Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 26, 1-13; http://www.math.u-szeged.hu/ejqtde/

Existence Results for Impulsive Neutral Functional Differential Equations With State-Dependent

Delay

M. Mallika Arjunan and V. Kavitha

Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore–641 114, Tamil Nadu, India. E-mails: arjunphd07@yahoo.co.in and kavi velubagyam@yahoo.co.in

Abstract In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray-Schauder Alternative fixed point theorem. Example is provided to illustrate the main result.

Keywords: Abstract Cauchy problem, impulsive neutral equations, state-dependent delay, semigroup of linear operators, unbounded delay. AMS-Subject Classification: 34A37, 34K30, 34K40.

1 Introduction

The purpose of this article is to establish the existence of mild solutions for a class of impulsive abstract neutral functional differential equations with state-dependent delay described by the form

\frac{d}{dt}[x(t) + G(t, x_{t})] = Ax(t) + F (t, x_{ρ(t,x_{t})}), t ∈ I = [0, a],  (1.1)
x_{0} = φ ∈ B,  (1.2)
Δx(t_{i}) = I_{i}(x_{t_{i}}), i = 1, 2, ..., n,  (1.3)

where A is the infinitesimal generator of a compact C_{0}-semigroup of bounded linear operators (T (t))_{t≥0} on a Banach space X; the function x_{s} : (−∞, 0] → X, x_{s}(θ) = x(s + θ), belongs to some abstract phase space B described axiomatically; 0 < t_{1}.... < t_{n} < a are prefixed numbers; F, G : I × B → X, ρ : I × B → (−∞, a], I_{i} : B × X → X, i = 1, 2, ..., n, are appropriate functions and Δξ(t) represents the jump of the function ξ at t, which is defined by Δξ(t) = ξ(t^{+}) − ξ(t^{−}).

Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process.

Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems. For more details on this theory and its applications we refer the monographs of Bainov and Simeonov [3], Lakshmikantham et al. [16] and Samoilenko and Perestyuk [19], where numerous properties of their solutions are studied and detailed bibliographies are given.

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 few years, see for instance [1, 4, 5, 8, 9, 10, 12, 13, 14, 20, 21] and the references therein. The literature related to impulsive partial functional differential equations with state-dependent delay is limited, to our knowledge, to the recent works [2, 11]. The study of impulsive partial neutral functional differential equations with statedependent delay described in the general abstract form (1.1)-(1.3) is an untreated topic in the literature, and this fact, is the main motivation of our paper.

2 Preliminaries

Throughout this article, A : D(A) ⊂ X → X is the infinitesimal generator of a compact C_{0}-semigroup of linear operators (T (t))_{t≥0} on a Banach space X and M̃ is a positive constant such that ∥ T (t) ∥≤ M̃ for every t ∈ I. For background information related to semigroup theory, we refer the reader to Pazy [18].

To consider the impulsive condition (1.3), it is convenient to introduce some additional concepts and notations. We say that a function u : [σ, τ] → X is a normalized piecewise continuous function on [σ, τ] if u is piecewise continuous and left continuous on (σ, τ]. We denote by PC([σ, τ]; X) the space formed by the normalized piecewise continuous functions from [σ, τ] into X. In particular, we introduce the space PC formed by all functions u : [0, a] → X such that u is continuous at t ≠ t_{i}, u(t_{i}^{−}) = u(t_{i}) and u(t_{i}^{+}) exists, for all i = 1, ⋯, n. In this paper we always assume that PC is endowed with the norm ∥ u ∥_{PC}= sup_{s∈I} ∥ u(s) ∥. It is clear that (PC, ∥ · ∥_{PC}) is a Banach space.

To simplify the notations, we put t_{0} = 0, t_{n+1} = a and for u ∈ PC we denote by ũ_{i} ∈ C([t_{i}, t_{i+1}]; X), i = 0, 1, ⋯, n, the function given by

ũ_{i}(t) = \{ \begin{matrix}u(t), & for t ∈ (t_{i}, t_{i+1}], \\ u(t_{i}^{+}), & for t = t_{i}. \end{matrix}

Moreover, for B ⊆ PC we denote by B̃_{i}, i = 0, 1, ⋯, n, the set B̃_{i} = {ũ_{i} : u ∈ B}.

Lemma 2.1 A set B ⊆ PC is relatively compact in PC if, and only if, the set B̃_{i} is relatively compact in C([t_{i}, t_{i+1}]; X), for every i = 0, 1, ⋯, n.

In this work we will employ an axiomatic definition for the phase space B which is similar to those introduced in [15]. Specifically, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm ∥ · ∥_{B}, and satisfies the following axioms:

(A) If x : (−∞, σ + b] → X, b > 0, is such that x|_{[σ,σ+b]} ∈ PC([σ, σ + b] : X) 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(·).

