\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 149, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/149\hfil Monotone iterative method] {Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces} \author[P. Chen, J. Mu\hfil EJDE-2010/149\hfilneg] {Pengyu Chen, Jia Mu} % in alphabetical order \address{Pengyu Chen \newline Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{chpengyu123@163.com} \address{Jia Mu \newline Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{mujia05@lzu.cn} \thanks{Submitted August 4, 2010. Published October 21, 2010.} \subjclass[2000]{34K30, 34K45, 35F25} \keywords{Initial value problem; lower and upper solution; \hfill\break\indent impulsive integro-differential evolution equation; $C_0$-semigroup; cone} \begin{abstract} We use a monotone iterative method in the presence of lower and upper solutions to discuss the existence and uniqueness of mild solutions for the initial value problem \begin{gather*} u'(t)+Au(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\\ \Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\\ u(0)=x_0, \end{gather*} where $A:D(A)\subset E\to E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)(t\geq 0)$ in $E$. Under wide monotonicity conditions and the non-compactness measure condition of the nonlinearity $f$, we obtain the existence of extremal mild solutions and a unique mild solution between lower and upper solutions requiring only that $-A$ generate a strongly continuous semigroup. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \section{Introduction} The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, chemical, biological, and engineering background; hence it has emerged as an important area of research in the previous decades, see for example \cite{l1}. Consequently, some basic results on impulsive differential equations have been obtained and applications to different areas have been considered by many authors; see \cite{b1,g1,l2,l5} and their references. In this article, we use a monotone iterative method in the presence of lower and upper solutions to discuss the existence of mild solutions to the initial value problem (IVP) of first order semilinear impulsive integro-differential evolution equations of Volterra type in an ordered Banach space $E$ $$\begin{gathered} u'(t)+Au(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\\ \Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\\ u(0)=x_0, \end{gathered} \label{e1}$$ where $A:D(A)\subset E\to E$ be a closed linear operator and $-A$ generates a strongly continuous semigroup ($C_0$-semigroup, in short) $T(t)(t\geq 0)$ in $E$; $f\in C(J\times E\times E, E)$, $J=[0,a]$, $a>0$ is a constant, $00$ and $\delta\in \mathbb{R}$ such that $$\| T(t)\|\leq Ce^{\delta t},\quad t\geq 0.