\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 36, pp. 1--7.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/36\hfil Existence of solutions] {Existence of solutions for impulsive neutral functional differential equations \\ with multiple delays} \author[M. Lakrib\hfil EJDE-2008/36\hfilneg] {Mustapha Lakrib} \address{ Mustapha Lakrib \newline Laboratoire de Math\' ematiques, Universit\' e Djillali Liab\es, B.P. 89, 22000 Sidi Bel Abb\es, Alg\'erie} \email{mlakrib@univ-sba.dz} \thanks{Submitted September 5, 2007. Published March 12, 2008.} \subjclass[2000]{34A37, 34K40, 34K45} \keywords{Neutral functional differential equations; impulses; multiple delay; \hfill\break\indent fixed point theorem} \begin{abstract} In this paper an existence result for initial value problems for first order impulsive neutral functional differential equations with multiple delay is proved under weak conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction} Impulsive differential equations have become more important in recent years in some mathematical models of processes and phenomena studied in physics, optimal control, chemotherapy, biotechnology, population dynamics and ecology. The reader is referred to monographs \cite{Ben,LBS,S} and references therein. In this paper we study the existence of solutions for initial value problems for first order neutral functional differential equations, with multiple delays and with impulsive effects, of the form \begin{gather}\label{e11} \begin{gathered} \frac{d}{dt}[x(t)- f(t, x_t)]=g(t, x_t)+\sum_{i=1}^px(t-\tau_i),\\ \mbox{a.e. } t\in J=[0,1], t\neq t_{k}, k=1,\dots,m, \end{gathered}\\ \label{e12} \Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad k=1,\dots,m, \\ \label{e13} x_0=\phi, \end{gather} where $f,g:J\times \mathcal{D} \to \mathbb{R}^n$ are given functions, $\mathcal{D}$ consists of functions $\psi:J_0\to\mathbb{R}^n$ such that $\psi$ is continuous everywhere except for a finite number of points $s$ at which $\psi(s^-)$ and $\psi(s^+)$ exist with $\psi(s^-)=\psi(s)$, $J_0=[-r,0]$, $r=\max\{\tau_i:\ i=1,\dots,p\}$, $\phi\in \mathcal{D}$, $0=t_{0}1\}$ is bounded, then $\Gamma$ has a fixed point. \end{theorem} \section{Existence result} In this section we state and prove our existence result for problem (\ref{e11})-(\ref{e13}), using the following conditions: \begin{itemize} \item [(H1)] The function $f:J\times \mathcal{D} \to \mathbb{R}^n$ is such that $$|f(t,x)|\leq c_1\|x\|_\mathcal{D}+c_2\,\,\,\mbox{for all}\,\,\, t\in J\,\, \hbox{and all}\,\,\, x\in \mathcal{D}$$ where $0\leq c_1<1$ and $c_2\geq 0$ are some constants. \item[(H2)] The function $g:J\times \mathcal{D} \to \mathbb{R}^n$ is Carath\'eodory, that is, \begin{itemize} \item[(i)] $t\mapsto g(t,x)$ is measurable for each $x\in \mathcal{D}$, \item[(ii)] $x\mapsto g(t,x)$ is continuous for a.e. $t\in J$. \end{itemize} \item [(H3)] There exist a function $q\in L^1(J,\mathbb{R})$ with $q(t)>0$ for a.e. $t\in J$ and a continuous and nondecreasing function $\psi:[0,\infty)\to [0,\infty)$ such that $$|g(t,x)|\le q(t)\psi(\|x\|_\mathcal{D})\quad \mbox{for a.e.  t\in J and each x\in \mathcal{D}}$$ with $$\label{inequ1} \int_{C}^{\infty}\frac{ds}{s+ \psi(s)}=\infty$$ where $C=\frac{1}{1-c_1}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p} \tau_i\Big)+2c_2\Big].