(B) The space B is complete.

Example 2.1 The phase spaces PC_{h}(X), PC_{g}^{0}(X).

As usual, we say that φ : (−∞, 0] → X is normalized piecewise continuous, if φ is left continuous and the restriction of φ to any interval [−r, 0] is piecewise continuous.

Let g : (−∞, 0] → [1, ∞) be a continuous, nonincreasing function with g(0) = 1, which satisfies the conditions (g-1), (g-2) of [15]. This means that lim_{θ→−∞} g(θ) = ∞ and that the function G(t) := sup_{−∞<θ≤−t} \frac{g(t+θ)}{g(θ)} is locally bounded for t ≥ 0. Next, we modify slightly the definition of the spaces C_{g}, C_{g}^{0} in [15]. We denote by PC_{g}(X) the space formed by the normalized piecewise continuous functions φ such that \frac{φ}{g} is bounded on (−∞, 0] and by PC_{g}^{0}(X) the subspace of PC_{g}(X) formed by the functions φ such that \frac{φ(θ)}{g(θ)} → 0 as θ → −∞. It is easy to see that PC_{g}(X) and PC_{g}^{0}(X) endowed with the norm ∥ φ ∥_{B}:= sup_{θ≤0} \frac{∥φ(θ)∥}{g(θ)}, are phase spaces in the sense considered in this work. Moreover, in these cases K(s) ≡ 1 for s ≥ 0.

Example 2.2 The phase space PC_{r} × L^{2}(g , X).

Let 1 ≤ p < ∞, 0 ≤ r < ∞ and g(·) be a nonnegative Borel measurable function on (−∞, r) which satisfies the conditions (g-5)-(g-6) in the terminology of [15]. Briefly, this means that g(·) is locally integrable on (−∞, −r) and that there exists a nonnegative and locally bounded function G on (−∞, 0] such that g(ξ + θ) ≤ G(ξ)g(θ) for all ξ ≤ 0 and θ ∈ (−∞, −r) \ N_{ξ}, where N_{ξ} ⊆ (−∞, −r) is a set with Lebesgue measure 0.

Let B := PC_{r} × L^{p}(g; X), r ≥ 0, p > 1, be the space formed of all classes of functions φ : (−∞, 0] → X such that φ |_{[−r,0]} ∈ PC([−r, 0], X), φ(·) is Lebesgue-measurable on (−∞, −r] and g∥φ∥^{p} is Lebesgue integrable on (−∞, −r]. The seminorm in ∥ · ∥_{B} is defined by

∥ φ ∥_{B}:= sup_{θ∈[−r,0]} ∥ φ(θ) ∥ + (∫_{−∞}^{−r} g(θ) ∥ φ(θ) ∥^{p} dθ)^{1/p} .

Proceeding as in the proof of [15, Theorem 1.3.8] it follows that B is a phase space which satisfies the axioms (A) and (B). Moreover, for r = 0 and p = 2 this space coincides with

C_{0} × L^{2}(g, X), H = 1; M(t) = G(−t)^{\frac{1}{2}} and K(t) = 1 + (∫_{−t}^{0} g(τ)dτ)^{\frac{1}{2}} , for t ≥ 0.

Remark 2.1 In retarded functional differential equations without impulses, the axioms of the abstract phase space B include the continuity of the function t → x_{t}, see [15, 7] for details. Due to the impulsive effect, this property is not satisfied in impulsive delay systems and, for this reason, has been unconsidered in our description of B.

Remark 2.2 Let φ ∈ B and t ≤ 0. The notation φ_{t} represents the function defined by φ_{t}(θ) = φ(t + θ). Consequently, if the function x(·) in axiom (A) is such that x_{0} = φ, then x_{t} = φ_{t}. We observe that φ_{t} is well defined for t < 0 since the domain of φ is (−∞, 0]. We also note that in general φ_{t} ∉ B; consider, for example, functions of the type x^{μ}(t) = (t − μ)^{−α}X_{(μ,0]}, μ > 0, where X_{(μ,0]} is the characteristic function of (μ, 0], μ < −r and αp ∈ (0, 1), in the space PC_{r} × L^{p}(g; X).

Additional terminologies and notations used in this paper are standard in functional analysis. In particular, for Banach spaces (Z, ∥ · ∥_{Z}), (W, ∥ · ∥_{W}), the notation L(Z, W) stands for the Banach space of bounded linear operators from Z into W and we abbreviate to L(Z) whenever Z = W. Moreover, B_{r}(x, Z) denotes the closed ball with center at x and radius r > 0 in Z, and for a bounded function ξ : I → Z and 0 ≤ t ≤ a we employ the notation ∥ξ∥_{Z,t} for

∥ξ(θ)∥_{Z,t} = sup{∥ξ(s)∥_{Z} : s ∈ [0, t]}.  (2.4)

We will simply write ∥ξ∥_{t} when no confusion arises. In particular, if M(·), K(·) are the functions in axiom (A), then M_{a} = sup_{t∈I} M(t) and K_{a} = sup_{t∈I} K(t).

