EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN BANACH SPACES

Abstract

This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using resolvent operators and Krasnoselski-Schaefer type fixed point theorem. An example is provided to illustrate the results.

1. Introduction

The aim of this paper is to establish the existence results for the following neutral functional integrodifferential equations with infinite delays:

where A is the infinitesimal generator of a compact, analytic resolvent operator R(t), t ≥ 0 in a Banach space X, a : D × Bh → X, g : J × Bh × X → X, k : D × Bh → X and f : J × Bh × X → X are given functions, where Bh is a phase space defined later and D = {(t, s) ∈ J × J : s ≤ t}.0 < t1 < t2 < ... < tm < b are bounded functions. B(t), t ∈ J is a bounded linear operator.

The histories xt : (−∞, 0] → X, xt(s) = x(t + s), s ≤ 0, belong to an abstract phase space Bh.

Neutral differential and integrodifferential equations arise in many areas of applied mathematics and for this reason these equations have been investigated extensively in the last decades. There are many contributions relative to this topic and we refer the reader to [1,2,3,8,9,10,11,12,13,14,15].

The theory of nonlinear functional differential or integrodifferential equations with resolvent operators is an important branch of differential equations, which has an extensive physical background, see for instance [16,17,18].

Since many control systems arsing from realistic models depend heavily on histories ( that is, the effect of infinite delay on the state equations [23]), there is real need to discuss the existence results for partial neutral functional integrodifferential equations with infinite delay. The development of the theory of functional differential equations with infinite delays depends on a choice of a phase space. In fact, various phase spaces have been considered and each different phase space has required a separate development of the theory [20]. The common space is the phase space B proposed by Hale and Kato in [19].

The main purpose of this paper is to deal with the existence of mild solutions for the problem (1)-(2). Here, we use an abstract phase space adopted in [6,24]. Sufficient conditions for the existence results are derived by means of the Krasnoselski-Schaefer type fixed point theorem combined with theories of analytic resolvent operators. The results generalise the results of [4,6,7,21].

 

2. Main results

Throughout this paper, we assume that (X, ∥·∥) is a Banach space, the notaion L(X, Y) stands for the Banach space of all linear bounded operators from X into Y , and we abbreviate this notation to L(X) when X = Y . R(t), t > 0 is compact, analytic resolvent operator generated by A.

Assume that

Definition 2.1. A family {R(t) : t ≥ 0} of continuous linear operators on X is called a resolvent operator for

if:

Theorem 2.1 ([21]).Let the assumptions (A1) and (A2) be satisfied. Then there exists a constant H = H(b) such that

where L(X) denotes the Banach space of continuous linear operators on X.

Next, if the C0-semigroup T(·) generated by A is compact ( that is, T(t) is a compact operator for all t > 0), then the corresponding resolvent operator R(·) is also compact ( that is, R(t) is a compact operator for all t > 0) and is operator norm continuous ( or continuous in the uniform operator topology) for t > 0.

Proof. Now, we define the abstract phase space Bh as given in [24,7].

Assume that h : (−∞, 0] → (0,+∞) is a continuous function with For any a > 0, we define

and equip the space B with the norm

Let us define

If Bh is endowed with the norm

then it is clear that (Bh, ∥ · ∥Bh) is a Banach space.

Now we consider the space

Set ∥ · ∥b be a semi norm in B′h defined by

Next, we introduce the basic definitions and lemmas which are used throughout this paper.

Let A : D(A) → X be the infinitesimal generator of a compact, analytic resolvent operator R(t), t ≥ 0. Let 0 ∈ ρ(A), then it is possible to define the fractional power (−A)α, for 0 < α ≤ 1, as closed linear operator on its domain D(−A)α. Further more, the subspace D(−A)α is dense in X, and the expression

defines a norm on D(−A)α.

Furthermore, we have the following properties appeared in [22]. □

Lemma 2.2. The following properties hold:

(i) If 0 < β < α ≤ 1, then Xα ⊂ Xβ and the imbedding is compact whenever the resolvent operator of A is compact.

