{T} \sup_{x\in \overline{D(A)},\|x\|\leq r} \|F(t,x)\|dt<\infty,$$ for every $r>0$, and there exist $R>0$, $b,c\in L^{1}(0,T)$, $c(t)\geq 0$, $t\in (0,T)$, $c\neq 0$, such that $$\label{e1.3} [x,y-F(t,x)]_{+}\geq c(t)\|x\|+b(t), \quad t\in (0,T),\; x\in D(A), \; \|x\|\geq R,\; y\in Ax,$$ where $[x,v]_{+}=\lim\limits_{h\downarrow 0}\frac 1h (\|x+hv\|-\|x\|)$, $x,v\in X$. Then \eqref{e1.1} admits at least one mild $T$-periodic solution. \end{theorem} This periodic problem has been intensively studied in the literature \cite{c1,h1,s1,s2,v1,v2}. The most common argument used is to apply a fixed point theorem for a suitable Poincar\'{e} operator. Generally, Schauder's fixed point theorem produced remarkable results for the general Banach space setting (Vrabie \cite{v2}, Shioji \cite{s1}). For example the main result in Shioji \cite{s1} has a statement similar to Theorem \ref{thm1.1}. In addition $X$ is separable, $\overline{D(A)}$ is convex, and condition \eqref{e1.3} is strengthened to $c(t)=c>0,$ $t\in [0,T]$. The earlier result of Vrabie \cite{v2} studies \eqref{e1.1} under the assumptions that $A-aI$ is $m$-accretive for some $a>0$ and the growth condition \eqref{e1.2} is replaced by $$\limsup_{r\to \infty } \frac{1}{r}\sup \{\|F(t,x)\|: t\geq 0,x\in \overline{D(A)},\|x\|\leq r\}=m0 such that \{x\in X; \|x\|=r\}\cap D(A) is nonempty and$$ \langle x,y-F(t,x)\rangle \geq 0, \quad \mbox{for every }x\in D(A),\; \|x\|=r,\; y\in Ax,\;0\leq t\leq T, $$where \langle \cdot ,\cdot \rangle  stands for the inner product of X. Also, in \cite{v1} the case m=a\geq 0 is studied in general Banach spaces under the condition: there exists r>0 such that$$ [x,y-F(t,x)]_{+}\geq 0, \quad \mbox{for } 0\leq t\leq T,\; x\in D(A),\; y\in Ax,\; \|x\|\geq r. $$Most of these new assumptions are needed for the invariance of a closed convex ball in the Schauder fixed point theorem argument. The method we use here relies on the Leray-Schauder topological degree applied for the Green operator and it is neither concerned with the initial value problem for \eqref{e1.1} nor involves the Poincar\'{e} operator. That is why, the convexity of \overline{D(A)} or the strong accretivity of A are no longer needed. Condition \eqref{e1.3} ensures some a priori'' estimates for the periodic solutions of \eqref{e1.1} and that the periodic solution'' operator % P_{A}:L^{1}(0,T;X)\to C([0,T];X), which associates to g\in L^{1}(0,T;X) all solutions y\in P_{A}g of the periodic problem y'(t)+Ay(t)\ni g(t), 0\leq t\leq T, y(0)=y(T), is compact. Next section is devoted to preliminaries and main notations. Section 3 contains the proof of our main result. This paper concludes with section 4 where two examples concerning nonlinear evolution equation of parabolic type are presented. \section{Preliminaries} For notions such as m-accretive operator, mild solution, or