Existence of mild solutions of nonlinear integrodifferential evolution equations with time varying delays

Abstract

Using the method of semigroup, existence and uniqueness of mild, strong and classical solutions of semilinear evolution equations have been discussed by Pazy [14] and the nonlocal Cauchy problem for the same equation has been studied by Byszewskii [5,6]. Balachandran and Chandrasekaran [1]
studied the nonlocal Cauchy problem for semilinear integrodifferential equation with deviating argument. Balachandran et.al [2] discussed the existence of nonlinear neutral differential equations with time varying delays. Sevaral authors [3,4,9] have discussed about the existence of solutions and controllability of nonlinear integrodifferential systems in Banach spaces. Grimmer [8] obtained the repre-
sentation of solutions of integrodifferential equations by using resolvent operators in a Banach space. Liu [11] discussed the Cauchy problem for integrodifferential evolution equations in abstract spaces and also in [12] he discussed nonautonomous integrodifferential equations. Lin and Liu [10] studied the nonlocal Cauchy problem for semilinear integrodifferential equations by using resolvent opera-
tors. Liu and Ezzinbi [13] investigated non-autonomous integrodifferential equations with nonlocal conditions. Byszewskii and Acka [7] studied the classical solution of nonlinear functional differential equation with time varying delays. The purpose of this paper is to prove the existene of mild solutions for time varying delay integrodifferential evolution equations with the help of Schaefer’s fixed point
theorem.