EXISTENCE RESULTS FOR A PARTIAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY

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).


Introduction
In this paper we study the existence of mild solutions for a class of abstract partial neutral integro-differential equations with state-dependent delay described in the form where t ∈ I = [0, b], A, B(t) for t ≥ 0 are closed linear operators defined on a common domain D(A) which is dense in X, N(t) (t ≥ 0) is bounded linear operators on X, the history x t : (−∞, 0] → X given by x t (θ) = x(t + θ) belongs to some abstract phase space B defined axiomatically and f : [0, b]×B → X and ρ : [0, b]×B → (−∞, b] are appropriate functions. Functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last years.The literature devoted to this subject is concerned fundamentally with first order functional differential equations for which 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 partial functional differential equations with state-dependent delay has been recently treated in the literature in [2,14,15,16,17].Our purpose in this paper is to establish the existence of mild solutions for the partial neutral system without using many of the strong restrictions considered in the literature (see [6] for details).† The work of this author was supported by FAPEMIG/Brazil, Grant CEX-APQ-00476-09.

preliminaries
In what follows we recall some definitions, notations and results that we need in the sequel.Throughout this paper, (X, • ) is a Banach space and A, B(t), t ≥ 0, are closed linear operators defined on a common domain D = D(A) which is dense in X.The notation [D(A)] represents the domain of A endowed with the graph norm.Let (Z, • Z ) and (W, • W ) be Banach spaces.In this paper, the notation L(Z, W ) stands for the Banach space of bounded linear operators from Z into W endowed with the uniform operator topology and we abbreviate this notation to L(Z) when Z = W. Furthermore, for appropriate functions K : [0, ∞) → Z the notation K denotes the Laplace transform of K .The notation, B r (x, Z) stands for the closed ball with center at x and radius r > 0 in Z.On the other hand, for a bounded function γ : [0, a] → Z and t ∈ [0, a], the notation γ Z, t is given by and we simplify this notation to γ t when no confusion about the space Z arises.
To obtain our results, we assume that the integro-differential abstract Cauchy problem has an associated resolvent operator of bounded linear operators (R(t)) t≥0 on X. Definition 2.1.A one parameter family of bounded linear operators (R(t)) t≥0 on X is called a resolvent operator of (2.2)-(2.3)if the following conditions are verified.
The existence of a resolvent operator for problem (2.2)-(2.3)was studied in [6].In this work we have considered the following conditions.In what follows, we always assume that the conditions (P1)-(P4) are verified.
We consider now the non-homogeneous problem with initial condition (2.3), where f : In [6,Theorem 2.4] we have established that the solutions of problem (2.6)-(2.3)are given by the variation of constants formula.
For additional details concerning phase space we refer the reader to [18].For completeness, we include the following well known result.