This paper has four sections. In Section 3 we establish the existence of mild solutions for system (1.1)-(1.3). Section 4 is reserved for examples.

To conclude the current section, we recall the following well-known result.

Theorem 2.1 [6, Theorem 6.5.4]. (Leray-Schauder Alternative) Let D be a closed convex subset of a Banach space Z and assume that 0 ∈ D. Let Γ : D → D be a completely continuous map. Then, either the set {z ∈ D : z = λΓ(z), 0 < λ < 1} is unbounded or the map Γ has a fixed point in D.

3 Existence Results

In this section we discuss the existence of mild solutions for the abstract system (1.1)- (1.3). To prove our results we always assume that φ ∈ B and that ρ : I × B → (−∞, a] is a continuous function and (Y, ∥ · ∥_{Y}) is a Banach space continuously included in X. Additionally, we introduce the following conditions.

H_{1} For every y ∈ Y, the function t → T (t)y is continuous from [0, ∞) into Y. Moreover, T (t)(Y) ⊂ D(A) for every t > 0 and there exists a positive function γ ∈ L^{1}([0, a]) such that ∥AT (t)∥_{L(Y;X)} ≤ γ(t), for every t ∈ I.

H_{2} Let R(ρ^{−}) = {ρ(s, ψ) : (s, ψ) ∈ I × B, ρ(s, ψ) ≤ 0}. The function t → φ_{t} is well defined from R(ρ^{−}) into B and there exists a continuous and bounded function J^{φ} : R(ρ^{−}) → ℝ such that ∥ φ_{t} ∥_{B}≤ J^{φ}(t) ∥ φ ∥_{B} for every t ∈ R(ρ^{−}).

H_{3} The function F : I × B → X satisfies the following conditions:

(i) Let x : (−∞, a] → X be such that x_{0} = φ and x|_{I} ∈ PC. The function t → F (t, x_{ρ(t,x_{t})}) is measurable on I and the function t → F (s, x_{t}) is continuous on R(ρ^{−}) ∪ I for every s ∈ I.

(ii) For each t ∈ I, the function F (t, ·) : B → X is continuous. (iii) There exists an integrable function m : I → [0, ∞) and a continuous nondecreas-

ing function W : [0, ∞) → (0, ∞) such that

∥ F (t, ψ) ∥ ≤ m(t)W (∥ ψ ∥_{B}),  (t, ψ) ∈ I × B.

H_{4} The function G is Y - valued, G : I × B → Y is continuous and there exist a positive constants c_{1}, c_{2} such that ∥G(t, ψ)∥_{Y} ≤ c_{1}∥ψ∥_{B} + c_{2}, ∀ (t, ψ) ∈ I × B.

H_{5} The function G is Y - valued, G : I × B → Y is continuous and there exists L_{G} > 0 such that

∥G(t, ψ_{1}) − G(t, ψ_{2})∥_{Y} ≤ L_{G}∥ψ_{1} − ψ_{2}∥_{B}, (t, ψ_{i}) ∈ I × B, i = 1, 2.

H_{6} The maps I_{i} are completely continuous and there are positive constants c_{i}^{j}, j = 1, 2, such that ∥I_{i}(ψ)∥ ≤ c_{i}^{1}∥ψ∥_{B} + c_{i}^{2}, i = 1, 2, ..., n, for every ψ ∈ B.

H_{7} The functions I_{i} : ℝ × B → X are continuous and there are positive constants L_{i}, i = 1, 2, ..., n, such that

∥I_{i}(ψ_{1}) − I_{i}(ψ_{2})∥ ≤ L_{i}∥ψ_{1} − ψ_{2}∥_{B}, ψ_{j} ∈ B, j = 1, 2, i = 1, 2, ..., n.

H_{8} Let S(a) = {x : (−∞, a] → X : x_{0} = 0; x|_{I} ∈ PC} endowed with the norm of uniform convergence on I and y : (−∞, a] → X be the function defined by y_{0} = φ on (−∞, 0] and y(t) = T (t)φ(0) on I. Then, for every bounded set Q ⊂ S(a), the set of functions {t → G(t, x_{t} + y_{t}) : x ∈ Q} is equicontinuous on I.