$$ Let $I=[t_0,T](t_0\geq 0)$, $T>t_0$ be a constant. It is well-known \cite[Chapter 4, Theorem 2.9]{p1} that for any $x_0\in D(A)$ and $h\in C^1(I,E)$, the initial value problem of the linear evolution equation $$\begin{gathered} u'(t)+Au(t)=h(t),\quad t\in I,\\ u(t_0)=x_0, \end{gathered} \label{e8}$$ has a unique classical solution $u\in C^1(I,E)\cap C(I,E_1)$ given by $$u(t)=T(t-t_0)x_0+\int_{t_0}^tT(t-s)h(s)ds, \quad t\in I.\label{e9}$$ If $x_0\in E$ and $h\in C(I,E)$, the function $u$ given by \eqref{e9} belongs to $C(I,E)$, we call it a mild solution \cite{p1} of IVP\eqref{e8}. To prove our main results, for any $h\in PC(J,E)$, we consider the initial value problem (IVP) of linear impulsive evolution equation in $E$ $$\begin{gathered} u'(t)+Au(t)=h(t),\quad t\in J',\\ \Delta u |_{t=t_k}=y_k,\quad k=1,2,\dots ,m,\\ u(0)=x_0, \end{gathered} \label{e10}$$ where $y_k\in E$, $k=1,2,\dots,m$, $x_0\in E$. \begin{lemma} \label{lem2} Let $T(t)(t\geq 0)$ be a $C_0$-semigroup in $E$ generated by $-A$, for any $h\in PC(J,E)$, $x_0\in E$ and $y_k\in E$, $k=1,2,\dots,m$, then the linear IVP \eqref{e10} has a unique mild solution $u\in PC(J,E)$ given by u(t)=T(t)x_0+\int_0^t T(t-s)h(s)ds + \sum _{00$such that $$f(t,u_2,v_2)-f(t,u_1,v_1)\geq-M(u_2-u_1),$$ for any$t\in J$, and$ v_0(t)\leq u_1\leq u_2\leq w_0(t)$,$Tv_0(t)\leq v_1\leq v_2\leq Tw_0(t)$. \item[(H2)]$I_k(u)$is increasing on order interval$[v_0(t),w_0(t)]$for$t\in J, k=1,2,\dots,m$. \end{itemize} Then the IVP\eqref{e1} has minimal and maximal mild solutions$\underline{u}$and$\overline{u}$between$v_0$and$w_0$. \end{theorem} \begin{proof} Let$\overline{M}=\sup_{t\in J}\| S(t)\|$, we define the mapping$Q:[v_0,w_0]\to PC(J,E)by \begin{aligned} Qu(t)&=S(t)x_0+\int_0^t S(t-s)(f(s,u(s),Tu(s))+Mu(s))ds\\ &\quad +\sum_{00 such that $$\| f(t,u(t),Tu(t))+Mu(t)\|\leq \overline{M_1},\quad u\in[v_0,w_0].$$ By the compactness of $S(\epsilon)$, $Y_\epsilon(t)=\{(W_\epsilon u)(t):u\in[v_0,w_0]\}$ is precompact in $E$. Since \begin{align*} \| (Wu)(t)-(W_\epsilon u)(t)\| & \leq \int_{t-\epsilon}^t\| S(t-s)\|\cdot\| f(s,u(s),Tu(s))+Mu(s)\| ds \\ &\leq \overline{M}~\overline{M_1}\epsilon, \end{align*} the set $Y(t)$ is totally bounded in $E$. Furthermore, $Y(t)$ is precompact in $E$. On the other hand, for any $0\leq t_1\leq t_2\leq a$, we have \begin{aligned} &\| (Wu)(t_2)-(Wu)(t_1)\|\\ &=\|\int_0^{t_1} (S(t_2-s)-S(t_1-s))(f(s,u(s),Tu(s))+Mu(s))ds \\ &\quad +\int_{t_1}^{t_2} S(t_2-s)(f(s,u(s),Tu(s))+Mu(s))ds\| \\ &\leq \overline{M_1}\int_0^{t_1} \| S(t_2-s)-S(t_1-s)\| ds+\overline{M}~\overline{M_1}(t_2-t_1) \\ & \leq \overline{M_1}\int_0^a\| S(t_2-t_1+s)-S(s)\| ds +\overline{M}~\overline{M_1}(t_2-t_1). \end{aligned} \label{e16} The right side of \eqref{e16} depends on $t_2-t_1$, but is independen of $u$. As $T(\cdot)$ is compact, $S(\cdot)$ is also compact and therefore $S(t)$ is continuous in the uniform operator topology for $t>0$. So, the right side of \eqref{e16} tends to zero as $t_2-t_1\to 0$. Hence $W([v_0,w_0])$ is equicontinuous function of cluster in $C(J,E)$. The same idea can be used to prove the compactness of $V$. For $0\leq t\leq a$, since $\{Qu(t):u\in[v_0,w_0]\}=\{S(t)x_0+(Wu)(t)+(Vu)(t):u\in[v_0,w_0]\}$, and $Qu(0)=x_0$ is precompact in $E$. Hence, $Q([v_0,w_0])$ is precompact in $C(J,E)$ by the Arzela-Ascoli theorem. So $Q:[v_0,w_0]\to[v_0,w_0]$ is completely continuous. Hence, $Q$ has minimal and maximal fixed points $\underline{u}$ and $\overline{u}$ in $[v_0,w_0]$, and therefore, they are the minimal and maximal mild solutions of the IVP\eqref{e1} in $[v_0,w_0]$, respectively. \end{proof} \begin{theorem} \label{thm2} Let $E$ be an ordered Banach space, whose positive cone $P$ is normal, $A:D(A)\subset E\to E$ be a closed linear operator and $-A$ generates a positive $C_0$-semigroup $T(t)(t\geq 0)$ in $E$, $f\in C(J\times E\times E, E)$ and $I_k\in C(E,E)$, $k=1,2,\dots,m$. If the IVP\eqref{e1} has a lower solution $v_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$ and an upper solution $w_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$ with $v_0\leq w_0$, conditions {\rm (H1)} and {\rm (H2)} hold, and satisfy \begin{itemize} \item[(H3)] There exist a constant $L>0$ such that for all $t\in J$, $$\alpha(\{f(t,u_n,v_n)\})\leq L(\alpha(\{u_n\}) +\alpha(\{v_n\})),$$ and increasing or decreasing sequences $\{u_n\}\subset[v_0(t),w_0(t)]$ and $\{v_n\}\subset[v_0(t),w_0(t)]$. \end{itemize} Then the IVP\eqref{e1} has minimal and maximal mild solutions between $v_0$ and $w_0$, which can be obtained by a monotone iterative procedure starting from $v_0$ and $w_0$ respectively. \end{theorem} \begin{proof} From Theorem \ref{thm1}, we know that $Q:[v_0,w_0]\to[v_0,w_0]$ is a continuously increasing operator. Now, we define two sequences $\{v_n\}$ and $\{w_n\}$ in $[v_0,w_0]$ by the iterative scheme $$v_n=Qv_{n-1},\quad w_n=Qw_{n-1},\quad n=1,2,\dots.\label{e17}$$ Then from the monotonicity of $Q$, it follows that $$v_0\leq v_1\leq v_2\leq\dots\leq v_n\leq \dots \leq w_n\leq \dots\leq w_2\leq w_1\leq w_0.\label{e18}$$ We prove that $\{v_n\}$ and $\{w_n\}$ are convergent in $J$. For convenience, let$B=\{v_n: n\in \mathbb{N}\}$ and $B_0=\{v_{n-1}: n\in \mathbb{N}\}$. Then $B=Q(B_0)$. Let $J_1'=[0,t_1]$, $J_k'=(t_{k-1},t_k]$, $k=2,3,\dots m+1$. From $B_{0}=B\bigcup\{v_0\}$ it follows that $\alpha(B_{0}(t))=\alpha(B(t))$ for $t\in J$. Let $\varphi(t):=\alpha(B(t)), t\in J$, Going from $J_1'$ to $J_{m+1}'$interval by interval we show that $\varphi(t)\equiv 0$ in $J$. For $t\in J$, there exists a $J_k'$ such that $t\in J_k'$. By \eqref{e2} and Lemma \ref{lem1}, we have that \begin{align*} \alpha(T(B_0)(t)) &=\alpha\Big(\Big\{\int_0^tK(t,s)v_{n-1}(s)ds: n\in \mathbb{N}\Big\} \Big) \\ &\leq \sum _{j=1}^{k-1}\alpha\Big(\Big\{\int_{t_{j-1}}^{t_j}K(t,s)v_{n-1}(s)ds: n\in \mathbb{N}\Big\}\Big)\\ &\quad +\alpha\Big(\Big\{\int_{t_{k-1}}^tK(t,s)v_{n-1}(s)ds: n\in \mathbb{N}\Big\}\Big)\\ &\leq 2K_0\sum _{j=1}^{k-1}\int_{t_{j-1}}^{t_j}\alpha(B_0(s))ds +2K_0\int_{t_{k-1}}^t\alpha(B_0(s))ds \\ &=2K_0\sum_{j=1}^{k-1}\int_{t_{j-1}}^{t_j}\varphi(s)ds +2K_0\int_{t_{k-1}}^t\varphi(s)ds \\ &=2K_0\int_0^t\varphi(s)ds, \end{align*} and therefore, $$\int_0^t\alpha(T(B_0)(s))ds\leq 2aK_0\int_0^t\varphi(s)ds.