Existence results for a partial neutral integro-differential equation with state-dependent delay

Abstract

Abstract. In this paper we study the existence of mild solutions for a class of abstractpartial neutral integro-differential equations with state-dependent delay. Keywords: Integro-differential equations, neutral equation, resolvent of operators, state-dependent delay.AMS-Subject Classification: 34K30, 35R10, 47D06.1. IntroductionIn this paper we study the existence of mild solutions for a class of abstract partialneutral integro-differential equations with state-dependent delay described in the formddt[x(t)+Z t−∞ N(t−s)x(s)ds] = Ax(t) +Z t−∞ (1.1) B(t− s)x(s)ds+f(t,x ρ(t,x t ) ),(1.2) x 0 = ϕ∈ B,where t∈ I= [0,b],A,B(t) for t≥ 0 are closed linear operators defined on a commondomain D(A) which is dense in X, N(t) (t≥ 0) is bounded linear operators on X, thehistory x t : (−∞,0] → Xgiven by x t (θ) = x(t+θ) belongs to some abstract phase spaceB defined axiomatically and f: [0,b]×B → Xand ρ: [0,b]×B → (−∞,b] are appropriatefunctions.Functional differential equations with state-dependent delay appear frequently in ap-plications as model of equations and for this reason the study of this type of equa-tions has received great attention in the last years. The literature devoted to thissubject is concerned fundamentally with first order functional differential equations forwhich the state belong to some finite dimensional space, see among another works,[1, 3, 4, 5, 7, 9, 10, 11, 12, 19, 21, 22]. The problem of the existence of solutions for partialfunctional differential equations with state-dependent delay has been recently treated inthe literature in [2, 14, 15, 16, 17]. Our purpose in this paper is to establish the exis-tence of mild solutions for the partial neutral system without using many of the strongrestrictions considered in the literature (see [6] for details).