Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 29, 1-12; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE RESULTS FOR A PARTIAL NEUTRAL

INTEGRO-DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY

JOSÉ PAULO C. DOS SANTOS^{†}

Abstract. In this paper we study the existence of mild solutions for a class of abstract partial neutral integro-differential equations with state-dependent delay.

Keywords: Integro-differential equations, neutral equation, resolvent of operators, statedependent delay. AMS-Subject Classification: 34K30, 35R10, 47D06.

1. 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

(1.1) \frac{d}{dt}[x(t) + ∫_{−∞}^{t} N(t − s)x(s)ds] = Ax(t) + ∫_{−∞}^{t} 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 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).

EJQTDE, 2010 No. 29, p. 1

† The work of this author was supported by FAPEMIG/Brazil, Grant CEX-APQ-00476-09.

2. 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 K̂ 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

∥ γ ∥_{Z, t}= sup{∥ γ(s) ∥_{Z}: s ∈ [0, t]}, (2.1)

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

(2.2)  \frac{d}{dt} [x(t) + ∫_{0}^{t} N(t − s)x(s) ds] = Ax(t) + ∫_{0}^{t} B(t − s)x(s) ds,
(2.3)  x(0) = z ∈ X.

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.

(a) The function R(·) : [0, ∞) → L(X) is strongly continuous, exponentially bounded and R(0)x = x for all x ∈ X.

(b) For x ∈ D(A), R(·)x ∈ C([0, ∞), [D(A)]) ⋂ C^{1}((0, ∞), X), and

(2.4) \frac{d}{dt} [R(t)x + ∫_{0}^{t} N(t − s)R(s)xds] = AR(t)x + ∫_{0}^{t} B(t − s)R(s)xds,
(2.5) \frac{d}{dt} [R(t)x + ∫_{0}^{t} R(t − s)N(s)xds] = R(t)Ax + ∫_{0}^{t} R(t − s)B(s)xds,

for every t ≥ 0.

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. (P1) The operator A : D(A) ⊆ X → X is the infinitesimal generator of an analytic

semigroup (T (t))_{t≥0} on X, and there are constants M_{0} > 0 and ϑ ∈ (π/2, π) such that ρ(A) ⊇ Λ_{ϑ} = {λ ∈ ℂ \ {0} : | arg(λ) |< ϑ} and ∥ R(λ, A) ∥≤ M_{0} | λ |^{−1} for all λ ∈ Λ_{ϑ}.

EJQTDE, 2010 No. 29, p. 2 (P2) The function N : [0, ∞) → L(X) is strongly continuous and N̂(λ)x is absolutely

convergent for x ∈ X and Re(λ) > 0. There exists α > 0 and an analytical extension of N̂(λ) (still denoted by N̂(λ)) to Λ_{ϑ} such that ∥ N̂(λ) ∥≤ N_{0} | λ |^{−α} for every λ ∈ Λ_{ϑ}, and ∥N̂(λ)x∥ ≤ N_{1}|λ|^{−1}∥x∥_{1} for every λ ∈ Λ_{ϑ} and x ∈ D(A).

(P3) For all t ≥ 0, B(t) : D(B(t)) ⊆ X → X is a closed linear operator, D(A) ⊆ D(B(t)) and B(·)x is strongly measurable on (0, ∞) for each x ∈ D(A). There exists b(·) ∈ L_{loc}^{1}(ℝ^{+}) such that b̂(λ) exists for Re(λ) > 0 and ∥ B(t)x ∥≤ b(t) ∥ x ∥_{1} for all t > 0 and x ∈ D(A). Moreover, the operator valued function B̂ : Λ_{π/2} → L([D(A)], X) has an analytical extension (still denoted by B̂) to Λ_{ϑ} such that ∥B̂(λ)x∥ ≤ ∥B̂(λ)∥ ∥x∥_{1} for all x ∈ D(A), and ∥B̂(λ)∥ → 0 as |λ| → ∞.

(P4) There exists a subspace D ⊆ D(A) dense in [D(A)] and positive constants C_{i}, i = 1, 2, such that A(D) ⊆ D(A), B̂(λ)(D) ⊆ D(A), N̂(λ)(D) ⊆ D(A), ∥AB̂(λ)x∥ ≤ C_{1}∥x∥ and ∥N̂(λ)x∥_{1} ≤ C_{2}|λ|^{−α}∥x∥_{1} for every x ∈ D and all λ ∈ Λ_{ϑ}.

