\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 84, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/84\hfil Quasilinear impulsive equations] {Existence of mild solutions for quasilinear integrodifferential equations with \\ impulsive conditions} \author[K. Balachandran, F. P. Samuel\hfil EJDE-2009/84\hfilneg] {Krishnan Balachandran, Francis Paul Samuel} % in alphabetical order \address{Krishnan Balachandran \newline Department of Mathematics, Bharathiar University, Coimbatore 641 046, India} \email{balachandran\_k@lycos.com} \address{Francis Paul Samuel \newline Department of Mathematics, Bharathiar University, Coimbatore 641 046, India} \email{paulsamuel\_f@yahoo.com} \thanks{Submitted February 24, 2009. Published July 10, 2009.} \subjclass[2000]{34A37, 34G60, 34G20, 47H10} \keywords{Semigroup; mild solution; impulsive conditions} \begin{abstract} We prove the existence and uniqueness of mild solutions of quasilinear integrodifferential equations with nonlocal and impulsive conditions in Banach spaces. The results are obtained by using a fixed point technique and semigroup theory. Examples are provided to illustrate the theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \section{Introduction} Many evolution process are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control model in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus differential equations involving impulsive effects appear as a natural description of observed evolution phenomena of several real world problems. Existence of solutions of impulsive differential equation of the form \begin{gather} u'(t) = Au(t)+f(t,u(t)), \quad t \in (0,a]\label{1e1}\\ u(0)+g(u)= u_0,\label{1e2}\\ \Delta u(t_i)=I_i(u(t_i)), \quad i=1,2,3,\dots , p,\; 0 \omega$every finite sequences$0 \leq t_1 \leq t_2 \leq \dots \leq t_k\leq T$,$b_j \in B,\ 1 \leq j\leq k$. The stability of$\{A(t, b)\}, (t, b) \in [0,T] \times B$implies (see \cite{P1}) that $\|\prod_{j =1}^k S_{t_j,b_j}(s_j)\| \leq M \exp \big\{\omega\sum_{j=1}^k s_j\big\},\quad s_j \geq 0$ and any finite sequences$0 \leq t_1 \leq t_2 \leq \dots \leq t_k\leq T$,$b_j \in B$,$1 \leq j\leq k$.$k = 1,2,\dots $\end{definition} \begin{definition} \label{def2.3} \rm Let$S_{t,b}(s), s\geq 0$be the$C_0$-semigroup generatated by$A(t, b)$,$(t ,b) \in J \times B$. A subspace$Y$of$X$is called$A(t, b)$-admissible if$Y$is invariant subspace of$S_{t,b}(s)$and the restriction of$S_{t,b}(s)$to$Y$is a$C_0$-semigroup in$Y$. \end{definition} Let$B\subset X$be a subset of$X$such that for every$(t, b) \in [0,T] \times B$,$A(t, b)$is the infinitesimal generator of a$C_0$-semigroup$S_{t,b}(s), s \geq0$on$X$. We make the following assumptions: \begin{itemize} \item[(H1)] The family$\{A(t, b)\},(t, b) \in [0,T] \times B$is stable. \item[(H2)]$Y$is$A(t, b)$-admissible for$(t, b)\in [0,T] \times B$and the family$\{\tilde{A}(t, b)\}, (t, b) \in [0,T] \times B$of parts$\tilde{A}(t, b)$of$A(t, b)$in$Y$, is stable in$Y$. \item[(H3)] For$(t, b)\in [0,T] \times B$,$D(A(t, b)) \supset Y$,$A(t, b)$is a bounded linear operator from$Y$to$X$and$t \to A(t, b)$is continuous in the$B(Y, X)$norm$\|.