$ \item [(H4)] The functions $I_k:\mathbb{R}^n\to \mathbb{R}^n$, $k=1,\dots,m$, are continuous. \end{itemize} \begin{theorem}\label{t2} Under assumptions {\rm (H1)--(H4)}, the initial value problem \eqref{e11}--\eqref{e13} has a solution on $J_1$. \end{theorem} \begin{proof} Transform the problem (\ref{e11})-(\ref{e13}) into a fixed point problem. Consider the operator $\Gamma:\Omega\to \Omega$ defined by $\Gamma x(t)=\begin{cases} \phi(t) &\mbox{for t\in J_0},\\[3pt] \phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t g(s,x_s)ds\\ +\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds +\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds \\ +\sum_{0 < t_k < t}I_{k}(x(t_{k}^{-})) &\mbox{for t\in J}. \end{cases}$ We shall show that the operator $\Gamma$ satisfies the conditions of Theorem \ref{t1} with $X=\Omega$. For better readability, we break the proof into a sequence of steps. \smallskip \noindent {\bf Step 1.} We show that $\Gamma$ has bounded values for bounded sets in $\Omega$. To show this, let $B$ be a bounded set in $\Omega$. Then there exists a real number $\rho>0$ such that $\|x\|\le \rho$, for all $x\in B$. Let $x\in B$ and $t\in J$. After some standard calculations we get \begin{align*} |\Gamma x(t)|&\le \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big) +2c_2+c_1\|x_t\|_{\mathcal{D}}+ \int_0^1q(s)\psi(\|x_s\|_\mathcal{D})ds\\ &\quad +p \int_0^1|x(s)|ds+\sum_{k=1}^{m}|I_{k}(x(t_{k}^{-}))|\\ &\le \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big)+2c_2+(c_1+p)\rho\\ &\quad + \psi(\rho)\|q\|_{L^1} +\sum_{k=1}^{m} \sup\{|I_k(u)|: |u|\leq \rho\}=:\eta. \end{align*} If $t\in J_0$, then $|\Gamma x(t)|\le\|\phi\|_{\mathcal{D}}$ and the previous inequality holds. Hence $$\|\Gamma x\|\le\eta, \quad \mbox{for all}\quad x\in B,$$ that is, $\Gamma$ is bounded on bounded subsets of $\Omega$. \smallskip \noindent {\bf Step 2.} Next we show that $\Gamma$ maps bounded sets into equicontinuous sets. Let $B$ be, as in Step 1, a bounded set and $x\in B$. Let $t$ and $h\not=0$ be such that $t,t+h\in J\backslash\{t_{1},\dots,t_{m}\}$. It is not difficult to get \begin{align*} &|\Gamma x(t+h)-\Gamma x(t)|\hfill\\ & \leq |f(t+h,x_{t+h})-f(t,x_{t})|+\psi(\rho)\int_{t}^{t+h}q(s)ds+ p\rho h+\sum_{t < t_k1\}$is bounded. Let$x\in \mathcal{E}$and let$\lambda>1$be such that$\lambda x=\Gamma x$. Then$x|_{[-r,t_1]}$satisfies, for each$t\in [0,t_1], \begin{align*} x(t)&= \lambda^{-1}\big[ \phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t g(s,x_s)ds\hfill\\ & \quad +\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds +\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds\big]. \end{align*} It is straightforward to verify that \label{equ15} \begin{aligned} |x(t)| & \leq \|\phi\|_{\mathcal{D}}\Big(1+c_1 +\sum_{i=1}^{p}\tau_i\Big)+2c_2 +c_1\|x_t\|_{\mathcal{D}}\\ &\quad +\int_0^t [q(s)\psi(\|x_s\|_{\mathcal{D}})ds+ p |x(s)|]ds. \end{aligned} Introduce the functionv_{1}(t)= \max\{|x(s)|: s\in[-r,t]\}$, for$t\in [0,t_1]$. We have$|x(t)|, \|x_t\|_\mathcal{D}\le v_{1}(t)$for all$t\in [0,t_1]$and there is$t^*\in [-r,t]$such that$v_{1}(t)=|x(t^*)|$. If$t^*<0$, we have$v_{1}(t)\leq\|\phi\|_{\mathcal{D}}$for all$t\in [0,t_1]$. Now, if$t^*\geq 0$, from (\ref{equ15}) it follows that, for$t\in [0,t_1]$, $v_{1}(t) \leq \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big) +2c_2+c_1v_{1}(t)+\int_0^t[q(s)\psi(v_{1}(s))+p v_{1}(s)]ds$ and hence $v_{1}(t)\leq C_1^{1}+C_1^{2}\int_0^t Q(s)[\psi(v_{1}(s))+v_{1}(s)]ds$ where $$C_1^{1}=C_1^{2}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1 +\sum_{i=1}^{p}\tau_i\Big)+2c_2\Big], \quad C_1^{2}=\frac{1}{1-c_1}$$ and$Q(t)=\max\{q(t),p\}$, for$t\in[0,t_1]$. Set $$w_{1}(t)=C_1^{1}+C_1^{2}\int_0^t Q(s)[\psi(v_{1}(s))+v_{1}(s)]ds,\quad \hbox{for } t\in [0,t_1].$$ Then we have$v_{1}(t)\le w_{1}(t)$for all$t\in [0,t_1]$. A direct differentiation of$w_{1}$yields \begin{gather*} w_1'(t)\leq Q(t)[\psi(w_1(t))+w_1(t)],\quad \mbox{a.e. } t\in [0,t_1]\\ w_1(0)=C_1^{(1)}. \end{gather*} By integration, this gives $$\label{inequ15} \int_0^t \frac{w_1'(s)}{\psi(w_1(s))+w_1(s)}\,ds\le \int_0^t Q(s)ds\leq \|Q\|_{L^1},\quad t\in [0,t_1].$$ By a change of variables, inequality (\ref{inequ15}) implies $$\int_{C_1^{1}}^{w_1(t)}\frac{ds}{\psi(s)+s}\le \|Q\|_{L^1},\quad t\in [0,t_1].$$ By (\ref{inequ1}) and the mean value theorem, there is a constant$M_1=M_1(t_1)>0$such that$ w_1(t)\le M_1$for all$t\in [0,t_1]$, and therefore$v_1(t)\le M_1$, for all$t\in[0,t_1]$. At last, we choose$M_1$such that$\|\phi\|_{\mathcal{D}}\leq M_1$to get $\max\{|x(t)|:t\in[-r,t_1]\}=v_1(t_1)\le M_1.$ Now, consider$x|_{[-r,t_2]}$. It satisfies, for each$t\in [0,t_2], \begin{align*} x(t)&= \lambda^{-1}\big[ \phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t g(s,x_s)ds\hfill\\ & \quad +\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds +\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds+I_1(x(t_1))\big]. \end{align*} Therefore, \label{equ17} \begin{aligned} |x(t)|&\leq \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p} \tau_i\Big)+2c_2+c_1\|x_t\|_{\mathcal{D}}\\ & \quad +\int_0^t [q(s)\psi(\|x_s\|_{\mathcal{D}})+ p |x(s)|]ds+\sup\{|I_1(u)|:|u|\leq M_1\}. \end{aligned} Denote v_2(t)= \max\{|x(s)|: s\in[-r,t]\}$, for$t\in [0,t_2]$. Then, for each$t\in [0,t_2]$, we have$|x(t)|, \|x_t\|_\mathcal{D}\le v_2(t)$. Let$t^*\in [-r,t]$be such that$v_2(t)=|x(t^*)|$. In the case$t^*<0$, we have$v_2(t)\leq\|\phi\|_{\mathcal{D}}$for all$t\in [0,t_2]$. Now, if$t^*\geq 0$, then by (\ref{equ17}) we have, for$t\in [0,t_2], \begin{align*} v_2(t)&\leq \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big) +2c_2+c_1v_2(t)+\int_0^t[ q(s)\psi(v_2(s)) +p v_2(s)]ds\\ & \quad +\sup\{|I_1(u)|:|u|\leq M_1\}; \end{align*} that is, $v_2(t)\leq C_2^{1}+C_2^{2}\int_0^t Q(s)[\psi(v_2(s))+v_2(s)]ds$ where $$C_2^{1}=C_2^{2}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p} \tau_i\Big)+2c_2+\sup\{|I_1(u)|:|u|\leq M_1\}\Big],$$C_2^{2}=1/(1-c_1)$, and$Q(t)=\max\{q(t),p\}$, for$t\in[0,t_2]$. If we set $$w_2(t)=C_2^{1}+C_2^{2}\int_0^t Q(s)[\psi(v_2(s))+v_2(s)]ds,\quad \hbox{for}\ t\in [0,t_2],$$ then$v_2(t)\le w_2(t)$for all$t\in [0,t_2]$and \begin{gather*} w_2'(t)\leq Q(t)[\psi(w_2(t))+w_2(t)],\quad \mbox{a.e. } t\in [0,t_2]\\ w_2(0)=C_2^{1}.\hfill \end{gather*} This yields $\int_0^t \frac{w_2'(s)}{\psi(w_2(s))+w_2(s)}\,ds\le \int_0^t Q(s)ds\leq \|Q\|_{L^1},\quad t\in [0,t_2]$ which implies $$\int_{C_2^{1}}^{w_2(t)}\frac{ds}{\psi(s)+s}\le \|Q\|_{L^1},\quad t\in [0,t_2].