The following result has been established in [6, Theorem 2.1].

Theorem 2.1. Assume that conditions (P1)-(P4) are fulfilled. Then there exists a unique resolvent operator for problem (2.2)-(2.3).

In what follows, we always assume that the conditions (P1)-(P4) are verified.

We consider now the non-homogeneous problem

(2.6) \frac{d}{dt} [x(t) + ∫_{0}^{t} N(t − s)x(s)ds] = Ax(t) + ∫_{0}^{t} B(t − s)x(s) ds + f(t), t ∈ I = [0, b],

with initial condition (2.3), where f : [0, b] → X is a continuous function.

Definition 2.2. A function x : [0, b] → X is called a classical solution of problem (2.6) -(2.3) on (0, b] if x ∈ C([0, b], [D(A)]) ∩ C^{1}((0, b], X), the condition (2.3) holds and the equation (2.6) is verified on [0, b]. If, in further, x ∈ C([0, b], [D(A)]) ∩ C^{1}([0, b], X) the function x is said a classical solution of problem (2.6)-(2.3) on [0, b].

It is clear from the preceding definition that R(·)z is a solution of problem (2.6)-(2.3) on (0, ∞) for z ∈ D(A).

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.

Theorem 2.2. Let z ∈ D(A). Assume that f ∈ C([0, b], X) and x(·) is a classical solution of problem (2.6) -(2.3) on (0, b]. Then

(2.7)  x(t) = R(t)z + ∫_{0}^{t} R(t − s)f(s) ds, t ∈ [0, b].

Theorem 2.3. ([6, Lemma 3.1.1]) If R(λ_{0}, A) is compact for some λ_{0} ∈ ρ(A), then R(t) is compact for all t > 0.

EJQTDE, 2010 No. 29, p. 3

We will herein define the phase space B axiomatically, using ideas and notations developed in [18]. More precisely, B will denote the vector space of functions defined from (−∞, 0] into X endowed with a seminorm denoted ∥ · ∥_{B} and such that the following axioms hold:

(A) If x : (−∞, σ + b) → X, b > 0, σ ∈ ℝ, is continuous on [σ, σ + b) and x_{σ} ∈ B, then for every t ∈ [σ, σ + b) the following conditions hold:

(i) x_{t} is in B.

(ii) ∥ x(t) ∥≤ H ∥ x_{t} ∥_{B}.

(iii) ∥ x_{t} ∥_{B}≤ K(t − σ) sup{∥ x(s) ∥: σ ≤ s ≤ t} + M(t − σ) ∥ x_{σ} ∥_{B},

where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K(·) is continuous, M(·) is locally bounded and H, K, M are independent of x(·).

(A1) For the function x(·) in (A), the function t → x_{t} is continuous from [σ, σ + b) into

B. (B) The space B is complete.

Example 2.1. The phase space C_{r} × L^{p}(g, X)

Let r ≥ 0, 1 ≤ p < ∞ and let g : (−∞, −r] → ℝ be a nonnegative measurable function which satisfies the conditions (g-5), (g-6) in the terminology of [18]. Briefly, this means that g is locally integrable and there exists a non-negative, locally bounded function γ on (−∞, 0] such that g(ξ + θ) ≤ γ(ξ)g(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ N_{ξ}, where N_{ξ} ⊆ (−∞, −r) is a set with Lebesgue measure zero. The space C_{r} × L^{p}(g, X) consists of all classes of functions φ : (−∞, 0] → X such that φ is continuous on [−r, 0], Lebesgue-measurable, and g ∥ φ ∥^{p} is Lebesgue integrable on (−∞, −r). The seminorm in C_{r} × L^{p}(g, X) is defined by

∥ φ ∥_{B}:= sup{∥ φ(θ)∥ : −r ≤ θ ≤ 0} + (∫_{−∞}^{−r} g(θ) ∥ φ(θ) ∥^{p} dθ)^{1/p} .

The space B = C_{r} × L^{p}(g; X) satisfies axioms (A), (A1), (B). Moreover, when r = 0

and p = 2, we can take H = 1, M(t) = γ(−t)^{1/2} and K(t) = 1 + (∫_{−t}^{0} g(θ) dθ)^{1/2}, for t ≥ 0. (see [18, Theorem 1.3.8] for details).

For additional details concerning phase space we refer the reader to [18].

For completeness, we include the following well known result.

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.

