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International Journal of Differential Equations

Volume 2014 (2014), Article ID 784956, 13 pages

http://dx.doi.org/10.1155/2014/784956

## Existence of Solution via Integral Inequality of Volterra-Fredholm Neutral Functional Integrodifferential Equations with Infinite Delay

^{1}Department of Mathematics, Shivaji University, Kolhapur, Maharashtra 416 004, India^{2}Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra 431 004, India

Received 25 January 2014; Accepted 17 April 2014; Published 14 May 2014

Academic Editor: Toka Diagana

Copyright © 2014 Kishor D. Kucche and Machindra B. Dhakne. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work we study existence results for mixed Volterra-Fredholm neutral functional integrodifferential equations with infinite delay in Banach spaces. To obtain a priori bounds of solutions required in Krasnoselski-Schaefer type fixed point theorem, we have used an integral inequality established by B. G. Pachpatte. The variants for obtained results are given. An example is considered to illustrate the obtained results.

#### 1. Introduction

In this paper we establish existence results for the mixed Volterra-Fredholm neutral functional integrodifferential equations with infinite delay of the form where is the infinitesimal generator of a compact analytic semigroup of bounded linear operators , in a Banach space , , , , , and are given functions, , and is a phase space defined later. The histories , , , belong to the abstract phase space .

Due to the importance of neutral functional differential and integrodifferential equations with infinite delay in diverse fields of applied mathematics, these equations have generated considerable interest among researchers. Excellent account on the work with infinite delay can be found in [1–5]. The work in partial neutral functional differential equations with unbounded delay was initiated by Hernández and Henríquez [6, 7] and they have investigated the results pertaining to existence of mild, strong, and periodic solutions to the neutral functional differential equations. Recently, several works reported on existence results and controllability problem for various special forms of (1) and their variants with impulse or inclusion. Hernández [8] proved existence results for special form of (1) with , , by using the Leray-Schauder alternative. Li et al. [9] investigated the controllability problem when and by applying Sadovskii fixed point theorem. Henríquez [10, 11] has studied approximation and regularity of solutions of functional differential equations with unbounded delay.Chang et al. [12] established existence results for neutral functional integrodifferential equations with infinite delay using the resolvent operators and Krasnoselski-Schaefer type fixed point theorem. The work related to existence and controllability results with the impulse effect and infinite delay can be found in [13–15] and some of the references cited therein. The recent investigations on this theme can also be found in the work of Henriquez and dos Santos [16].

The authors [17–20] have studied existence, uniqueness, continuous dependence, and other properties of the solution of special forms of (1) with finite delay.

In this paper we investigate the existence results for (1) by using Krasnoselski-Schaefer type fixed point theorem via integral inequality by Pachpatte. We further prove existence results for the same equation without using integral inequality with different assumptions on the functions involved in the equation. To study (1), we use an abstract phase space given by Yan [21] instead of seminormed space, introduced by Hale and Kato in [3].

The paper is organized as follows. In Section 2, we present the preliminaries. Section 3 is concerned with main results and proof. In Section 4, we present an example to illustrate the application of our results.

#### 2. Preliminaries

We give some preliminaries from [21, 22] that will be used in our subsequent discussion. Assume that is a continuous function with . For any , we define and equip the space with the norm Let us define If is endowed with the norm then it is clear that is a Banach space.

Now we consider the space Set to be a seminorm in defined by

Let be the infinitesimal generator of a compact analytic semigroup of bounded linear operators , on a Banach space with the norm , and let ; then it is possible to define the fractional power , for , as closed linear invertible operator with domain dense in . The closedness of implies that endowed with the graph norm is a Banach space. Since is invertible, its graph norm is equivalent to the norm . Thus equipped with the norm is a Banach space which we denote by .

The following lemmas play an important role in our further discussions.