Remark 3.3 The condition (H_{2}) is frequently satisfied by functions that are continuous and bounded. In fact, assume that the space of continuous and bounded functions C_{b}((−∞, 0], X) is continuously included in B. Then, there exists L > 0 such that

∥ ψ_{t} ∥_{B}≤ L\frac{sup_{θ≤0} ∥ ψ(θ) ∥}{∥ ψ ∥_{B}} ∥ ψ ∥_{B}, t ≤ 0, ψ ≠ 0, ψ ∈ C_{b}((−∞, 0] : X).

It is easy to see that the space C_{b}((−∞, 0], X) is continuously included in PC_{g}(X) and PC_{g}^{0}(X). Moreover, if g(·) verifies (g-5)-(g-6) in [15] and g(·) is integrable on (−∞, −r], then the space C_{b}((−∞, 0], X) is also continuously included in PC_{r}×L^{p}(g; X). For complementary details related this matter, see Proposition 7.1.1 and Theorems 1.3.2 and 1.3.8 in [15].

Motivated by general semigroup theory, we adopt the following concept of mild solution.

Definition 3.1 A function x : (−∞, a] → X is called a mild solution of the abstract Cauchy problem (1.1)-(1.3) if x_{0} = φ; x_{ρ(s,x_{s})} ∈ B for every s ∈ I; the function t → AT (t−s)G(s, x_{s}) is integrable on [0, t), for every t ∈ [0, a]; and

x(t) = T (t)[φ(0) + G(0, φ)] − G(t, x_{t}) − ∫_{0}^{t} AT (t − s)G(s, x_{s})ds + ∫_{0}^{t} T (t − s)F (s, x_{ρ(s,x_{s})})ds

+ ∑_{0<t_{i}<t} T (t − t_{i})I_{i}(x_{t_{i}}), t ∈ I.

Remark 3.4. Let x(·) be a function as in axiom (A). Let us mention that the conditions (H_{1}), (H_{4}), (H_{5}) are linked to the integrability of the function s → AT (t − s)G(s, x_{s}). In general, except for the trivial case in which A is a bounded linear operator, the operator function t → AT (t) is not integrable over I. However, if condition (H_{1}) holds and G satisfies either assumption (H_{4}) or (H_{5}), then it follows from Bochner’s criterion and the estimate

∥AT (t − s)G(s, x_{s})∥ ≤ ∥AT (t − s)∥_{L(Y;X)}∥G(s, x_{s})∥_{Y}

≤ γ(t − s) sup_{s∈I} ∥G(s, x_{s})∥_{Y} ,

that s → AT (t − s)G(s, x_{s}) is integrable over [0, t), for every t ∈ I.

In the next Lemma, M_{a}, K_{a} are defined using the notation introduced in (2.4).

Lemma 3.1 [13, Lemma 2.1] Let x : (−∞, a] → X be a function such that x_{0} = φ and x|_{I} ∈ PC. Then

∥ x_{s} ∥_{B}≤ (M_{a} + J_{0}^{φ}) ∥ φ ∥_{B} +K_{a} sup{ ∥ x(θ) ∥; θ ∈ [0, max{0, s}] }, s ∈ R(ρ^{−}) ∪ I,

where J_{0}^{φ} = sup_{t∈R(ρ^{−})} J^{φ}(t).

Theorem 3.1 Let conditions (H_{1}) − (H_{3}), (H_{5}) and (H_{7}) be hold. If

K_{a}[L_{G} (1 + ∫_{0}^{a} γ(s)ds) + M̃lim inf_{ξ→∞^{+}} \frac{W (ξ)}{ξ} ∫_{0}^{a} m(s)ds + M̃∑_{i=1}^{n} L_{i}] < 1,  (3.1)

then there exists a mild solution of (1.1)-(1.3).

Proof: Consider the space Y = {u ∈ PC : u(0) = φ(0)} endowed with the norm ∥u∥_{a} = sup_{s∈I} ∥u(s)∥, and define the operator Γ : Y → Y by

Γx(t) = T (t)[φ(0) + G(0, φ)] − G(t, x̄_{t}) − ∫_{0}^{t} AT (t − s)G(s, x̄_{s})ds

+ ∫_{0}^{t} T (t − s)F (s, x̄_{ρ(s,x̄_{s})})ds + ∑_{0<t_{i}<t} T (t − t_{i})I_{i}(x̄_{t_{i}}), t ∈ I,

where x̄ : (−∞, a] → X is such that x̄_{0} = φ and x̄ = x on I. From our assumptions it is easy to see that Γx ∈ PC.