compact semigroup the reader is referred to Vrabie \cite{v3} and the references therein. We suggest Lloyd \cite{l1} for the theory and notations of the topological degree. Let (X,\Vert \cdot \Vert ) be a real Banach space and A:D(A)\subset X\to 2^{X}. If A-\omega I is m-accretive for some \omega \in \mathbb{R}  then for each (\xi ,f)\in \overline{D(A)}\times L^{1}(0,T;X) the initial value problem $$\label{e2.1} y'(t)+Ay(t)\ni f(t), \quad 0\leq t\leq T, \; y(0)=\xi ,$$ has a unique mild solution denoted by M(\xi ,f). For y_{i}=M(\xi _{i},f_{i}), where (\xi _{i},f_{i})\in \overline{D(A)}\times L^{1}(0,T;X), i=1,2 we have the mild solution inequality $$\label{e2.2} e^{\omega t}\Vert y_{1}(t)-y_{2}(t)\Vert \le e^{\omega s}\Vert y_{1}(s)-y_{2}(s)\Vert +\int_{s}^{t}e^{\omega \tau }\Vert f_{1}(\tau )-f_{2}(\tau )\Vert d\tau ,$$ for 0\le s\le t\le T. The norm of a function f in L^{p}(0,T;X), 1\leq p<\infty , is denoted by$$ \|f\|_{L^{p}}=(\int_{0}^{T}\|f(t)\|^{p}dt)^{1/p}. We use the norm \|y\|_{\infty }=\underset{t\in [0,T]}{\sup }\|y(t)\|, for y\in C([0,T];X). A family \mathcal{G}\subset L^{1}(0,T;X) is called uniformly integrable if for each \varepsilon >0 there exists \delta (\varepsilon )>0 such that for every measurable set E in [0,T] whose Lebesgue measure is less than \delta (\varepsilon ) we have \int_{E}\Vert f(t)\Vert dt<\varepsilon , uniformly for f\in \mathcal{G}. Every bounded subset of L^{p}(0,T;X), 10, and -A generates a compact semigroup then P_{A}(\mathcal{G}) is relatively compact in C([0,T];X). \end{theorem} \begin{proof} The mild solution inequality and the periodicity offer us the estimate $$\label{e2.3} \|y\|_{\infty }\leq \frac{e^{2\omega T}}{e^{\omega T}-1} \|f\|_{L^{1}},\quad\mbox{for every }y\in P_{A}f\,,$$ where, without loss of generality, we assume that 0\in A0. Since \mathcal{G} is bounded, this shows that the set of initial-final data B=\{y(0); y\in P_{A}(\mathcal{G})\} is bounded in X. According to Theorem \ref{thmVrabie}, this implies that B is relatively compact in X and P_{A}(\mathcal{G}) is relatively compact in C([0,T];X). \end{proof} \section{Proof of Theorem \ref{thm1.1}} In the sequel we assume that all the assumptions in Theorem \ref{thm1.1} hold. Without loss of generality we may assume that, in \eqref{e1.3}, R>\|b\|_{L^{1}}/\|c\|_{L^{1}}. Let K>C:=\|b\|_{L^{1}}+R+2 and \rho \in C^{\infty }(\mathbb{R}) such that 0\leq \rho \leq 1, \rho (u)=1 for |u|\leq K, \rho (u)=0 for |u|\geq K+1. \begin{lemma} \label{lem3.1} Let y\in C([0,T];X) be a mild T-periodic solution of $$\label{e3.1} y'(t)+Ay(t)\ni \rho (\|y(t)\|)F(t,y(t)), \quad t\in [0,T].$$ Then \|y\|_{\infty }\leq C or \|y\|_{\infty }\geq K. \end{lemma} \begin{proof} Assume by contradiction that C<\|y\|_{\infty}0 there exist \begin{itemize} \item[(i)] 0=t_{0}R. For \varepsilon <\min \{K-\|y\|_{\infty },\inf\limits_{[0,T]}\|y\|-R\}, we have R\leq \|y_{k}\|\leq K, k=1,\dots ,n. From \eqref{e1.3} and (iii) we find $$\label{e3.2} [y_{k},f_{k}-\frac{y_{k}-y_{k-1}}{t_{k}-t_{k-1} }-F(t,y_{k})]_{+}\geq c(t)\|y_{k}\|+b(t), \quad\mbox{a.e. }t\in [0,T].