3. 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 f_{1}(t) = − ∫_{−∞}^{0} N(t − s)φ(s)ds and f_{2}(t) = ∫_{−∞}^{0} B(t − s)φ(s)ds. We adopt the notion of mild solutions for (1.1)-(1.2) from the one given in [6].

Definition 3.3. A function u : (−∞, b] → X is called a mild solution of the neutral system (1.1)-(1.2) on [0, b] if u_{0} = φ, u_{ρ(t,u_{t})} ∈ B, f_{1} is differentiable on [0, b], f′_{1}, f_{2} ∈ L^{1}([0, b], X), u_{|_{[0,a]}} ∈ C([0, b], X) and

u(t) = R(t)φ(0) + ∫_{0}^{t} R(t − s)f(s, u_{ρ(s,u_{s})})ds + ∫_{0}^{t} R(t − s)(f′_{1}(s) + f_{2}(s))ds, t ∈ [0, b].

To prove our results we always assume that ρ : I × B → (−∞, b] is continuous and that φ ∈ B. If x ∈ C([0, b]; X) we define \overline{x} : (−∞, b] → X is the extension of x to (−∞, b] such that \overline{x}_{0} = φ. We define x̃ : (−∞, b] → X such that x̃ = x + y where y : (−∞, b] → X is the extension of φ ∈ B such that y(t) = R(t)φ(0) for t ∈ I

In the sequel we introduce the following conditions: (H_{1}) The function f : [0, b] × B → X verifies the following conditions.

(i) The function f(t, ·) : B → X is continuous for every t ∈ [0, b], and for every ψ ∈ B, the function f(·, ψ) : [0, b] → X is strongly measurable.

(ii) There exist m_{f} ∈ C([0, b], [0, ∞)) and a continuous non-decreasing function Ω_{f} : [0, ∞) → (0, ∞) such that ∥ f(t, ψ) ∥≤ m_{f}(t)Ω_{f}(∥ ψ ∥_{B}), for all (t, ψ) ∈ [0, b] × B.

(H_{φ}) The function t → φ_{t} is well defined and continuous from the set

R(ρ^{−}) = {ρ(s, ψ) : (s, ψ) ∈ I × B, ρ(s, ψ) ≤ 0}

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,

∥ φ_{t} ∥_{B}≤ L\frac{sup_{θ≤0} ∥ φ(θ) ∥}{∥ φ ∥_{B}} ∥ φ ∥_{B}

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.

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).

Lemma 3.1 ([14, Lemma 2.1]). Let x : (−∞, b] → X is continuous on [0, b] and x_{0} = φ. If (H_{φ})be hold, then

∥ x_{s} ∥_{B}≤ (M_{b} + J^{φ}) ∥ φ ∥_{B} +K_{b} sup{∥ x(θ) ∥; θ ∈ [0, max{0, s}] },

EJQTDE, 2010 No. 29, p. 5 s ∈ R(ρ^{−}) ∪ I, where J^{φ} = sup_{t∈R(ρ^{−})} J^{φ}(t).

Theorem 3.5. Let conditions (H_{1}) and (H_{φ}) be hold and assume that R(·) ∈ C((0, b]; L(X)). If M lim inf_{ξ→∞} \frac{Ω_{f} (ξ)}{ξ} ∫_{0}^{b} m_{f}(s)ds < 1, then there exists a mild solution of (1.1)-(1.2) on [0, b].

Proof: Let φ̄ : (−∞, b] → X be the extension of φ to (−∞, b] such that φ̄(θ) = φ(0) on I = [0, b]. Consider the space S(b) = {u ∈ C(I; X) : u(0) = φ(0)} endowed with the uniform convergence topology and define the operator Γ : S(b) → S(b) by

Γx(t) = R(t)φ(0) + ∫_{0}^{t} R(t − s)(f′_{1}(s) + f_{2}(s))ds + ∫_{0}^{t} R(t − s)f(s, \overline{x}_{ρ(s,\overline{x}_{s})})ds,

for t ∈ [0, b]. It is easy to see that ΓS(b) ⊂ S(b). We prove that there exists r > 0 such that Γ(B_{r}(φ̄|_{I}, S(b))) ⊆ B_{r}(φ̄|_{I}, S(b)). If this property is false, then for every r > 0 there exist x^{r} ∈ B_{r}(φ̄|_{I}, S(b)) and t^{r} ∈ I such that r < ∥Γx^{r}(t^{r}) − φ(0)∥. Then, from Lemma 3.1 we find that