We claim that there exists r > 0 such that Γ(B_{r}(0, Y)) ⊂ B_{r}(0, Y). If we assume this property is false, then for every r >∥ φ ∥ there exist x^{r} ∈ B_{r}(0, Y) and t^{r} ∈ I such that r <∥ Γx^{r}(t^{r}) ∥. Then, by using Lemma 3.1 we find that

r <∥ Γx^{r}(t^{r}) ∥

≤ M̃H ∥ φ ∥_{B} + ∥ T (t^{r})G(0, φ) − G(t^{r}, φ) ∥ + ∥ G(t^{r}, \overline{(x^{r})}_{t^{r}}) − G(t^{r}, φ) ∥

+ M̃∫_{0}^{t^{r}} m(s)W (∥ \overline{x^{r}}_{ρ(s,(\overline{x^{r}})_{s})} ∥_{B})ds + ∫_{0}^{t^{r}} ∥AT (t^{r} − s)∥_{L(Y;X)}∥G(s, \overline{(x^{r})}_{s}) − G(s, φ)∥ds

+ ∫_{0}^{t^{r}} ∥AT (t^{r} − s)∥_{L(Y;X)}∥G(s, φ)∥ds + M̃∑_{i=1}^{n} (L_{i} ∥ \overline{x}_{t_{i}} ∥_{B} + ∥ I_{i}(0) ∥)

≤ M̃H ∥ φ ∥_{B} + ∥ T (t^{r})G(0, φ) − G(t^{r}, φ) ∥ +L_{G}(K_{a}r + (M_{a} + 1)∥φ∥)

+L_{G}(K_{a}r + (M_{a} + 1)∥φ∥) ∫_{0}^{a} γ(s)ds + ∥G(s, φ)∥_{a} ∫_{0}^{a} γ(s)ds

+ M̃W ((M_{a} + J_{0}^{φ}) ∥ φ ∥_{B} +K_{a}r) ∫_{0}^{t^{r}} m(s)ds

+ M̃∑_{i=1}^{n} (L_{i}(K_{a}r + M_{a}∥φ∥)+ ∥ I_{i}(0) ∥) ,

and hence

1 ≤ K_{a} [L_{G} (1 + ∫_{0}^{a} γ(s)ds) + M̃lim inf_{ξ→∞} \frac{W (ξ)}{ξ}∫_{0}^{a} m(s)ds + M̃∑_{i=1}^{n} L_{i}] ,

which is contrary to our assumption.

Let r > 0 be such that Γ(B_{r}(0, Y)) ⊂ B_{r}(0, Y). Next, we will prove that Γ is a condensing map on B_{r}(0, Y). Consider the decomposition Γ = Γ_{1} + Γ_{2} where

Γ_{1}x(t) = T (t)[φ(0) + G(0, φ)] − G(t, x̄_{t}) − ∫_{0}^{t} AT (t − s)G(s, x̄_{s})ds

+ ∑_{0<t_{i}<t} T (t − t_{i})I_{i}(x̄_{t_{i}}), t ∈ I,

Γ_{2}x(t) = ∫_{0}^{t} T (t − s)F (s, x̄_{ρ(s,x̄_{s})})ds, t ∈ I.

Proceeding as in the proof of [11, Theorem 3.1] we can conclude that Γ is continuous and that Γ_{2} is completely continuous. Moreover, from the estimate

∥Γ_{1}u − Γ_{1}v∥_{PC} ≤ K_{a}[L_{G} (1 + ∫_{0}^{a} γ(s)ds) + M̃∑_{i=1}^{n} L_{i}]∥u − v∥_{PC}, u, v ∈ B_{r}(0, Y),

it follows that Γ_{1} is a contraction on B_{r}(0, Y).

These remarks prove that Γ is a condensing operator from B_{r}(0, Y) into B_{r}(0, Y). Now, the existence of a mild solution is a consequence of [17, Theorem 4.3.2]. The proof is complete.

Theorem 3.2 Assume that conditions (H_{1}) − (H_{4}), (H_{6}) and (H_{8}) are satisfied. Further, assume that ρ(t, ψ) ≤ t for every (t, ψ) ∈ I × B and that G : I × B → X is completely continuous. If μ = [1 − c_{1}K_{a}(1 + ∫_{0}^{a} γ(s)ds) − M̃K_{a} ∑_{i=1}^{n} c_{i}^{1}] > 0 and

\frac{M̃K_{a}}{μ} ∫_{0}^{a} m(s)ds < ∫_{D}^{∞} \frac{ds}{W (s)},

where D = (M_{a}+J_{0}^{φ}+M̃HK_{a})∥φ∥_{B}+\frac{K_{a}C}{μ} and C = M̃∥G(0, φ)∥+M̃∑_{i=1}^{n} c_{i}^{2}+(M̃∑_{i=1}^{n} c_{i}^{1}+ c_{1})(M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} + c_{1}(M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} ∫_{0}^{a} γ(s)ds + c_{2}(1 + ∫_{0}^{a} γ(s)ds), then there exists a mild solution of (1.1)-(1.3).