\|$for every$b\in B$. \item[(H4)] There is a constant$L > 0$such that $\|A(t, b_1) - A(t, b_2)\|_Y{_\to}_ X\ \leq L \|b_1 - b_2\|_X$ holds for every$b_1, b_2 \in B$and$0\leq t \leq T$. \end{itemize} Let$B$be a subset of$X$and$\{A(t, b)\}, (t, b)\in [0,T] \times B$be a family of operators satisfying the conditions (H1)--(H4). If$u \in PC([0,T] : X)$has values in$B$then there is a unique evolution system$U(t,s;u), 0 \leq s \leq t \leq T$, in$X$satisfying, (see \cite[Theorem 5.3.1 and Lemma 6.4.2, pp. 135, 201-202]{P1} \begin{itemize} \item[(i)]$\|U(t,s;u)\| \leq M e^{\omega(t-s)}$for$0 \leq s \leq t \leq T$. where$M$and$\omega$are stability constants. \item[(ii)]$\frac{\partial^+}{\partial t} U(t,s;u)y =A(s, u(s))U(t,s;u)y$for$y\in Y$, for$0 \leq s \leq t \leq T$. \item[(iii)]$\frac{\partial}{\partial s} U(t,s;u)y = -U(t,s;u)A(s, u(s))y$for$y\in Y$, for$0 \leq s\leq t \leq T$. \end{itemize} Further we assume that \begin{itemize} \item[(H5)] For every$u \in PC([0,T] : X)$satisfying$u(t)\in B$for$0 \leq t \leq T$, we have $U(t,s;u)Y\subset Y,\quad 0 \leq s\leq t \leq T$ and$U(t,s;u)$is strongly continuous in$Y$for$0 \leq s\leq t \leq T$. \item[(H6)] Closed bounded convex subsets of$Y$are closed in$X$. \item[(H7)] For every$(t,b) \in J \times B$,$f(t, b) \in Y$and$((t,s),b) \in \Omega \times B$,$g(t,s, b) \in Y$. \item[(H8)]$h: PC([0,T]: B) \to Y$is Lipschitz continuous in$X$and bounded in$Y$, that is, there exist constant$H > 0$such that \begin{gather*} \|h(u) - h(v)\|_Y \leq H\|u - v\|_{PC},\ \ u,v\in PC([0,T];X). \end{gather*} For the conditions (H9) and (H10) let$Z$be taken as both$X$and$Y$. \item[(H9)]$g:\Omega \times Z \to Z$is continuous and there exist constants$G > 0$and$G_1>0$such that \begin{gather*} \int_0^t\|g(t,s,u)- g(t,s,v)\|_Z ds \leq G \|u - v\|_Z),\ \ u,v \in X, \\ G_1= \max\{\int_0^t\|g(t,s,0)\|_Z \ ds :(t,s)\in \Omega \}. \end{gather*} \item[(H10)]$f : [0,T] \times Z \to Z$is continuous and there exist constants$F > 0$and$F_1 > 0$such that \begin{gather*} \|f(t,u) - f(t,v)\|_Z \leq F \|u - v\|_Z , \ u,v\in X, \\ F_1 = \max_{t\in [0,T]} \|f(t,0)\|_Z. \end{gather*} \end{itemize} Let us take$M_0 = \max \{\|U(t,s;u)\|_{B(Z)}, 0 \leq s\leq t \leq T, \ u\in B\}$. \begin{itemize} \item[(H11)]$I_i: X \to X$is continuous and there exist constant$l_i > 0,\\ i=1,2,3,\dots,m$such that \begin{eqnarray*} \|I_i(u) - I_i(v)\| &\leq& l_i\|u - v\|, \ \ u,v\in