Existence Results
In this section we study the existence of mild solutions for system (1.1)-(1.2).Throughout this section M is a positive constant such that R(t) ≤ M for every t ∈ I.In the EJQTDE, 2010 No. 29, p. 4 rest of this work, ϕ is a fixed function in B and f i : [0, b] → X, i = 1, 2, will be the functions defined by We adopt the notion of mild solutions for (1.1)-(1.2) from the one given in [6].
To prove our results we always assume that ρ : In the sequel we introduce the following conditions: The function t → ϕ t is well defined and continuous from the set into B and there exists a continuous and bounded function J ϕ : R(ρ) → (0, ∞) such that ϕ t B ≤ J ϕ (t) ϕ B for every t ∈ R(ρ).
Remark 1.The condition (H ϕ ) is frequently verified by continuous and bounded functions.In fact, if B verifies axiom C 2 in the nomenclature of [18], then there exists L > 0 such that ϕ B ≤ L sup θ≤0 ϕ(θ) for every ϕ ∈ B continuous and bounded, see [18, Proposition 7.1.1]for details.Consequently, for every continuous and bounded function ϕ ∈ B \ {0} and every t ≤ 0. We also observe that the space C r × L p (g; X) verifies axiom C 2 , see [18, p.10] for details.If (H ϕ )be hold, then It is easy to see that ΓS(b) ⊂ S(b).We prove that there exists r > 0 such that Γ(B r ( φ| I , S(b))) ⊆ B r ( φ| I , S(b)).If this property is false, then for every r > 0 there exist x r ∈ B r ( φ| I , S(b)) and t r ∈ I such that r < Γx r (t r ) − ϕ(0) .Then, from Lemma 3.1 we find that which contradicts our assumption.Let r > 0 be such that Γ(B r ( φ| I , S(b))) ⊆ B r ( φ| I , S(b)), in the sequel, r * is the number defined by r * := (M b + J ϕ ) ϕ B + K b (r + ϕ(0) ).To prove that Γ is a condensing operator, we introduce the decomposition Γ = Γ 1 + Γ 2 , where It is easy to see that Γ The case t = 0 is trivial.Let 0 < ǫ < t < b.From the assumptions, we can fix numbers the first term of the left-hand side belong to a compact set in X and diam(C ǫ ) → 0 when ǫ → 0. This proves that Γ 2 (B r ( φ| I , S(b)))(t) is totally bounded and hence relatively compact in X for every t ∈ [0, b].
Step 2. The set Γ 2 (B r ( φ| Theorem 3.6.Let conditions (H 1 ), (H ϕ ) be hold, ρ(t, ψ) ≤ t for every (t, ψ) ∈ I × B and assume that R( In the sequel, we prove that Γ verifies the conditions of Theorem 2.4.We next establish an a priori estimate for the solutions of the integral equation x = λΓx for λ ∈ (0, 1).Let x λ be a solution of x = λΓx, λ ∈ (0, 1).By using Lemma 3.1, the notation and the fact that ρ(s, (x λ ) s ) ≤ s, for each s ∈ I, we find that Denoting by β λ (t) the right hand side of the last inequality, we obtain that and hence, This inequality and (3.1) permit us to conclude that the set of functions {β λ : λ ∈ (0, 1)} is bounded, which in turn shows that {x λ : λ ∈ (0, 1)} is bounded in BS(b).A procedure similar to the proof of Theorem 3.5 allows us to show that Γ is completely continuous on BS(b).By the Theorem 2.4 the proof is ended.EJQTDE, 2010 No. 29, p. 8
Under the above conditions we can represent the system We next consider the problem of the existence of mild solutions for the system (4.1)-(4.3).To this end, we introduce the following conditions.Proof: From the condition (a) it is easy to see that f is a bounded linear operator with f L(B,X) ≤ L f , and from condition (b) it follows that f 1 and f 2 are continuous.If the condition (4.8) is valid, then f 1 is differentiable and − s)x(s)ds] = Ax(t) + t −∞ B(t − s)x(s)ds + f (t, x ρ(t,xt) ), (1.1)

Theorem 2 . 4 .
( Leray-Schauder Alternative ) [8, Theorem 6.5.4]Let D be a closed convex subset of a Banach space Z with 0 ∈ D. Let G : D → D be a completely continuous map.Then, G has a fixed point in D or the set {z ∈ D : z = λG(z), 0 < λ < 1} is unbounded.

Remark 2 .
In the rest of this section, M b and K b are the constants M b = sup s∈[0,b] M(s) and K b = sup s∈[0,b] K(s).
t+h t m f (s)ds which shows that the set of functions Γ 2 (B r ( φ| I , S(b))) is right equicontinuous at t ∈ (0, b).A similar procedure permit to prove the right equicontinuity at zero and the left equicontinuity at t ∈ (0, b].Thus, Γ 2 (B r ( φ| I , S(b))) is equicontinuous.By using a EJQTDE, 2010 No. 29, p. 7 procedure similar to the proof of [14, Theorem 2.3], we prove that that Γ 2 (•) is continuous on B r ( φ| I , S(b)), which completes the proof that Γ 2 (•) is completely continuous.The existence of a mild solution for (1.1)-(1.2) is now a consequence of [20, Theorem 4.3.2].This completes the proof.