\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 111, 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-2005/111\hfil Existence and uniqueness solutions] {Existence and uniqueness of mild and classical solutions of impulsive evolution equations} \author[A. Anguraj, M. M. Arjunan\hfil EJDE-2005/111\hfilneg] {Annamalai Anguraj, Mani Mallika Arjunan} \address{Annamalai Anguraj \hfill\break Department of Mathematics\\ P.S.G. College of Arts and Science\\ Coimbatore - 641 014, Tamilnadu, India} \email{angurajpsg@yahoo.com} \address{Mani Mallika Arjunan \hfill\break Department of Mathematics\\ P.S.G. College of Arts and Science\\ Coimbatore- 641 014, Tamilnadu, India} \email{arjunphd07@yahoo.co.in} \date{} \thanks{Submitted June 15, 2005. Published October 17, 2005.} \subjclass[2000]{34A37, 34G60, 34G20} \keywords{Semigroups; evolution equations; impulsive conditions} \begin{abstract} We consider the non-linear impulsive evolution equation \begin{gather*} u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), \quad 00,\; t\neq t_i, $$with impulsive condition in \eqref{e2}-\eqref{e3}, where A  is sectorial operator with some conditions given on the fractional operators  A^\alpha,\alpha \geq 0  . Liu \cite{l2} studied the existence of mild solutions of the impulsive evolution equation$$ u'(t) =Au(t)+f(t,u(t)), \quad 00$,$h_i >0$,$i =1,2,3,\dots,p. such that \begin{gather} \begin{aligned} &\|f(t,u_1,u_2,u_3)-f(t,v_1,v_2,v_3)\| \\ & \leq L_1\|u_1-v_1\|+L_2\|u_2-v_2\| + L_3\|u_3-v_3\|, \quad t\in [0,T_0],\; u,v \in X; \end{aligned}\label{e10}\\ \|I_i(u)-I_i(v)\| \leq h_i\|u-v\|, \quad u,v \in X.\label{e11} \end{gather} \end{itemize} Let G(\cdot) $be the$ C_0 $semigroup generated by the unbounded operator$ A $. Let$ B(X) $be the Banach space of all linear and bounded operators on$ X $. Let \begin{gather*} M =\max_{t\in [0,T_0]} \|G(t)\|_{B(X)}, \quad L = \max\{L_1,L_2,L_3\}\,. \\ K^* =\sup_{t\in [0,T_0]} \int_0^t{|K(s,t)|}\mathrm{d}t <\infty,\quad H^* = \sup_{t\in [0,T_0]} \int_0^{T_0}{|H(s,t)|}\mathrm{d}t <\infty \end{gather*} \begin{itemize} \item[(H2)] The constants$ L,L_1,L_2,L_3,K^*,H^* $satisfy the inequality $$M \Big[LT_0(1+K^*+H^*)+ \sum_{i=1}^p h_i\Big]<1$$ \end{itemize} \section{Existence Theorems} \subsection{Mild solution} A function$ u(\cdot)\in PC([0,T_0],X) \$ is a mild solution of equations \eqref{e5}--\eqref{e7} if it satisfies \begin{aligned} u(t)& =G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s\\ &\quad + \sum_{0