Proof: On the space BPC = {u : (−∞, a] → X, u_{0} = 0, u|_{I} ∈ PC} endowed with the norm ∥ · ∥_{PC}, we define the operator Γ : BPC → BPC by (Γu)_{0} = 0 and

Γx(t) = T (t)G(0, φ) − G(t, x̄_{t}) − ∫_{0}^{t} AT (t − s)G(s, x̄_{s})ds

+ ∫_{0}^{t} T (t − s)F (s, x̄_{ρ(s,x̄_{s})})ds + ∑_{0<t_{i}<t} T (t − t_{i})I_{i}(x̄_{t_{i}}), t ∈ I,

where x̄ = x + y on (−∞, a] and y(·) is the function introduced in (H_{8}). In order to use Theorem 2.1, we establish a priori estimates for the solutions of the integral equation z = λΓz, λ ∈ (0, 1). By using Lemma 3.1, the notation α^{λ}(s) = sup_{θ∈[0,s]} ∥x^{λ}(θ)∥, and the fact that ρ(s, (\overline{x^{λ}})_{s}) ≤ s, for each s ∈ I, we find that

∥x^{λ}(t)∥ ≤ ∥T (t)G(0, φ)∥ + c_{1}∥(\overline{x^{λ}})_{t}∥_{B} + c_{2} + ∫_{0}^{t} γ(t − s)(c_{1}∥(\overline{x^{λ}})_{s}∥_{B} + c_{2})ds

+ M̃∫_{0}^{t} m(s)W (∥(\overline{x^{λ}})_{s}∥)ds + M̃ ∑_{0<t_{i}≤t} c_{i}^{1}[∥(\overline{x^{λ}})_{t_{i}}∥_{B}] + M̃∑_{i=1}^{n} c_{i}^{2}

≤ M̃∥G(0, φ)∥ + c_{1}∥φ∥_{B}[M_{a} + J_{0}^{φ} + K_{a}M̃H + (M_{a} + J_{0}^{φ} + K_{a}M̃H) ∫_{0}^{a} γ(s)ds]

+ c_{2}(1 + ∫_{0}^{a} γ(s)ds) + c_{1}K_{a}α^{λ}(t)(1 + ∫_{0}^{a} γ(s)ds)

+ M̃∫_{0}^{t} m(s)W ((M_{a} + J_{0}^{φ} + K_{a}M̃H)∥φ∥_{B} + K_{a}α^{λ}(s))ds + M̃∑_{i=1}^{n} c_{i}^{2}

+ M̃ ∑_{0<t_{i}≤t} c_{i}^{1}((M_{a} + J_{0}^{φ} + K_{a}M̃H)∥φ∥_{B} + K_{a}α^{λ}(t)).

Consequently,

α^{λ}(t) ≤ \frac{C}{μ} + \frac{M̃}{μ} ∫_{0}^{t} m(s)W ((M_{a} + J_{0}^{φ} + K_{a}M̃H)∥φ∥_{B} + K_{a}α^{λ}(s))ds,

where

C = M̃∥G(0, φ)∥ + M̃∑_{i=1}^{n} c_{i}^{2} + (M̃∑_{i=1}^{n} c_{i}^{1} + c_{1})(M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B}
+ c_{1}(M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} ∫_{0}^{a} γ(s)ds + c_{2}(1 + ∫_{0}^{a} γ(s)ds).

If ζ^{λ}(t) = (M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} + K_{a}α^{λ}(t),

ζ^{λ}(t) ≤ (M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} + K_{a}[\frac{C}{μ} + \frac{M̃}{μ} ∫_{0}^{t} m(s)W (ζ^{λ}(s))ds]

≤ (M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} + \frac{K_{a}C}{μ} + \frac{K_{a}M̃}{μ} ∫_{0}^{t} m(s)W (ζ^{λ}(s))ds

Denoting by β_{λ}(t) the right-hand side of the last inequality, it follows that,

β′_{λ}(t) ≤ \frac{K_{a}M̃}{μ}m(t)W (β_{λ}(t))

and hence

∫_{β_{λ}(0)=D}^{β_{λ}(t)} \frac{ds}{W (s)} ≤ \frac{M̃K_{a}}{μ} ∫_{0}^{a} m(s)ds < ∫_{D}^{∞} \frac{ds}{W (s)}, t ∈ I

where D = (M_{a} + J_{0}^{φ} + M̃HK_{a})∥φ∥_{B} + \frac{K_{a}C}{μ}, which implies that the set of functions \{β_{λ}(·) : λ ∈ (0, 1)} is bounded in C(I; R). Thus, {x^{λ}(·) : λ ∈ (0, 1)} is bounded on BPC.