\label{e19}$$ For $t\in J_1'$, from \eqref{e13}, using Lemma \ref{lem1}, assumption (H3) and \eqref{e19}, we have \begin{align*} \varphi(t)&=\alpha(B(t))=\alpha(Q(B_0)(t))\\ &=\alpha\Big(\Big\{S(t)x_0+\int_0^t S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s))+Mv_{n-1}(s))ds\Big\}\Big)\\ &\leq2\overline{M}\int_0^t\alpha(\{f(s,v_{n-1}(s),Tv_{n-1}(s)) +Mv_{n-1}(s)\})ds \\ &\leq 2\overline{M}\int_0^t(L(\alpha(B_{0}(s))+\alpha(Q(B_{0})(s))) +M\alpha(B_{0}(s)))ds \\ &\leq2\overline{M}(L+M+2aLK_0)\int_0^t\varphi(s)ds. \end{align*} Hence by the Bellman inequality, $\varphi(t)\equiv 0$ in $J_1'$. In particular, $\alpha(B(t_1))=\alpha(B_{0}(t_1))=\varphi(t_1)=0$, this implies that $B(t_1)$ and $B_{0}(t_1)$ are precompact in $E$. Thus $I_1(B_{0}(t_1))$ is precompact in $E$, and $\alpha(I_1(B_{0}(t_1)))=0$. Now, for $t\in J_2'$, by \eqref{e13} and the above argument for $t\in J_1'$, we have \begin{align*} \varphi(t)&=\alpha(B(t))=\alpha(Q(B_0)(t)) \\ &=\alpha\Big(\Big\{S(t)x_0+\int_0^t S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s)) +Mv_{n-1}(s))ds\\ &\quad +S(t-t_1)I_1(v_{n-1}(t_1))\Big\}\Big) \\ &\leq2\overline{M}(L+M+2aLK_0)\int_0^t\varphi(s)ds\\ &=2\overline{M}(L+M+2aLK_0)\int_{t_1}^t\varphi(s)ds. \end{align*} Again by Bellman inequality, $\varphi(t)\equiv 0$ in $J_2'$, from which we obtain that $\alpha(B_{0}(t_2))=0$ and $\alpha(I_2(B_{0}(t_2)))=0$. Continuing such a process interval by interval up to $J_{m+1}'$, we can prove that $\varphi(t)\equiv 0$ in every $J_k', k=1,2,\dots,m+1$. Hence, for any $t\in J, \{v_n(t)\}$ is precompact, and $\{v_n(t)\}$ has a convergent subsequence. Combing this with the monotonicity \eqref{e18}, we easily prove that $\{v_n(t)\}$ itself is convergent, i.e., $\lim_{n\to \infty}v_n(t)=\underline{u}(t)$, $t\in J$. Similarly$, \lim_{n\to \infty}w_n(t)=\overline{u}(t)$, $t\in J$. Evidently $\{v_n(t)\}\in PC(J,E)$, so $\underline{u}(t)$ is bounded integrable in every $J_k$, $k=1,2,\dots,{m+1}$. Since for any $t\in J_k$, \begin{align*} v_n(t)&=Qv_{n-1}(t)\\ &=S(t)x_0+\int_0^t S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s))+Mv_{n-1}(s))ds\\ &\quad +\sum_{0n, by (H1) and (H4), \begin{align*} \theta&\leq f(t,u_m,v_m)-f(t,u_n,v_n)+M(u_m-u_n) \\ &\leq (M+\overline{C})(u_m-u_n)+\overline{L}(v_n-v_m). \end{align*} By this and the normality of coneP, we have \begin{align*} &\| f(t,u_m,v_m)-f(t,u_n,v_n)\|\\ &\leq N\| (M+\overline{C})(u_m-u_n)+\overline{L}(v_n-v_m)\| +M\| u_m-u_n\| \\ &\leq (N(M+\overline{C})+M)\| u_m-u_n\|+N\overline{L}\| v_n-v_m\|. \end{align*} From this inequality and the definition of the measure of noncompactness, it follows that \begin{align*} \alpha(\{f(t,u_n,v_n)\}) &\leq (N(M+\overline{C})+M)\alpha(\{u_n\}) +N\overline{L}\alpha(\{v_n\})\\ &\leq L_1(\alpha(\{u_n\})+\alpha(\{v_n\})), \end{align*} whereL_1=\max\{(N(M+\overline{C})+M),N\overline{L}\}$. If$\{u_n\}$and$\{v_n\}$are