$$ Integrate on [t_{k-1},t_{k}] and add from k=1 to n, to obtain \label{e3.3} \begin{aligned} &\|y_{n}\|+\sum_{k=1}^n \|y_{k}\|\int_{t_{k-1}}^{t_{k}}c(t)dt\\ &\leq \|y_{0}\|+\sum_{k=1}^n \int_{t_{k-1}}^{t_{k}}\|f_{k}-F(t,y_{k})\| +\int_{0}^{t_{n}}|b(t)|dt \\ &\leq \|y_{0}\|+\|b\|_{L^{1}}+\sum_{k=1}^{n}\int_{t_{k-1}}^{t_{k}}\Vert \rho (\|y(t)\|)F(t,y(t))-f_{k}\Vert dt\\ &\quad +\int_{0}^{T}\|F(t,y(t))-F_{\varepsilon }(t)\|dt, \end{aligned} where F_{\varepsilon }(t)=F(t,y_{k}) if t\in [t_{k-1},t_{k}), k=1,\dots ,n, F_{\varepsilon }(t)=F(t,y(t)), t\in [t_{n},T]. According to (ii), this yields $$\label{e3.4} \|y_{n}\|+R\int_{0}^{t_{n}}c(t)dt\leq \|y_{0}\|+\|b\|_{L^{1}}+\varepsilon +\int_{0}^{T}\|F(t,y(t))-F_{\varepsilon }(t)\|dt.$$ Since F is Carath\'{e}odory, from \eqref{e1.2} and the Lebesgue dominated convergence theorem, we have \lim\limits_{\varepsilon \to 0} \int_{0}^{T}\|F(t,y(t))-F_{\varepsilon }(t)\|dt=0. Let \varepsilon \to 0, c\to T in \eqref{e3.4}. We find R\leq \|b\|_{L^{1}}/\|c\|_{L^{1}} which is a contradiction. Therefore, \inf\limits_{[0,T]}\|y\|\leq R. Eventually shifting the time, we may assume without loss of generality, that there exists t_{+}\in [0,T], such that \|y(0)\|=R+1, \|y(t_{+})\|=\|y\|_{\infty }, and \|y(t)\|\geq R+1, for every t\in [0,t_{+}]. By the same argument used above we obtain \|y\|_{\infty }\leq R+1+\|b\|_{L^{1}}\leq C which is a contradiction. The proof is complete. \end{proof} For 0\leq \lambda \leq 1, define the operators L_{\lambda}:C([0,T];X)\to C([0,T];X), L_{\lambda }v=P_{A+\omega I}(\lambda \rho (\|v\|)F(\cdot ,v)+\lambda\omega v),\quad v\in C([0,T];X), $$i.e., y=L_{\lambda }v is the unique T-periodic solution of $$\label{e3.5} y'(t)+Ay(t)+\omega y(t)\ni \lambda \rho (\|v(t)\|)F(t,v(t))+\lambda \omega v(t),\quad 0\leq t\leq T.$$ where \omega >0 is specified below. Note that L_{0}=0 and that Lemma \ref{lem3.1} contains an a priori'' estimate for the fixed points of L_{1}. Similarly, we provide an a priori'' estimate for the fixed points of L_{\lambda }, 0<\lambda <1. \begin{lemma} \label{lem3.2} For \omega >0 big enough, every T-periodic solution y\in C([0,T];X), of $$\label{e3.6} y'(t)+Ay(t)+\omega (1-\lambda )y(t)\ni \lambda \rho (\|y(t)\|)F(t,y(t)),\quad 0\leq t\leq T,\; 0<\lambda <1,$$ satisfies \|y\|_{\infty }\leq C or \|y\|_{\infty }\geq K. \end{lemma} \begin{proof} Consider a_{K}(t)=\sup\{ \|F(t,x)\|;\; x\in \overline{D(A)},\|x\|\leq K+1\}, t\in [0,T]. \noindent According to \eqref{e1.2} a_{K}\in L^{1}(0,T)  and \rho (\|x\|)\|F(t,x)\|\leq a_{K}(t), a.e. t\in [0,T], x\in \overline{D(A)}. Assume by contradiction that C<\|y\|_{\infty }R. Reasoning as above we find $$\label{e3.7} \omega (1-\lambda )RT+R\|c\|_{L^{1}}\leq (1-\lambda )\|a_{K}\|_{L^{1}}+\|b\|_{L^{1}},$$ which is absurd if \omega >\|a_{K}\|_{L^{1}}/RT, since R\|c\|_{L^{1}}>\|b\|_{L^{1}}. Therefore, \inf\limits_{[0,T]}\|y\|\leq R, and we may assume that there exist 0\leq s_{0}0, where \delta _{0} depends only on a_{K}. If \omega >\|a_{K}\|_{L^{1}}/R\delta _{0}, then \eqref{e3.9} provides us with a contradiction. The proof is complete. \end{proof} \begin{lemma} \label{lem3.3} H(\lambda )=L_{\lambda }, 0\leq \lambda \leq 1, defines a homotopy of compact transformations in C([0,T];X). \end{lemma} \begin{proof} Condition \eqref{e1.2} ensures the fact that for every 0\leq \lambda \leq 1, the operator C([0,T];X)\ni v\mapsto \lambda \rho (\|v\|)F(t,v)+\lambda \omega v transforms bounded subsets of C([0,T];X) into locally integrable subsets of L^{1}(0,T;X). According to Theorem \ref{thm2.1}, this implies that L_{\lambda }:C([0,T];X)\to C([0,T];X) is compact, for every 0\leq \lambda \leq 1. For 0\leq \lambda, \mu \leq 1 and v\in C([0,T];X), we have, from the mild solution inequality combined with the periodicity, that $$\label{e3.10} \|H(\lambda )v-H(\mu )v\|_{\infty }\leq |\lambda -\mu |e^{2\omega T}/(e^{\omega T}-1)\|a_{K}\|_{L^{1}}$$ which shows that H is a homotopy, thereby completing the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Pick r_{0}\in (C,K) and let B=\{v\in C([0,T];X); \|v\|_{\infty }0, p>(n-2)/2 such that \rho'(r)\geq c|r|^{p-1} for every r\neq 0, and f:\mathbb{R}\times \Omega \times \mathbb{R}\to \mathbb{R}, f=f(t,x,u) is T-periodic in t, f(t,x,\cdot ) is continuous for almost every (t,x)\in \mathbb{R}\times \Omega , f(\cdot ,\cdot,u) is measurable for every u\in \mathbb{R}, and there exist M>0, a,c\in L^{1}(0,T), c(t)\geq 0, t\in (0,T), c\neq 0, b,d\in L^{1}((0,T)\times \Omega ) such that $$\label{e4.2} |f(t,x,u)|\leq a(t)|u|+b(t,x), \quad (t,x,u)\in [0,T]\times \Omega \times \mathbb{R},$$ and $$\label{e4.3} \begin{gathered} f(t,x,u)u\leq 0, \quad |f(t,x,u)|\geq c(t)|u|+d(t,x), \\ (t,x)\in [0,T]\times \Omega ,\quad |u|\geq M. \end{gathered}$$ Then \eqref{e4.1} has at least one T-periodic solution. \end{theorem} \begin{remark} \label{rmk4.2} \rm For the problem above, Shioji \cite{s2} considers \eqref{e4.2} and the condition$$ \limsup_{|u|\to \infty } \mathop{\rm ess\,sup} _{(t,x)\in \mathbb{R}\times \Omega } \frac{f(t,x,u)}{u}<0, which is equivalent to there exist \delta ,M>0, f(t,x,u)/u\leq -\delta  for (t,x,u)\in \mathbb{R}\times \Omega \times \mathbb{R} with |u|\geq M. This clearly has the form of \eqref{e4.3} with c(t)=\delta , t\in [0,T], d=0. \end{remark} \begin{proof}[Proof of Theorem \ref{thm4.1}] The operator A:D(A)=\{u\in L^{1}(\Omega ); \rho (u)\in W_{0}^{1,1}(\Omega ), \Delta \rho (u)\in L^{1}(\Omega )\}\subset L^{1}(\Omega )\to L^{1}(\Omega ) given by Au:=-\Delta \rho (u), u\in D(A), is m-accretive in X=L^{1}(\Omega ), \overline{D(A)}=L^{1}(\Omega ), and it generates a compact semigroup (Vrabie \cite{v3}). Define F:\mathbb{R}\times L^{1}(\Omega )\to L^{1}(\Omega ) by F(t,u)(x):=f(t,x,u(x)), t\in \mathbb{R}, u\in L^{1}(\Omega ). >From \eqref{e4.2} F is well defined, Carath\'{e}odory, and satisfies \eqref{e1.2}. Since A0=0, for every u\in D(A) we have $$\label{e4.4} [u,\Delta \rho (u)-F(t,u)]_{+}\geq [u,\Delta \rho (u)]_{+}-[u,F(t,u)]_{+}\geq -[u,F(t,u)]_{+}.$$ Denote by \{u<0\}=\{x\in \Omega ;\; u(x)<0\}, \{u=0\}=\{x\in \Omega ;\; u(x)=0\}, \{u>0\}=\{x\in \Omega ;\; u(x)>0\}, and |\Omega | the measure of \Omega . >From \eqref{e4.2} and \eqref{e4.3}, for every u\in L^{1}(\Omega ) we obtain \label{e4.5} \begin{aligned} &[u,F(t,u)]_{+}\\ &=\int_{\{u>0\}} f(t,x,u(x))dx- \int_{\{u<0\}} f(t,x,u(x))dx+\int_{\{u=0\}} f(t,x,0)dx\\ &\leq 2Ma(t)+3\int_{\Omega } b(t,x)dx +\int_{\{u>M\}} f(t,x,u(x))dx-\int_{\{u<-M\}} f(t,x,u(x))dx\\ &\leq -c(t)\|u\|_{L^{1}}+Mc(t)|\Omega |+2Ma(t) +3\int_{\Omega } b(t,x)dx+\int_{\Omega } |d(t,x)|dx. \end{aligned} Relations \eqref{e4.3} and \eqref{e4.4} combined prove that \eqref{e1.3} is fulfilled with R=0. According to Theorem \ref{thm1.1}, \eqref{e4.1} has at least one T-periodic solution. \end{proof} Next, we consider the periodic problem for the nonlinear heat equation $$\label{e4.6} \begin{gathered} \frac{\partial y}{\partial t}-\Delta _{p}(y)=f(t,x,y(t,x)) \quad\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \Omega, \\ y=0\quad\mbox{a.e. }(t,x)\in \mathbb{R}_{+}\times \partial\Omega, \\ y(t,x)=y(t+T,x)\quad \mbox{for every }t\geq 0,\mbox{ and a.e. } x\in \Omega , \end{gathered}$$ where \Omega  is a bounded domain of \mathbb{R}^{n}, with smooth boundary \partial \Omega , p\geq 2, and \Delta _{p}y=\sum_{i=1}^n \frac{\partial }{\partial x_{i}} (|\frac{\partial y}{\partial x_{i}}|^{p-2} \frac{\partial y}{\partial x_{i}}), $$is the pseudo-Laplace operator. \begin{theorem} \label{thm4.2} Suppose that f:\mathbb{R}\times \Omega \times \mathbb{R}\to \mathbb{R}, f=f(t,x,u) is T-periodic in t, f(t,x,\cdot ) is continuous for almost every (t,x)\in \mathbb{R}\times \Omega , f(\cdot ,\cdot ,u) is measurable for every u\in \mathbb{R}, and there exist M>0, a,k\in L^{1}(0,T), a,k\geq 0, k\neq 0, b,d\in L^{1}(0,T;L^2(\Omega )) such that $$\label{e4.7} |f(t,x,u)|\leq a(t)|u|+b(t,x), \quad (t,x,u)\in [0,T]\times \Omega \times \mathbb{R},$$ and $$\label{e4.8} f(t,x,u)u\leq [c-k(t)]|u|^{p}+d(t,x)|u|, \quad (t,x)\in [0,T]\times \Omega ,\; |u|\geq M,$$ where c>0 is such that $$\label{e4.9} \int_{\Omega }|\nabla u(x)|^{p}dx\geq c\int_{\Omega }|u(x)|^{p}dx, \quad u\in W_{0}^{1,p}(\Omega ).