∥ Γx^{r}(t^{r}) − φ(0) ∥ ≤ ∥ R(t^{r})φ(0) − φ(0) ∥ +M ∥ f′_{1} + f_{2} ∥_{L^{1}([0,b],X)}
+M ∫_{0}^{t^{r}} m_{f}(t^{r} − s)Ω_{f}(∥\overline{x^{r}}_{ρ(s,(\overline{x^{r}})_{s})}∥_{B})ds
≤ (M + 1)H ∥ φ ∥_{B} +M ∥ f′_{1} + f_{2} ∥_{L^{1}([0,b],X)}
+Ω_{f} ((M_{b} + J^{φ})∥φ∥_{B} + K_{b}(r + ∥φ(0)∥)) ∫_{0}^{b} m_{f}(s)ds.

Therefore

1 ≤ M lim inf_{ξ→∞} \frac{Ω_{f}(ξ)}{ξ} ∫_{0}^{b} m_{f}(s)ds,

which contradicts our assumption.

Let r > 0 be such that Γ(B_{r}(φ̄|_{I}, S(b))) ⊆ B_{r}(φ̄|_{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

Γ_{1}x(t) = R(t)φ(0) + ∫_{0}^{t} R(t − s)(f′_{1}(s) + f_{2}(s))ds,
Γ_{2}x(t) = ∫_{0}^{t} R(t − s)f(s, \overline{x}_{ρ(s,\overline{x}_{s})})ds,

for t ∈ I.

It is easy to see that Γ_{1}(·) is continuous and a contraction on B_{r}(φ̄|_{I}, S(b)). Next we prove that Γ_{2}(·) is completely continuous from B_{r}(φ̄|_{I}, S(b)) into B_{r}(φ̄|_{I}, S(b)). Step 1. The set Γ_{2}(B_{r}(φ̄|_{I}, S(b))(t) is relatively compact on X for every t ∈ [0, b].

The case t = 0 is trivial. Let 0 < ϵ < t < b. From the assumptions, we can fix numbers 0 = t_{0} < t_{1} < ⋯ < t_{n} = t − ϵ such that ∥ R(t − s) − R(t − s′) ∥≤ ϵ if s, s′ ∈ [t_{i}, t_{i+1}], for

EJQTDE, 2010 No. 29, p. 6 some i = 0, 1, 2, ⋯, n − 1. Let x ∈ B_{r}(φ̄|_{I}, S(b)), under theses conditions, from the mean value theorem for the Bochner Integral (see [20, Lemma 2.1.3]) we see that

Γ_{2}x(t) =

∑_{i=1}^{n} ∫_{t_{i−1}}^{t_{i}} R(t − t_{i})f(s, \overline{x}_{ρ(t,\overline{x}_{s})})ds

+ ∑_{i=1}^{n} ∫_{t_{i−1}}^{t_{i}}(R(t − s) − R(t − t_{i}))f(s, \overline{x}_{ρ(t,\overline{x}_{s})})ds

+ ∫_{t_{n}}^{t} R(t − s)f(s, \overline{x}_{ρ(t,\overline{x}_{s})})ds

∈ ∑_{i=1}^{n}(t_{i} − t_{i−1})\overline{co(\{R(t − t_{i})f(s, ψ) : ψ ∈ B_{r^{∗}}(0, B), s ∈ [0, b]\})}

+ϵ r^{∗∗} + MΩ_{f}(r^{∗}) ∫_{t−ϵ}^{t} m_{f}(s)ds

∈ ∑_{i=1}^{n}(t_{i} − t_{i−1})\overline{co(\{R(t − t_{i})f(s, ψ) : ψ ∈ B_{r^{∗}}(0, B), s ∈ [0, b]\})} + ϵB_{r^{∗∗}}(0, X) + C_{ϵ},

the first term of the left-hand side belong to a compact set in X and diam(C_{ϵ}) → 0 when ϵ → 0. This proves that Γ_{2}(B_{r}(φ̄|_{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}(φ̄|_{I}, S(b))) is equicontinuous on [0, b].