two decreasing sequences, the above inequality is also valid. Hence (H3) holds. Therefore, by Theorem \ref{thm2}, the IVP\eqref{e1} has minimal and maximal mild solutions$\underline{u}$and$ \overline{u}$between$v_0$and$w_0$. By the proof of Theorem \ref{thm2}, \eqref{e17} and \eqref{e18} are valid. Going from$J_1'$to$J_{m+1}'$interval by interval we show that$\underline{u}(t)\equiv\overline{u}(t)$in every$J_k'$. For$t\in J_1', by \eqref{e13} and assumption (H4), we have \begin{align*} \theta &\leq \overline{u}(t)-\underline{u}(t) =Q\overline{u}(t)-Q\underline{u}(t) \\ &=\int_0^t S(t-s)\big[f(s,\overline{u}(s),T\overline{u}(s)) -f(s,\underline{u}(s),T\underline{u}(s)) +M(\overline{u}(s)-\underline{u}(s))\big]ds \\ &\leq \overline{M}(M+\overline{C} +a\overline{L}K_0)\int_0^t(\overline{u}(s)-\underline{u}(s))ds. \end{align*} From this and the normality of coneP$it follows that $$\| \overline{u}(t)-\underline{u}(t)\| \leq N\overline{M}(M+\overline{C}+a\overline{L}K_0) \int_0^t\|\overline{u}(s)-\underline{u}(s)\| ds.$$ By this and Bellman inequality, we obtained that$\underline{u}(t)\equiv\overline{u}(t)$in$J_1'$. For$t\in J_2'$, since$I_1(\overline{u}(t_1))=I_1(\underline{u}(t_1))$, using \eqref{e13} and completely the same argument as above for$t\in J_1', we can prove that \begin{align*} \| \overline{u}(t)-\underline{u}(t)\| &\leq N\overline{M}(M+\overline{C}+a\overline{L}K_0) \int_0^t\|\overline{u}(s)-\underline{u}(s)\| ds \\ & =N\overline{M}(M+\overline{C}+a\overline{L}K_0) \int_{t_1}^t\|\overline{u}(s)-\underline{u}(s)\| ds. \end{align*} Again, by the Bellman inequality, we obtain that\underline{u}(t)\equiv\overline{u}(t)$in$J_2'$. Continuing such a process interval by interval up to$J_{m+1}'$, we see that$\underline{u}(t)\equiv\overline{u}(t)$over the whole of$J$. Hence,$\widetilde{u}:=\underline{u}=\overline{u}$is the unique mild solution of the IVP\eqref{e1} in$[v_0,w_0]$, which can be obtained by the monotone iterative procedure \eqref{e18} starting from$v_0$or$w_0$. \end{proof} If lower solution and upper solutions for the IVP\eqref{e1} do not exist, then we have the following result. \begin{theorem} \label{thm4} Let$E$be an ordered Banach space, whose positive cone$P$is normal,$A:D(A)\subset E\to E$be a closed linear operator and$-A$generates a positive$C_0$-semigroup$T(t)(t\geq 0)$in$E$,$f\in C(J\times E\times E, E)$and$I_k\in C(E,E)$,$k=1,2,\dots,m$. If there exist$a>0$,$x_0\in D(A)$,$x_0\geq \theta$,$y_k\in D(A)$,$y_k\geq \theta$,$k=1,2,\dots,m$,$h\in PC(J,E)\cap C^1(J',E)$and$h(t)\geq \theta$, such that \begin{gather*} f(t,x,Tx)\leq ax+h(t), \quad I_k(x)\leq y_k, x\geq \theta; \\ f(t,x,Tx)\geq ax-h(t), \quad I_k(x)\geq-y_k, x\leq \theta. \end{gather*} Then we have: \begin{itemize} \item[(i)] If the$C_0$-semigroup$T(t)(t\geq 0)$generated by$-A$is compact in$E$, and conditions {\rm (H1)} and {\rm (H2)} are satisfied, then the IVP\eqref{e1} has minimal and