$$ Then \eqref{e4.6} has at least one T-periodic solution. \end{theorem} \begin{remark} \label{rmk4.4} \rm In \cite{v2} the existence of periodic solutions for the nonlinear heat equation governed by the pseudo-Laplace operator is showed under the conditions$$ |f(t,x,u)|\leq a|u|+b,\quad f(t,x,u)u\leq \alpha |u|^{p}+\beta ,\;\; (t,x,u)\in \mathbb{R}\times \Omega \times \mathbb{R},  where $a,b,\alpha ,\beta >0$ and $\alpha 0$, $(t,x)\in \mathbb{R}\times \Omega$, since for $M=\beta /d$ and $|u|\geq M$, $d|u|\geq \beta$. \end{remark} \begin{proof}[Proof of Theorem \ref{thm4.2}] The operator $A:D(A)=\{u\in W_{0}^{1,2}(\Omega )$, $\Delta _{p}u\in L^2(\Omega )\}\subset L^2(\Omega )\to L^2(\Omega )$ given by $Au:=-\Delta_{p}u$, $u\in D(A)$, is maximal monotone in $X=L^2(\Omega )$, and it generates a compact semigroup (Vrabie \cite{v3}). Define $F:\mathbb{R}\times L^2(\Omega )\to L^2(\Omega )$ by $F(t,u)(x):=f(t,x,u(x))$, $t\in \mathbb{R}$, $u\in L^2(\Omega )$. >From \eqref{e4.7} $F$ is well defined, Carath\'{e}odory, and satisfies \eqref{e1.2}. For $u\in D(A)$, according to \eqref{e4.7}, \eqref{e4.8}, we have \label{e4.10} \begin{aligned} &\|u\|_{L^2}[u,Au-F(t,u)]_{+}\\ &=\langle u,-\Delta _{p}u-F(t,u)\rangle _{L^2} \\ &=\int_{\Omega }|\nabla u(x)|^{p}dx-\int_{\Omega }f(t,x,u(x))u(x)dx\\ &\geq c\int_{\Omega }|u|^{p}dx-\int_{\{|u|\geq M\}} f(t,x,u(x))u(x)dx -\int_{\{|u|0$and$\|u\|_{L^2}\geq R$, $$\label{e4.11} [u,Au-F(t,u)]_{+}\geq c(t)\|u\|_{L^2}-\alpha (t),$$ where$c(t)=\min \{c,k(t)\}R^{p-2}|\Omega |^{(2-p)/2}$,$t\in [0,T] $. Clearly,$c\geq 0$,$c\neq 0$and all the conditions of Theorem \ref{thm1.1} are fulfilled. This provides us with a$T\$-periodic solution of \eqref{e4.6}. \end{proof} \begin{thebibliography}{2} \bibitem{c1} R. Ca\c{s}caval and I. I. Vrabie; \emph{Existence of periodic solutions for a class of nonlinear evolution equations}, Rev. Mat. Univ. Complutense Madrid \textbf{7} (1994), 325-338. \bibitem{h1} N. Hirano and N. Shioji; \emph{Existence of periodic solutions under saddle point type conditions}, J. Nonlinear Convex Anal. \textbf{1} (2000), 115-128. \bibitem{l1} N. G. Lloyd; \emph{Degree theory}, Cambridge University Press (1978). \bibitem{s1} N. Shioji; \emph{Periodic solutions for nonlinear evolution equations in Banach spaces}, Funkc. Ekvacioj, Ser. Int. \textbf{42} (1999), 157-164. \bibitem{s2} N. Shioji; \emph{Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators}, Proc. Am. Math. Soc. \textbf{125} (1997), 2921-2929. \bibitem{v1} M. D. Voisei; \emph{Existence of periodic solutions for nonlinear equations of evolution in Banach spaces}, Math. Sci. Res. Hot-Line \textbf{3} (1999), 1-7. \bibitem{v2} I. I. Vrabie; \emph{Periodic solutions for nonlinear evolution equations in a Banach space}, Proc. Am. Math. Soc. \textbf{109}, (1990), 653-661. \bibitem{v3} I. I. Vrabie; \emph{Compactness methods for nonlinear evolutions. 2nd ed}, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman (1995). \end{thebibliography} \end{document}