Let 0 < ϵ < t < b and 0 < δ < ϵ such that ∥ R(s) − R(s′) ∥≤ ϵ for every s, s′ ∈ [ϵ, b] with | s − s′ |≤ δ. Under these conditions, for x ∈ B_{r}(φ̄|_{I}, S(b)) and 0 < h ≤ δ with t + h ∈ [0, b], we get

∥ Γ_{2}x(t + h) − Γ_{2}x(t) ∥
≤ ∫_{0}^{t−ϵ}[R(t + h − s) − R(t − s)]f(s, \overline{x}_{ρ(s,\overline{x}_{s})})ds
+ ∫_{t−ϵ}^{t}[R(t + h − s) − R(t − s)]f(s, \overline{x}_{ρ(s,\overline{x}_{s})})ds
+ ∫_{t}^{t+h} R(t + h − s)f(s, \overline{x}_{ρ(s,\overline{x}_{s})})ds
≤ ϵ r^{∗∗} + 2MΩ_{f}(r^{∗}) ∫_{t−ϵ}^{t} m_{f}(s)ds + MΩ_{f}(r^{∗}) ∫_{t}^{t+h} m_{f}(s)ds

which shows that the set of functions Γ_{2}(B_{r}(φ̄|_{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}(φ̄|_{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}(φ̄|_{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.

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(·) ∈ C((0, b]; L(X)). If

K_{b}M ∫_{0}^{b} m_{f}(s)ds < ∫_{c}^{∞} \frac{1}{Ω_{f}(s)}ds, (3.1)

where c = (M_{b} + K_{b}MH) ∥ φ ∥_{B} +K_{b}M ∥ f′_{1} + f_{2} ∥_{L^{1}([0,b],X)}, then there exists a mild solution of (1.1)-(1.2) on [0, b].

Proof: Let on the space BS(b) = {u : (−∞, b] → X; u_{0} = 0, u|_{I} ∈ C(I; X)} endowed with the uniform convergence topology. We define the operator Γ : BS(b) → BS(b) by

Γx(t) = \{ \begin{matrix}∫_{0}^{t} R(t − s)f(s, x̃_{ρ(s,x̃_{s})})ds + ∫_{0}^{t} R(t − s)(f′_{1}(s) + f_{2}(s))ds t ∈ I = [0, b], \\ 0, t ∈ (−∞, 0], \end{matrix}

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

α^{λ}(s) = (M_{b} + K_{b}MH) ∥ φ ∥_{B}) + K_{b}∥x^{λ}∥_{s},

and the fact that ρ(s, ˜(x^{λ})_{s}) ≤ s, for each s ∈ I, we find that

α^{λ}(t) ≤ (M_{b} + K_{b}MH) ∥ φ ∥_{B} +K_{b}M ∥ f′_{1} + f_{2} ∥_{L^{1}([0,b],X)}
+K_{b}M ∫_{0}^{t} m_{f}(s)Ω_{f}(α^{λ}(s))ds.

Denoting by β_{λ}(t) the right hand side of the last inequality, we obtain that

β′_{λ}(t) ≤ K_{b}Mm_{f}(t)Ω_{f}(β_{λ}(t))

and hence,

∫_{c}^{β_{λ}(t)} \frac{1}{Ω_{f}(s)}ds ≤ K_{b}M ∫_{0}^{b} m_{f}(s)ds.

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

4. Example

To finish this paper, we discuss the existence of solutions for the partial integrodifferential system

\frac{∂}{∂t} [u(t, ξ) + ∫_{−∞}^{t}(t − s)^{α}e^{−ω(t−s)}u(s, ξ)ds] = \frac{∂^{2}u(t, ξ)}{∂ξ^{2}} + ∫_{−∞}^{t} e^{−γ(t−s)} \frac{∂^{2}u(s, ξ)}{∂ξ^{2}}ds

+ ∫_{−∞}^{t} a(s − t)u(s − ρ_{1}(t)ρ_{2}(∥ u(t) ∥), ξ)ds, (t, ξ) ∈ I × [0, π], (4.1)

(4.2)  u(t, 0) = u(t, π) = 0,  t ∈ [0, b],
(4.3)  u(θ, ξ) = φ(θ, ξ),  θ ≤ 0, ξ ∈ [0, π].

In this system, α ∈ (0, 1), ω, γ are positive numbers, and a : [0, ∞) → ℝ is an appropriate function. Moreover, we have identified φ(θ)(ξ) = φ(θ, ξ).

To represent this system in the abstract form (1.1)-(1.2), we choose the spaces X = L^{2}([0, π]) and B = C_{0} × L^{2}(g, X), see Example 2.1 for details. We also consider the operators A, B(t) : D(A) ⊆ X → X, t ≥ 0, given by Ax = x′′, B(t)x = e^{−γt}Ax for x ∈ D(A) = {x ∈ X : x′′ ∈ X, x(0) = x(π) = 0} and N(t)x = t^{α}e^{−ωt}x for x ∈ X.