$$ Again, by (\ref{inequ1}) and the mean value theorem, there is a constant \mbox{$M_2=M_2(t_1,t_2)>0$} such that$ w_2(t)\le M_2$for all$t\in [0,t_2]$, and then$v_2(t)\le M_2$, for all$t\in[0,t_2]$. Finally, if we choose$M_2$such that$\|\phi\|_{\mathcal{D}}\leq M_2$, we get $\max\{|x(t)|:t\in[-r,t_2]\}=v_2(t_2)\le M_2.$ Continue this process for$x|_{[-r,t_3]}, \ldots, x|_{J_1}$, we obtain that there exists a constant$M=M(t_1,\ldots,t_m)>0$such that $$\|x\|\le M.$$ This finish to show that the set$\mathcal{E}$is bounded in$\Omega$. As a result the conclusion of Theorem \ref{t1} holds and consequently the initial value problem (\ref{e11})-(\ref{e13}) has a solution$x$on$J_1$. This completes the proof. \end{proof} We conclude this paper with a discussion on two special cases. In each one, some of the conditions in Theorem \ref{t2} can be either removed or weakened. \smallskip \noindent\textbf{Case 1:} Consider the initial value problem for first order impulsive functional differential equations with multiple delays \begin{gather}\label{e111} x'(t)=g(t, x_t)+\sum_{i=1}^px(t-\tau_i),\ \mbox{a.e. } t\in J=[0,1], t\neq t_{k}, k=1,\dots,m,\\ \label{e112} \Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad k=1,\dots,m, \\ \label{e113} x_0=\phi, \end{gather} derived from problem (\ref{e11})-(\ref{e13}) when$f\equiv 0$. In this case one obtains the next existence result which is an immediate corollary of Theorem \ref{t2}. \begin{theorem} Under conditions {\rm (H2)--(H4)}, the initial value problem \eqref{e111}--\eqref{e113} has a solution on$J_1$if constant$C$in {\rm (H3)} is replaced by $C=\|\phi\|_{\mathcal{D}}\Big(1+\sum_{i=1}^{p}\tau_i\Big).$ \end{theorem} \noindent\textbf{Case 2:} Without the second term in the right hand side of (\ref{e111}), problem (\ref{e111})-(\ref{e113}) is an initial value problem for first order impulsive functional differential equations \begin{gather}\label{e1111} x'(t)=g(t, x_t),\ \mbox{a.e. } t\in J=[0,1], t\neq t_{k}, k=1,\dots,m, \\ \label{e1112} \Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad k=1,\dots,m, \\ \label{e1113} x_0=\phi. \end{gather} The corresponding existence result is as bellow. Its proof is omitted because it is the same as the proof of Theorem \ref{t2}. \begin{theorem} Under conditions {\rm (H2)--(H4)}, the initial value problem \eqref{e1111}--\eqref{e1113} has a solution on$J_1\$ if relation \eqref{inequ1} in {\rm (H3)} is replaced by $\int_{\|\phi\|_{\mathcal{D}}}^{\infty}\frac{ds}{\psi(s)}=\infty.$ \end{theorem} \begin{thebibliography}{00} \bibitem{Ben} M. Benchohra, J. Henderson and S. Ntouyas, \emph{Inpulsive Differential Equations and Inclusions}, Series Contemporary Mathematics and Its Applications, Vol. 2, Hindawi Publ. Corp., 2006. \bibitem{LBS} V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, \textit{Theory of Impulsive Differential Equations}, {World Scientific Pub. Co., Singapore, 1989.} \bibitem{S} A. Samoilenko and N. Peresyuk, \textit{Differential Equations with Impusive Effectes}, {World Scientific Pub. Co., Singapore, 1995.} \bibitem{Sch} H. Schaefer, \"Uber die Methode der a priori-Schranken, \emph{Mathematische Annalen}, 129 (1955), 415-416 (German). \bibitem{Smart} D. R. Smart, \emph{Fixed Point Theorems}, Cambridge Tracts in Mathematics, vol. 66, Cambridge University Press, Cambridge, 1980. \end{thebibliography} \end{document}