maximal mild solutions. \item[(ii)] If conditions {\rm (H1), (H2), (H3)} are satisfied, then the IVP\eqref{e1} has minimal and maximal mild solutions. \item[(iii)] If the positive cone$P$is regular, and conditions {\rm (H1)} and {\rm (H2)} are satisfied, then the IVP\eqref{e1} has minimal and maximal mild solutions. \item[(iv)] If conditions {\rm (H1), (H2), (H4)} are satisfied, then the IVP\eqref{e1} has a unique mild solution. \end{itemize} \end{theorem} \begin{proof} Firstly, we consider the IVP of linear impulsive evolution equation in$E$$$\begin{gathered} u'(t)+Au(t)-au(t)=h(t),\quad t\in J',\\ \Delta u |_{t=t_k}=y_k,\quad k=1,2,\dots ,m,\\ u(0)=x_0. \end{gathered} \label{e20}$$ Since$-A+aI$generates a positive$C_0$-semigroup$S(t)=e^{at}T(t)(t\geq 0)$in$E$. So, by \cite[Chapter 4, Theorem 2.9]{p1} and Lemma \ref{lem2}, we know that IVP\eqref{e20} has a unique positive classical solution$ \widetilde{u}\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$. Let$v_0=-\widetilde{u}$,$w_0=\widetilde{u}$, it is easy to see that$v_0$and$w_0$are lower solution and upper solution of the IVP\eqref{e1} respectively. So, our conclusions (i), (ii), (iii) and (iv) follow from the Theorem \ref{thm1}, Theorem \ref{thm2}, Corollary \ref{coro1} and Theorem \ref{thm3} respectively. \end{proof} \section{Applications} Consider the IVP of impulsive parabolic partial differential equation $$\begin{gathered} \frac{\partial u}{\partial t}+A(x,D)u(t) = f(x,t,u(t),Tu(t)),\quad x\in\Omega, t\in J, t\neq t_k,\\ \Delta u |_{t=t_k}=I_k(u(x,t_k)) ,\quad x\in\Omega, k=1,2,\dots ,m,\\ Bu=0,\quad (x,t)\in \partial\Omega\times J,\\ u(x,0)=\varphi(x),\quad x\in\Omega, \end{gathered} \label{e21}$$ where$J=[0,a]$,$0N+2$,$P=\{u\in L^p(\Omega): u(x)\geq 0, a.e.~x\in \Omega\}$, and define the operator$A$as follows: $$D(A)=\{u\in W^{2,p}(\Omega): Bu=0\}, \quad Au=A(x,D)u.$$ Then$E$is a Banach space,$P$is a regular cone of$E$, and$-A$generates a positive and analytic$C_0$-semi-group$T(t)(t\geq 0)$in$E$(see \cite{l3,l4,p1}). So, the problem \eqref{e21} can be transformed into the IVP \eqref{e1}. To solve the IVP\eqref{e21}, we also need following assumptions: (a) Let$f(x,t,0,0)\geq 0$,$I_k(0)\geq 0$,$\varphi(x)\geq 0$,$x\in\Omega$, and there exists a function$w=w(x,t)\in PC(J,E)\cap C^{2,1}(\overline{\Omega}\times J)$, such that \begin{gather*} \frac{\partial w}{\partial t}+A(x,D)w\geq f(x,t,w,Tw),\quad (x,t)\in\Omega\times J, t\neq t_k,\\ \Delta w|_{t=t_k}\geq I_k(w(x,t_k)) ,\quad x\in\Omega,\; k=1,2,\dots ,m,\\ Bw=0,\quad (x,t)\in\partial\Omega\times J,\\ w(x,0)\geq\varphi(x),\quad x\in\Omega. \end{gather*} (b) There exists a constant$M>0$such that $$f(x,t,x_2,y_2)-f(x,t,x_1,y_1) \geq-M(x_2-x_1),$$for any$t\in J$, and$ 0\leq x_1\leq x_2\leq w(x,t), 0\leq y_1\leq y_2\leq Tw(x,t)$. (c) For any$u_1, u_2\in [0,w(x,t)]$with$u_1\leq u_2$, we have $$I_k(u_1(x,t_k))\leq I_k(u_2(x,t_k)),\quad x\in\Omega,\; k=1,2,\dots ,m.