(ii) For every 0 < α ≤ 1 there exists Cα > 0 such that

Lemma 2.3 ([5]). Let Փ1,Փ2 be two operators satisfying Փ1 is contraction and Փ2 is completely continuous. Then either

(i) the operator equation Փ1x + Փ2x = x has a solution, or

(ii) the set is unbounded for λ ∈ (0, 1).

Lemma 2.4 ([12]). Let v(·),w(·) : [0, b] → [0,∞) be continuous functions. If w(·) is nondecreasing and there are constants θ > 0, 0 < α < 1 such that

then

for every t ∈ [0, b] and every n ∈ N such that nα > 1, and Γ(·) is the Gamma function.

Lemma 2.5 ([6]). Assume x ∈ B′h, then for t ∈ J, xt ∈ Bh. Moreover,

where

Definition 2.2. A function x : (−∞, b] → X is called a mild solution of problem (1)-(2) if the following holds: x0 = ϕ ∈ Bh on (−∞, 0]; the restriction of x(·) to the interval J is continuous, and for each s ∈ [0, t), the function is integrable and the integral equation

is satisfied.

Definition 2.3. A map f : J × Bh × X → X is said to be an L1-Caratheodory if

(i) For each t ∈ J, the function f(t, ·, ·) : Bh × X → X is continuous.

(ii) For each (ϕ, x) ∈ Bh × X ; the function f(·, ϕ, x) : J → X is strongly measurable.

(iii) For every positive integer q > 0, there exists αq ∈ L1(J,R+) such that

 

3. Existence Results

In this section, we shall present and prove our main result. For the proof of the main result, we will use the following hypotheses:

(H1) ([see Lemma 2.2]) A is the infinitesimal generator of a compact analytic resolvent operator R(t), t > 0 and 0 ∈ ρ(A) such that

(H2) There exist a constant N1 > 0 such that

(H3) There exist constants 0 < β < 1,C0,c1,c2,N2 such that g is Xβ -valued, (−A)βg is continuous, and

(H4) (i) For each (t, s) ∈ D, the function k(t, s, ·) : Bh → X is continuous and for each x ∈ Bh, the function k(·, ·, x) : D → X is strongly measurable.

(H5) The function f : J × Bh × X → X satisfies the following caratheodory conditions:

(H6) ∥f(t, x, y)∥ ≤ p(t)Ψ(∥x∥Bh +∥y∥) for almost all t ∈ J and all x ∈ Bh, y ∈ X, where

p ∈ L1(J,R+) and Ψ : R+ → (0,∞) is continuous and increasing with

where

with lM0N2(1 + N1) < 1,

We consider the operator Փ : Bh′ → Bh′ defined by

From hypothesis (H1), (H2) and Lemma 2.3, the following inequality holds:

Then from Bochner theorem [25], it follows that is integrable on [0, t).

For ϕ ∈ Bh, we defined by by

and then It is easy to see that x satisfies (3) if and only if y satisfies y0 = 0 and

Let

thus is a Banach space. Set for some q ≥ 0, then is uniformly bounded, and for y ∈ Bq, from Lemma 2.5, we have

Define the operator

Now we decompose where

Obviously the operator Փ has a fixed point is equivalent to has one. Now, we shall show that the operators satisfy all the conditions of Lemma 2.3.

Lemma 3.1. If assumptions (H1)-(H6) hold, then is a contraction and is completely continuous.

Proof. First we show that is a contraction on From (H1)-(H3) and Lemma 2.5, we have

Since ∥u0∥Bh = 0, ∥v0∥Bh = 0. Taking supremum over t,

where Thus is a contraction on □

Next we show that the operator is completely continuous. First we prove that maps bounded sets into bounded sets in

Indeed, it is enough to show that there exists a positive constant Λ such that for each Now for each t ∈ J,

By (H1)-(H6) and (5), we have for t ∈ J,

Then for each

Next we show that maps bounded sets into equicontinuous sets of

Let 0 < r1 < r2 ≤ b, for each Let r1, r2 ∈ J − {t2, t2, ..., tm}. Then we have

The right-hand side from Theorem 2.1 of the above inequality tends to zero as r2 → r1 and for ϵ sufficiently small. Thus the set is equicontinuous. Here we consider only the case 0 < r1 < r2 ≤ b, since the other cases r1 < r2 ≤ 0 or r1 ≤ 0 ≤ r2 ≤ b are very simple.