To prove that Γ is completely continuous, we introduce the decomposition Γ = Γ_{1} +Γ_{2} + Γ_{3} where (Γ_{i}x)_{0} = 0, i = 1, 2, 3. and

Γ_{1}x(t) = T (t)G(0, φ) − G(t, x̄_{t}) + ∫_{0}^{t} T (t − s)F (s, x̄_{ρ(s,x̄_{s})})ds, t ∈ I,
Γ_{2}x(t) = − ∫_{0}^{t} AT (t − s)G(s, x̄_{s})ds, t ∈ I,
Γ_{3}x(t) = ∑_{0<t_{i}<t} T (t − t_{i})I_{i}(x̄_{t_{i}}), t ∈ I.

From the proof of [11, Theorem 3.1] and our assumptions on G we infer that Γ_{1} is completely continuous and easily we can prove that Γ_{2} is continuous. It remains to show that Γ_{2} is compact and that Γ_{3} is completely continuous. Now, by using the proof of [14, Theorem 3.2] together with the Arzela-Ascoli theorem we conclude that Γ_{2} is completely continuous. Next, by using Lemma 2.1, the continuiuty of Γ_{3} can be proven using phase space axioms. On the other hand for r > 0, t ∈ [t_{i}, t_{i+1}] ∩ (0, a], i ≥ 1, and u ∈ B_{r} = B_{r}(0, BPC), we find that

Γ̃_{3}u(t) ∈ \begin{cases}∑_{j=1}^{i} T (t − t_{j})I_{j}(B_{r^{∗}}(0, X)),  t ∈ (t_{i}, t_{i+1}), \\ ∑_{j=0}^{i} T (t_{i+1} − t_{j})I_{j}(B_{r^{∗}}(0, X)),  t = t_{i+1}, \\ ∑_{j=1}^{i−1} T (t_{i} − t_{j})I_{j}(B_{r^{∗}}(0, X)) + I_{i}(B_{r^{∗}}(0, X)), t = t_{i}, \end{cases}

where r^{∗} = (M_{a} + HM̃)∥φ∥_{B} + K_{a}r, which proves that [{ \atop Γ_{3}(B_{r})}]_{i}(t)] is relatively compact in X, for every t ∈ [t_{i}, t_{i+1}], since the maps I_{j} are completely continuous. Moreover, using the compactness of the operators I_{i} and the strong continuity of (T (t))_{t≥0}, we can prove that [{ \atop Γ_{3}(B_{r})}]_{i}(t)] is equicontinuous at t, for every t ∈ [t_{i}, t_{i+1}]. Now, from Lemma 2.1, we conclude that Γ_{3} is completely continuous.

These remarks, in conjunction with Theorem 2.1 show that Γ has a fixed point x ∈ BPC. Clearly, the function u = x + y is a mild solution of (1.1)-(1.3). The proof is now complete.

4 Example

In this section, we consider an applications of our abstract results. At first we introduce the required technical framework. In the rest of this secion, X = L^{2}([0, π]) and A be the operator Au = u′′ with domain D(A) = {u ∈ X : u′′ ∈ X, u(0) = u(π) = 0}. It is well known that A is the infinitesimal generator of an analytic semigroup on X. Furthermore, A has a discrete spectrum with eigen values of the form −n^{2}, n ∈ N, whose corresponding (normalized) eigen functions are given by z_{n}(ζ) = \sqrt{\frac{2}{π}} sin(nζ). In addition, the following properties hold.

(a) {z_{n} : n ∈ N} is an orthonormal basis of X;

(b) For u ∈ X, T (t)u = ∑_{n=1}^{∞} e^{−n^{2}t} < u, z_{n} > z_{n} and Au = − ∑_{n=1}^{∞} n^{2} < u, z_{n} > z_{n}, for u ∈ D(A);

(c) It is possible to define the fractional power (−A)^{α}, α ∈ (0, 1), as a closed linear operator over its domain D((−A)^{α}). More precisely, the operator (−A)^{α} : D((−A)^{α}) ⊆ X → X is given by (−A)^{α}u = ∑_{n=1}^{∞} n^{2α} < u, z_{n} > z_{n}, for all u ∈ D(−A)^{α}, where D(−A)^{α} = {u ∈ X : ∑_{n=1}^{∞} n^{2α} < u, z_{n} > z_{n} ∈ X};

(d) If X_{α} is the space D(−A)^{α} endowed with the graph norm ∥ · ∥_{α}, then X_{α} is a Banach space. Moreover, for 0 < β ≤ α ≤ 1, X_{α} ⊂ X_{β}; the inclusion X_{α} → X_{β} is completely continuous and there are constans C_{α} > 0 such that ∥T (t)∥_{L(X_{α};X)} ≤ \frac{C_{α}}{t^{α}} for t ≥ 0.