$$ Assumption (a) implies that$v_0\equiv 0$and$w_0\equiv w(x,t)$are lower and upper solutions of the IVP\eqref{e1} respectively, and from (b) and (c), it is easy to verify that conditions (H1) and (H2) are satisfied. So, from Corollary \ref{coro1}, we have the following result. \begin{theorem} \label{thm5} If the assumptions (a), (b) and (c) are satisfied, then the IVP\eqref{e21} has minimal and maximal mild solutions between$0$and$w(x,t)$, which can be obtained by a monotone iterative procedure starting from$0$and$w(x,t)\$ respectively. \end{theorem} \subsection*{Acknowledgements} The author is deeply indebted to the anonymous referee for his/her valuable suggestions which improve the presentation of this paper. This work is supported by grants 10871160 from the NNSF of China, 0710RJZA103 from the the NSF of Gansu Province, and by Project of NWNU-KJCXGC-3-47. \begin{thebibliography}{00} \bibitem{b1} M. Benchohra, J. Henderson, S. Ntouyas; Impulsive differential equations and inclusions, Contemporary Mathematics and its Applications, vol. 2, Hindawi Publ.Corp., 2006. \bibitem{d1} K. Deiling; Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. \bibitem{d2} S. Du, V. Lakshmikantham; Monotone iterative technique for differential equtions in Banach spaces, J. Anal. Math. Anal. 87(1982), 454--459. \bibitem{g1} D. Guo, X. Liu; Extremal solutions of nonlinear impulsive integro differential equations in Banach spaces, J. Math. Anal. Appl. 177(1993), 538--552. \bibitem{h1} H. P. Heinz; On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7(1983), 1351--1371. \bibitem{l1} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov; Theory of impulsive differential equations, World Scientific, Singapore, 1989. \bibitem{l2} Y. Li, Z. Liu; Monotone iterative technique for addressing impulsive integro-differential equtions in Banach spaces, Nonlinear Anal. 66(2007), 83--92. \bibitem{l3} Y. Li; Positive solutions of abstract semilinear evolution equations and their applications, Acta. Math. Sinica. 39(5)(1996), 666--672 (in Chinese). \bibitem{l4} Y. Li; Global solutions of inition value problems for abstract semilinear evolution equations, Acta. Analysis Functionalis Applicata. 3(4)(2001), 339--347 (in Chinese). \bibitem{l5} J. Liu, Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impulsive Systems. 6(1)(1999), 77--85. \bibitem{p1} A. Pazy; Semigroups of linear operators and applications to partial differential equations, Springer--verlag, Berlin, 1983. \bibitem{s1} J. Sun, Z. Zhao; Extremal solutions initial value problem for integro-differential of mixed type in Banach spaces, Ann. Diff. Eqs. 8(1992), 465--474. \bibitem{s2} J. Sun, X. Zhang; The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta. Math. Sinica. 48(3)(2005), 439--446 (in Chinese). \bibitem{z1} X. Zhang; Solutions of semilinear evolution equations of mixed type in Banach spaces, Acta. Analysis Functionalis Applicata. 11(4)(2009), 363--368 (in Chinese). \end{thebibliography} \end{document}