The operator A is the infinitesimal generator of an analytic semigroup, ρ(A) = ℂ\{−n^{2} : n ∈ ℕ} and for all ϑ ∈ (π/2, π) there exists M_{ϑ} > 0 such that ∥ R(λ, A) ∥≤ M_{ϑ} | λ |^{−1} for all λ ∈ Λ_{ϑ}. Moreover, it is easy to see that conditions (P2)-(P4) in Section 2 are satisfied with N̂(λ) = \frac{Γ(α+1)}{(λ+ω)^{α+1}} , b(t) = e^{−γt} and D = C_{0}^{∞}([0, π]), where Γ is the gamma function and C_{0}^{∞}([0, π]) is the space of infinitely differentiable functions that vanish at ξ = 0 and ξ = π.

Under the above conditions we can represent the system

(4.4) \frac{∂}{∂t} [u(t, ξ) + ∫_{0}^{t}(t − s)^{α}e^{−ω(t−s)}u(s, ξ)ds] = \frac{∂^{2}u(t, ξ)}{∂ξ^{2}} + ∫_{0}^{t} e^{−γ(t−s)} \frac{∂^{2}u(s, ξ)}{∂ξ^{2}}ds,
(4.5)  u(t, π) = u(t, 0) = 0,

in the abstract form (2.2)-(2.3).

The Proposition 4.1 below, is a consequence of [6, Theorem 2.1] and [6, Lemma 3.11] .

Proposition 4.1. There exists a compact operator resolvent for (4.4)-(4.5).

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.

(a) The function a(·) is continuous and L_{f} = (∫_{−∞}^{0} \frac{|a(−s)|^{2}}{g(s)} ds)^{\frac{1}{2}} < ∞.

(b) The functions φ, Aφ belong to B and the expressions sup_{t∈[0,b]} [∫_{−∞}^{0} \frac{(t−τ)^{2α}}{g(τ)} e^{2ωτ}dτ]^{\frac{1}{2}}

and (∫_{−∞}^{0} \frac{e^{2γτ}}{g(τ)} dτ)^{\frac{1}{2}} are finite. EJQTDE, 2010 No. 29, p. 9 Under the conditions (a) and (b), the functions f : [0, b]×B → X, f_{i} : [0, b] → X, i = 1, 2, given by

f(t, φ)(ξ) = ∫_{−∞}^{0} a(−s)φ(s, ξ)ds, f_{1}(t)(ξ) = ∫_{−∞}^{0}(t − s)^{α}e^{−ω(t−s)}φ(s, ξ)ds,
f_{2}(t)(ξ) = ∫_{−∞}^{0} e^{−γ(t−s)}Aφ(s, ξ)ds, ρ(s, ψ) = ρ_{1}(s)ρ_{2}(∥ φ(0) ∥),

are well defined, which us to permit re-write the system (4.1)-(4.3) in the abstract form

\frac{d}{dt}[ x(t) + ∫_{0}^{t}N(t − s)x(s)ds + f_{1}(t) ]

= Ax(t) + ∫_{0}^{t} B(t − s)x(s)ds + f_{2}(t) + f(t, x_{ρ(t,x_{t})}), t ∈ [0, b], (4.6)

(4.7)  x_{0} = φ ∈ B.

We said that a function u ∈ C([0, b]; X) is a mild solution of (4.1)-(4.3) if u(·) is a mild solution of the associated abstract system (4.6)-(4.7).

Proposition 4.2. Let φ ∈ B be such that condition (H_{φ}) holds, the functions ρ_{1}, ρ_{2} are bounded and assume that the above conditions are fulfilled. If L_{f} < 1 and

sup_{t∈[0,a]} [∫_{−∞}^{0} \frac{1}{ρ(τ)}[\frac{e^{−ω(t−τ)}}{(t − τ)^{1−α}}]^{2}dτ]^{\frac{1}{2}} < ∞, (4.8)

then there exists a mild solution of (4.1)-(4.3) on [0, b].

