EXISTENCE AND REGULARITY FOR SEMILINEAR NEUTRAL DIFFERENTIAL EQUATIONS IN HILBERT SPACES

Abstract

Abstract. In this paper, we construct some results on the existence andregularity for solutions of neutral functional di erential equations withunbounded principal operators in Hilbert spaces. In order to establishthe existence and regularity for solutions of the neutral system by usingfractional power of operators and the local Lipschtiz continuity of non-linear term without using many of the strong restrictions considering inthe previous literature. 1. IntroductionLet H and V be real Hilbert spaces such that V is a dense subspace inH. In this paper, we are concerned with the global existence of solution andthe approximate controllability for the following abstract neutral functionaldi erential system in a Hilbert space H:( ddt [(x (t) + (Bx)( )] = Ax ) + f t;x )) + k ); t2(0;T];x(0) = x 0 ; (Bx)(0) = y 0 ;(1.1)where Ais an operator associated with a sesquilinear form on V V satisfyingGarding’s inequality, fis a nonlinear mapping of [0;T] Vinto Hsatisfying thelocal Lipschitz continuity, B: L