X. \end{eqnarray*} \item[(H12)] There exist a positive constant$r>0such that \begin{gather*} M_0\Big[\|u_0\|_Y +Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]+\sum_{i = 1}^m (l_ir+\|I_i(0)\|)\Big] \leq r \ \ \mbox{and}\\ \begin{aligned} q &=\Big\{KT\Big[\|u_0\|_Y +Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]+\sum_{i = 1}^m (l_ir+\|I_i(0)\|)\Big]\\ &\quad+M_0\Big[H+T(F+G)+\sum_{i = 1}^m l_i\Big]<1. \end{aligned} \end{gather*} \end{itemize} \begin{definition} \label{def2.4} \rm A functionu\in PC([0,T] : X)is a mild solution of equations \eqref{1e6}--\eqref{1e8} if it satisfies \begin{aligned} u(t)&=U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^tU(t,s;u)\Big[f(s,u(s))\\ &\quad+\int_0^s g(s,\tau,u(\tau))d\tau \Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)), \ \ 0\leq t\leq T \end{aligned} \label{2e1} \end{definition} \begin{definition} \label{def2.5} \rm A functionu\in PC([0,T] : X)$such that$u(t)\in D(A(t,u(t)) \ $for$t\in (0,T], u\in C^1((0,T]\backslash\{t_1,t_2,\dots,t_m\}:X)$and satisfies \eqref{1e6}--\eqref{1e8} in X is called a classical solution of \eqref{1e6}--\eqref{1e8} on$[0,T]$, \end{definition} Further there exists a constant$K > 0$such that for every$u, v \in PC([0,T] : X)$and every$y \in Y$we have $$\label{2e2} \|U(t,s;u)y - U(t,s;v)y\| \leq K T\|y\|_Y \|u - v\|_{PC}.$$ \section{Existence Result} \begin{theorem} \label{thm3.1} Let$u_0 \in Y$and let$B = \{u\in X: \|u\|_X \leq r\}$,$r > 0$. If the assumptions {\rm (H1)--(H12)} are satisfied, then \eqref{1e6}--\eqref{1e8} has a unique mild solution$u\in PC([0,T] : Y)$. \end{theorem} \begin{proof} Let$S$be a nonempty closed subset of$PC([0,T]:X)$defined by $$S = \{u:u\in PC([0, T]: X),\|u(t)\|_{PC} \leq r \mbox{ for}\ 0\leq t\leq T\}.$$ Consider a mapping$\Phi$on$Sdefined by \begin{aligned} (\Phi u)(t) &= U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^t U(t,s;u)\Big[f(s,u(s))\\ &\quad +\int_0^sg(s,\tau, u(\tau))d\tau\Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)). \end{aligned} \label{3e1} We claim that\Phi$maps$S$into$S$. For$u\in S, we have \begin{align*} & \|\Phi u(t)\|_Y \\ &\leq \|U(t,0;u)u_0\|+\|U(t,0;u)h(u)\|\\ &\quad + \int_0^t\ \|U(t,s;u)\|\Big[\|f(s,u(s))- f(s,0)\| + \|f(s,0)\|\\ &\quad + \|\int_0^s [g(s,\tau,u(\tau))- g(s,\tau,0)] d\tau\|+ \|\int_0^s g(s,\tau,0)d\tau\|\Big]ds\\ &\quad+\sum_{0 < t_i < t}\|U(t,t_i;u)I_i(u(t_i))\|\\ &\leq M_0 \|u_0\|_Y+M_0\Big[H\|u\|+\|h(0)\|\Big]+M_0\Big[\int_0^tF\|u(s)\|ds+F_1T \\ &\quad + \int_0^tG\|u(s)\|ds+G_1T\Big] +M_0\sum_{i = 1}^m\Big(l_i\|u\|+\|I_i(0)\|\Big) \\ &\leq M_0\Big[\|u_0\|_Y +Hr+\|h(0)\|+T\Big[r(F+G)+F_1+G_1\Big]+\sum_{i = 1}^m\Big(l_ir+\|I_i(0)\|\Big). \end{align*} From assumption (H12), one gets\|\Phi u(t)\|_Y \leq r$. Therefore$\Phi$maps$S$into itself. Moreover, if$u,v \in S, then \begin{align*} &\|\Phi u(t)- \Phi v(t)\|\\ &\leq \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)h(u)-U(t,0;v)h(v)\|\\ &\quad + \int_0^t\|U(t,s;u)\Big[f(s,u(s)) + \int_0^s g(s,\tau,u(\tau))d\tau\Big]+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i))\\ &\quad - U(t,s;v)\Big[f(s,v(s)) + \int_0^s g(s,\tau,v(\tau))d\tau\Big]-\sum_{0 < t_i < t}U(t,t_i;v) I_i(v(t_i))\|ds \end{align*} Using assumptions (H8)-(H12), one can get \begin{align*} &\|\Phi u(t)- \Phi v(t)\|\\ &\leq KT\|u_0\|_Y\|u - v\|_{PC} +KT\Big[H\|u\|+\|h(0)\|\Big]\|u - v\|_{PC} \end{align*} \begin{align*} &\quad + M_0H\|u - v\|_{PC} + KT\|u - v\|_{PC}\Big[\int_0^t(F\|u(s)\|+F_1)ds\\ &\quad+\int_0^t(G\|u(s)\|+G_1)ds\Big] + KT\|u - v\|_{PC}\sum_{i = 1}^m\Big(l_ir+\|I_i(0)\|\Big)\\ &\quad + M_0\Big[\int_0^tF\|u(s)-v(s)\|ds +\int_0^tG\|u(s)- v(s)\|ds\Big]+M_0\sum_{i = 1}^ml_i\|u - v\|_{PC}\\ &\leq \Big\{KT\Big[\|u_0\|_Y+Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]\\ &\quad + \sum_{i = 1}^m(l_ir+\|I_i(0)\|)\Big] +M_0\Big[H+T(F+G)+\sum_{i = 1}^ml_i\Big]\Big\}\|u - v\|_{PC}\\ &= q \|u - v\|_{PC}, \ \ \ u,v\in PC([0,T];X) \end{align*} where0 < q < 1$. From this inequality it follows that for any$t\in [0,T]$, $\|\Phi u(t)- \Phi v(t)\|\leq q \|u - v\|_{PC},$ so that$\Phi$is a contraction on$S$. From the contraction mapping theorem it follows that$\Phi$has a unique fixed point$u\in S$which is the mild solution of \eqref{1e6}--\eqref{1e8} on$[0,T]$. Note that$u(t)$is in$PC([0,T] : Y)$by (H6) see \cite[pp. 135, 201-202 lemma 7.4]{P1}. In fact,$u(t)$is weakly continuous as a$Y$-valued function. This implies that$u(t)$is separably valued in$Y$, hence it is strongly measurable. Then$\|u(t)\|_ {PC}$is bounded and measurable function in$t$. Using the relation$u(t)= \Phi u(t)$, we conclude that$u(t)$is in$PC([0,T] :Y)$. \end{proof} \noindent\textbf{Remark.} %3.1 Using the additional assumption$A(t,b)u_0, b\in B$is bounded in$Y$one can establish a unique local classical solution for the equations \eqref{1e6}--\eqref{1e8}. \section{Quasilinear Delay Integrodifferential Equation} Next we consider the following quasilinear delay integrodifferential equation with impulsive nonlocal conditions \eqref{1e7} and \eqref{1e8} \begin{gather} u'(t)+A(t,u)u(t)= f(t,u(\alpha(t)))+\int_0^t g(t,s,u(\beta(s)))ds, \quad t\in [0,T],\label{4e1} \end{gather} where$A, f$and$h$are as before. Assume the following additional conditions: \begin{itemize} \item[(H13)]$\alpha,\beta$:$[0,T] \to [0,T]$are absolutely continuous and there exists constants$\delta_1, \delta_2 > 0$and such that$\alpha^\prime(t) \geq \delta_1$and$\ \beta^\prime(t) \geq \delta_2$for$0 < t \leq T$. \item[(H14)] There exist a positive constant$k>0such that \begin{gather*} M_0\Big[\|u_0\|_Y +Hk+\|h(0)\|+T\Big[k/\delta_1\delta_2(F\delta_2+G\delta_1)+F_1+G_1\Big]\\ +\sum_{i = 1}^m\Big(l_ik+\|I_i(0)\|\Big) \leq k \\ \begin{aligned}\mbox{and}\ \ \ p &=\{KT\Big[\|u_0\|_Y+Hk+\|h(0)\|+T[k/\delta_1\delta_2(F\delta_2+G\delta_1)+F_1+G_1]\\ &\quad + \sum_{i = 1}^m(l_ik+\|I_i(0)\|)\Big] +M_0\Big[H+T/\delta_1\delta_2(F\delta_2+G\delta_1)+\sum_{i = 1}^ml_i\Big]\Big\}<1. \end{aligned} \end{gather*} \end{itemize} For a mild solution of the equation \eqref{4e1} and \eqref{1e7}-\eqref{1e8} we mean a functionu\in PC([0,T]:X)$and$u_0 \in Xsatisfying the integral equation \begin{aligned} u(t)&=U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^tU(t,s;u)\Big[f(s,u(\alpha(s)))\\ &\quad+\int_0^s g(s,\tau,u(\beta(\tau)))d\tau \Big]ds +\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)), \quad 0\leq t\leq T. \end{aligned} \label{4e4} \begin{theorem} \label{thm4.1} If the assumptions {\rm (H1)--(H11)} and {\rm (H13)--(H14)} are satisfied, then the equation \eqref{4e1} with nonlocal and impulsive conditions \eqref{1e7}-\eqref{1e8} has a unique mild solutionu\in PC([0,T] : Y)$. \end{theorem} \begin{proof} Let$S$be a nonempty closed subset of$PC([0,T]:X)$defined by$ S = \{u:u\in PC([0, T]: X), \|u(t)\|_{PC} \leq k\text{ for } 0\leq t\leq T\}$. Consider a mapping$\Psi$on$Sdefined by \begin{align*} (\Psi u)(t) &= U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^t U(t,s;u)\Big[f(s,u(\alpha(s)))\\ &\quad +\int_0^sg(s,\tau, u(\beta(\tau)))d\tau\Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)). \end{align*} Obviously\Psi$maps$S$into$S$, by (H14) and $\|\Psi u(t)- \Psi v(t)\|\leq p \|u - v\|_{PC}.$ Since$p < 1$,$\Psi$is a contraction on$S$and so$\Psi$has a unique fixed point$u\in S$which is the mild solution of the problem \eqref{4e1} and \eqref{1e7}-\eqref{1e8} on$[0,T]$.\\ \end{proof} \noindent\textbf{Remark.} %4.1 Using the additional assumption$A(t,b)u_0, b\in B$is bounded in$Ya unique local classical solution for the equations \eqref{4e1}, \eqref{1e7}, \eqref{1e8} can be established. \section{Examples} In this section we shall give two examples to illustrate the theorems. \begin{example} \label{exa1} \rm Consider the nonlinear partial integrodifferential equation \begin{gather} \begin{aligned} &\frac{\partial}{\partial t}z(t,y)+ \frac{\partial^3}{\partial y^3}z(t,y)+z(t,y) \frac{\partial}{\partial y}z(t,y) \\ &=k_0(y)\sin z(t,y)+k_1\int_0^t e^{-z(s,y)}ds, \end{aligned} \label{5e1}\\ z(0,y)+\sum_{i=1}^mc_iz(t_i,y)=z_0(y),\quad y \in \mathbb{R},\label{5e2}\\ \Delta z|_{t=t_i}=I_i(z(y))=(\alpha_i|z(y)|+ t_i)^{-1}, \quad 1\leq i\leq m \label{5e3} \end{gather} where the constantsc_i$and$\alpha_i$are small and$k_0(y)$is continuous on$\mathbb{R}$, and$k_1>0$. \end{example} Let$H^s$be the Hilbert space introduced in \cite{P1}. Take$X= L^2(R)=H^0(R)$and$Y=H^s(R)$,$s\geq 3$. Define an operator$A_0$by$D(A_0)=H^3(R)$and$A_0z=D^3z$for$z\in D(A_0)$where$D=d/dy$. Then$A_0$is the infinitesimal generator of a$C_0$-group of isometries on$X$. Next we define for every$v\in Y$an operator$A_1(v)$by$D(A_1(v))=H^1(R)$and$z\in D(A_1(v)), A_1(v)z=vDz$. Then we have for every$v\in Y$the operator$A(v)=A_0+A_1(v)$is the infinitesimal generator of$C_0$semigroup$U(t,0;v)$on$X$satisfying$\|U(t,0;v)\|\leq e^{\beta t}$for every$\beta\geq c_0\|v\|_s$where$c_0$is a constant independent of$v\in Y$. Let$B_r$be the ball of radius$r>0$in$Y$and it is proved that the family of operators$A(v),v\in B_r$satisfies the conditions (H1)--(H7) (see \cite {P1}). Put$u(t)=z(t,\cdot)$,\ \$h(u)=\sum_{i=1}^mc_iz(t_i,\cdot)$and $$f(t,u)=k_0(\cdot)\sin z(t,\cdot),\quad g(t,s,u)=k_1e^{-z(s,\cdot)}.$$ With this choice of$A(u)$,$I_i$,$f$,$g$,$h$we see that the equation \eqref{5e1}--\eqref{5e3} is an abstract formulation of \eqref{1e6}--\eqref{1e8}. Further other conditions (H8)--(H11) are obviously satisfied and it is possible to choose$c_i$,$\alpha_i$,$k_0, k_1$in such a way that the constant$q<1$. Hence by Theorem \ref{thm3.1} the equation \eqref{5e1}--\eqref{5e3} has a unique mild solution on$J . \begin{example} \label{exa2} \rm Consider the delay partial integrodifferential equation \begin{aligned} &\frac{\partial}{\partial t}z(t,y)+ \frac{\partial^3}{\partial y^3}z(t,y)+z(t,y) \frac{\partial}{\partial y}z(t,y) \\ &=k_0(y)\arctan z(\sin t,y)+k_1\int_0^t e^{-z(\sin s,y)}ds, \end{aligned}\label{5e4} with the same impulsive and nonlocal conditions as in Example \ref{exa1}. Here f(t,u)=k_0(\cdot)\arctan z(\sin t,\cdot)$and$\alpha(t)=\beta(t)=\sin t$. With the same$A(u), I_i, g, h$we see that the equations \eqref{5e4} with \eqref{5e2}--\eqref{5e3} is an abstract formulation of \eqref{4e1} with \eqref{1e7}--\eqref{1e8}. Note that (H1)--(H11) are already satisfied and it is possible to choose the constants so that the conditions (H13) and (H14) are also satisfied. Now by Theorem \ref{thm4.1} the equation \eqref{5e4} has a unique mild solution on$J\$. \end{example} \begin{thebibliography}{20} \bibitem {B1} D. Bahuguna; \emph{Quasilinear integrodifferential equations in Banach spaces,} Nonlinear Analysis, 24 (1995), 175-183. \bibitem {B2} K. Balachandran and F.Paul Samuel; \emph{Existence of solutions of quasilinear delay integrodifferential equations with nonlocal conditions,} Electronic Journal of Differential Equations, 2009(2009) No.6, 1-7. \bibitem {B3} K. Balachandran and K. Uchiyama; \emph{Existence of solutions of quasilinear integrodifferential equations with nonlocal condition,} Tokyo Journal of Mathematics, 23(2000), 203-210. \bibitem {C1} M. 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