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Abstract. In this paper we study the existence of mild solutions for a class of abstractpartial neutral integro-dierential equations with state-dependent delay. Keywords: Integro-dierential equations, neutral equation, resolvent of operators, state-dependent delay.AMS-Subject Classication: 34K30, 35R10, 47D06.1. IntroductionIn this paper we study the existence of mild solutions for a class of abstract partialneutral integro-dierential 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 dened 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 dened axiomatically and f: [0,b]×B → Xand ρ: [0,b]×B → (−∞,b] are appropriatefunctions.Functional dierential 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 rst order functional dierential equations forwhich the state belong to some nite 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 dierential 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).

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