Next, we show that is continuous.

Let Then there is a number q > 0 such that |y(n)(t)| ≤ q for all n and a.e. t ∈ J, so y(n) ∈ Bq and y ∈ Bq. In view of (5), we have

By (H3), (H5) and Definition 2.2,

We have by the dominated convergence theorem that

Thus is continuous.

Next we show that maps Bq into a precompact in X. Let 0 < t ≤ b be fixed and ϵ be a real number satisfying 0 < ϵ < t. For y ∈ Bq, we define the operators

From Theorem 2.1 and the compactness of the operator R(ϵ), the set is precompact in X, for every ϵ, 0 < ϵ < t. Moreover, by Theorem 2.1 and for each y ∈ Bq, we have

So the set is precompact in X by using the total boundedness. Applying this idea again and observing that

Therefore,

and there are precompact sets arbitrarily close to the set . Thus the set is precompact in X.

Therefore from Arzela-Ascoli theorem, we can conclude that the operator is completely continuous. In order to study the existence results for the problem (1)-(2), we introduce a parameter λ ∈ (0, 1) and consider the following nonlinear operator equation

where Փ is already defined. The following lemma proves that an a priori bound exists for the solution of the above equation.

Lemma 3.2. If hypotheses (H1)-(H6) are satisfied, then there exists an a priori bound K > 0 such that ∥xt∥Bh ≤ K, t ∈ J, where K depends only on b and on the functions

Proof. From the equation (6), we have

Thus from this proof and Lemma 2.4 it follows that

Let μ(t) = sup{∥xs∥Bh : 0 ≤ s ≤ t}, then the function μ(t) is nondecreasing in J, and we have

By using lemma 2.5, we have

where

Let us take the right hand side of the above inequality as v(t). Then v(0) = B0K1, μ(t) ≤ v(t), 0 ≤ t ≤ b and

Since Ψ and Ω are nondecreasing.

Let Then w(0) = v(0) and v(t) ≤ w(t).

This implies that

This implies that v(t) < ∞. So there is a constant K such that v(t) ≤ K, t ∈ J. So ∥xt∥Bh ≤ μ(t) ≤ v(t) ≤ K, t ∈ J, where K depends only on b and on the functions □

Theorem 3.3. Assume that the hypotheses (H1)-(H6) hold. Then the problem (1)-(2) has at least one mild solution on J.

Proof. Let us take the set

Then for any we have by Lemma 3.2 that ∥xt∥Bh ≤ K, t ∈ J, and we have

which implies that the set G is bounded on J.

Consequently, by Krasnoselski-Schaefer type fixed point theorem and Lemma 3.2 the operator has a fixed point Then x is a fixed point of the operator Փ which is a mild solution of the problem (1)-(2). □

 

4. Example

Consider the following partial neutral integrodifferential equation of the form

where ϕ ∈ Bh. We take X = L2[0, π] with the norm | · | L2 and define A : X → X by Aw = w′′ with the domain

D(A) = {w ∈ X : w,w′ are absolutely continuous, w′′ ∈ X, w(0) = w(π) = 0}.Then

where n = 1, 2, . . . .. is the orthogonal set of eigen vectors of A. It is well known that A generates a strongly continuous semigroup that is analytic, and resolvent operator R(t) can be extracted this analytic semigroup and given by

Since the analytic semigroup R(t) is compact, there exists a constant M1 > 0 such that ∥R(t)∥ ≤ M1. Especially, the operator (−A)½ is given by

with the domain

Let and define

Hence for (t, ϕ) ∈ [0, b] × Bh, where ϕ(θ)(x) = ϕ(θ, x), (θ, x) ∈ (−∞, 0] × [0, π]. Set

and

where

Then, the system (7)-(9) is the abstract formulation of the system (1)-(2). Further, we can impose some suitable conditions on the above defined functions to verify the assumptions on Theorem 3.3. We can conclude that system (7)-(9) has at least one mild solution on J.