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

f′_{1}(t)(ξ) = ∫_{−∞}^{0}[(t − s)^{α−1} + ω(t − s)^{α}]e^{−ω(t−s)}φ(s, ξ)ds, ∀ (t, ξ) ∈ [0, b] × [0, π]. (4.9)

Moreover, using this expression we can prove that f_{1} ∈ C^{1}([0, b], X). Finally, from Theorem 3.5 we can assert that there exists a unique mild solution for the system (4.1)-(4.3) on [0, b].

Acknowledgment. The author wishes to thank the referee for comments and suggestions.

EJQTDE, 2010 No. 29, p. 10

References

[1] O. Arino, K. Boushaba, A. Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system. Spatial heterogeneity in ecological models (Alcalá de Henares, 1998). Nonlinear Analysis RWA. 1 (2000) 69-87.

[2] M. M. Arjunan and V. Kavitha, Existence results for impulsive neutral functional differential equations with state-dependent delay, E. J. Qualitative Theory of Diff. Equ. 26 (2009) 1-13.

[3] M. Bartha, Periodic solutions for differential equations with state-dependent delay and positive feedback. Nonlinear Analysis TMA. 53 (6) (2003) 839-857.

[4] Y. Cao, J. Fan, T. C. Gard, The effects of state-dependent time delay on a stage-structured population growth model. Nonlinear Analysis TMA. 19 (2) (1992) 95-105.

[5] A. Domoshnitsky, M. Drakhlin, E. Litsyn, On equations with delay depending on solution. Nonlinear Analysis TMA. 49 (5) (2002) 689-701.

[6] J. P. C. Dos Santos, H. Henriquez, E. Hernández, Existence results for neutral integro-differential equations with unbounded delay. to appear in Journal Integral Eq. and Applictions.

[7] C. Fengde, S. Dexian, S. Jinlin, Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl. 288 (1) (2003) 136-146.

[8] A. Granas, J. Dugundji, Fixed Point Theory. Springer-Verlag, New York, 2003.

[9] F. Hartung, Linearized stability in periodic functional differential equations with statedependent delays. J. Comput. Appl. Math. 174 (2) (2005) 201-211.

[10] F. Hartung, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study. Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000).Nonlinear Analysis TMA. 47 (7) (2001) 4557-4566.

[11] F. Hartung, T. Herdman, J. Turi, Parameter identification in classes of neutral differential equations with state-dependent delays. Nonlinear Analysis TMA. 39 (3) (2000) 305-325.

[12] F. Hartung, J. Turi, Identification of parameters in delay equations with state-dependent delays. Nonlinear Analysis TMA. 29 (11) (1997) 1303-1318.

[13] H. Henriquez, E. Hernández, J. P. C. dos Santos, Existence results for abstract partial neutral integro-differential equation with unbounded delay, E. J. Qualitative Theory of Diff. Equ. 29 (2009) 1-23.

[14] E. Hernández, L. Ladeira, A. Prokopczyk, A Note on state dependent partial functional differential equations with unbounded delay. Nonlinear Analysis, R.W.A. 7 (4) (2006) 510-519.

[15] E. Hernández, M. McKibben, On state-dependent delay partial neutral functional differential equations. Applied Mathematics and Computation. 186 (1) (2007) 294-301.

[16] E. Hernández, M. McKibben, H. Henriquez, Existence results for partial neutral functional differential equations with state-dependent delay. Mathematical and Computer Modelling. (49) (2009) 1260-1267.

[17] E. Hernández, M. Pierri, G. União. Existence results for a impulsive abstract partial differential equation with state-dependent delay. Comput. Math. Appl. 52 (2006) 411-420.

[18] Y. Hino, S. Murakami, T. Naito, Functional-differential equations with infinite delay. Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991.

[19] Y. Kuang, H. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations. Nonlinear Analysis TMA. 19 (9) (1992) 855-872.

[20] R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces. Robert E. Krieger Publ. Co., Florida, 1987.

[21] R. Torrejón, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation. Nonlinear Analysis TMA. 20 (12) (1993) 1383-1416.

EJQTDE, 2010 No. 29, p. 11

[22] L. Yongkun, Periodic solutions for delay Lotka-Volterra competition systems. J. Math. Anal. Appl. 246 (1) (2000) 230-244.

(Received February 6, 2010)

José Paulo C. dos Santos Departamento de Ciências Exatas-Universidade Federal de Alfenas. Rua Gabriel Monteiro da Silva, 700. 37130-000 Alfenas- MG, Brazil

E-mail address: zepaulo@unifal-mg.edu.br

EJQTDE, 2010 No. 29, p. 12