Consider the differential system

\frac{d}{dt}[u(t, ζ) + ∫_{−∞}^{t} ∫_{0}^{π} b(t − s, η, ζ)u(s, η)dηds] = \frac{∂^{2}}{∂ζ^{2}} u(t, ζ)

+ ∫_{−∞}^{t} a(s − t)u(s − ρ_{1}(t)ρ_{2}(∥u(t)∥), ζ)ds, t ∈ I, ζ ∈ [0, π]  (4.1)
u(t, 0) = u(t, π) = 0, t ∈ I  (4.2)
u(τ, ζ) = φ(τ, ζ), τ ≤ 0, 0 ≤ ζ ≤ π  (4.3)
Δu(t_{j}, ζ) = ∫_{−∞}^{t_{j}} γ_{j}(s − t_{j})u(s, ζ)ds, j = 1, 2, …, n.  (4.4)

where φ ∈ B = PC_{0} × L^{2}(g, X) and 0 < t_{1} < t_{2} < ⋯ < t_{n} < a are prefixed.

To treat this system, we will assume that g(·) satisfies the conditions (g − 5) − (g − 7) in [15]. We know from Theorem 1.37 and 7.1.1 in [15] that C_{b}((−∞, 0]; X) is continuously included in B. Additionally we assume that the functions ρ_{i} : [0, ∞) → [0, ∞), i = 1, 2. a :

R → R are continuous; L_{F} = (∫_{−∞}^{0} \frac{(a^{2}(s))}{g(s)}ds)^{\frac{1}{2}} < ∞ and that the following conditinos holds.

(a) The funtions γ_{i} : R → R, i = 1, 2, …, n, are continuous, bounded and for every

i = 1, 2, …, n, L_{i} = (∫_{−∞}^{0} \frac{(γ_{i}(s))^{2}}{g(s)} ds)^{\frac{1}{2}} < ∞.

(b) The functions b(s, η, ζ), \frac{∂b(s,η,ζ)}{∂ζ} are measurable, b(s, η, π) = b(s, η, 0) = 0 and

L_{g} = max \{(∫_{0}^{π} ∫_{−∞}^{0} ∫_{0}^{π} \frac{1}{g(s)} (\frac{∂^{i}b(s,η,ζ)}{∂ζ^{i}})^{2} dηdsdζ)^{\frac{1}{2}} : i = 0, 1\} < ∞.

Under these conditions, we can define the operators, ρ, G, F : I × B → X and I_{i} : B → X by

ρ(t, ψ) = ρ_{1}(t)ρ_{2}(∥ψ(0)∥),
G(ψ)(ζ) = ∫_{−∞}^{0} ∫_{0}^{π} b(s, ν, ζ)ψ(s, ν)dνds,
F (ψ)(ζ) = ∫_{−∞}^{0} a(s)ψ(s, ζ)ds
I_{i}(ψ)(ζ) = ∫_{−∞}^{∞} γ_{i}(s)ψ(s, ζ)ds, i = 1, 2, …, n,

which permit to transform system (4.1)-(4.4) into the system (1.1)-(1.3). Moreover, the maps, G, F, I_{i}, i = 1, 2, …, n are bounded linear operators with ∥G∥_{L(X)} ≤ L_{G} and ∥F ∥_{L(X)} ≤ L_{F} and ∥I_{i}∥_{L(X)} ≤ L_{i}, for every j = 1, 2, …, n.

Moreover, a straightforward estimation invloving (a) enables us to prove that G is D(−A)^{\frac{1}{2}} -valued with ∥(−A)^{\frac{1}{2}} G∥ ≤ L_{G}, which implies that G is completely continuous from I × B into X since the inclusion i : X_{\frac{1}{2}} → X is completely continuous. Thus, the assumptions (H_{1}), (H_{4}) and (H_{5}) are hold with Y = X_{\frac{1}{2}} .

From the Theorem 3.1 and Remark 3.3, we deduce the following propositions immediately.

Proposition 4.1 Assume that condition (H_{2}) holds and that the functions ρ_{1}, ρ_{2} are bounded. If

K_{a} (L_{G} + 2C_{1}\sqrt{a} + aL_{F} + ∑_{i=1}^{n} L_{i}) < 1,

there exists a mild solution of (4.1)-(4.4).

Proposition 4.2 Assume that φ ∈ C_{b}((−∞, 0); X). If

K_{a} (L_{G} + 2C_{1}\sqrt{a} + aL_{F} + ∑_{i=1}^{n} L_{i}) < 1,

there exists a mild solution of (4.1)-(4.4).

5 Acknowledgements

The authors wish to thank Dr. Paul Dhinakaran, Chancellor, Dr. Paul P. Appasamy, Vice–Chancellor, and Dr. Anne Mary Fernandez, Registrar, of Karunya University, Coimbatore, for their constant encouragements and support for this research work.

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(Received November 19, 2008)