Lemma 1 (see [22]). *The following properties hold.*(i)*If , then and the imbedding is compact whenever the resolvent operator of is compact.*(ii)*For every , there exists such that
*

Lemma 2 (see [23]). *Let be a Banach space, and let , be two operators on such that*(a)* is contraction, and*(b)* is completely continuous.** Then either*(i)*the operator equation has a solution or*(ii)*the set is unbounded.*

Lemma 3 (see [8]). *Let , : be continuous functions. If is nondecreasing and there are constants , such that
*

*then*

*for every and every such that , and is the Gamma function.*

Lemma 4 (see [24]). *Assume ; then for , . Moreover,
**
where .*

Lemma 5 (see [25], p-47). *Let , , , , and be a real constant and
**
If
**
then
*

*Definition 6. *A function is called a mild solution of the problem (1) if on , the restriction of to the interval is continuous, and for each the function is integrable and the integral equation
is satisfied.

*Definition 7. *A map : is said to be an -Caratheodory if (i)for each , the function is continuous;(ii)for each ; the function is strongly measurable;(iii)for each positive integer , there exists such thatand for almost all .

#### 3. Existence Results

In this section we state and prove our main results. We list the following hypotheses for our convenience.(H1) is the infinitesimal generator of a compact analytic semigroup of bounded linear operators , in and such that where .(H2)There exist constants , , such that is -valued, is continuous, and(i), , , , ,(ii), , , with (H3)There exist integrable functions , , : such that(i), , ,(ii), , ,(iii), for each .(H4)For each , the functions , : are continuous and for each , the functions , : are strongly measurable. is an -Caratheodory.(H6) The condition holds, where We set and .

Using the hypotheses (H1), (H2) and Lemma 1, we have the following inequality: Thus from Bochner theorem, it follows that is integrable on .

Throughout this paper, for brevity we set

In the following theorem we establish a priori bound for the mild solution of the following system by using Pachpatte inequality: where . By Definition 6, the mild solution of the system (22) is given by

Theorem 8. *If hypotheses (H1)–(H6) are satisfied and letting be a mild solution of the system (22), then , , where
*

* Proof. *Using the hypotheses (H1)–(H3) in (23), we get
From inequality (25) and Lemma 4, we have
Define the function : , ; then is nondecreasing on , and we get
Therefore,
Using Lemma 3, we have
where
Thanks to Pachpatte’s inequality given in Lemma 5 and applying it with and using hypothesis (H6), we obtain
This implies that , .

Now we define the operator by
For , define by
then *.* Let , . It is easy to see that satisfies (15) if and only if satisfies and
Let *;* then for any we have
thus is a Banach space. Define for some ; then is uniformly bounded, and for , from Lemma 4, we have
Define the operator by
In the view of Krasnoselski-Schaefer type fixed point theorem, we decompose as , where and are defined on , respectively, by
Observe that the operator having a fixed point is equivalent to having one. Next, our aim is to prove that the operator is a contraction, while is a completely continuous operator.

Theorem 9. *If the hypotheses (H1) and (H2) are satisfied, then is a contraction on .*

* Proof. *Let any , ; then by using the hypotheses (H1), (H2) and Lemma 4, from (38) for each , we have
since , .

This implies that
Since , is contraction on .

Theorem 10. *If the hypotheses (H1), (H3)–(H5) are satisfied, then is completely continuous operator.*

* Proof. *We give the proof in the following steps. *Step **1. * maps bounded sets into bounded sets in .

Let any . Then it is enough to prove that for some constant . By using the hypotheses (H1), (H3) and condition (36) from (39), we have
Thus for each , we have . *Step **2. * maps bounded sets into equicontinuous sets of .

Let and , . Then from (39), using the hypotheses (H1) and (H3) and condition (36), we have the following three cases. *Case **1.* Let . Then, we have
*Case **2.* Let . Then, we have
*Case **3. *If , then .

From Cases 1–3, we deduce that the right-hand side of the above inequality tends to zero as tends to for sufficiently small, since the compactness of , , implies the continuity in the uniform operator topology. Thus the set is equicontinuous. *Step **3. * maps into a precompact set in .

Together with Arzela-Ascoli theorem and Steps 1–2 to prove is precompact in , it is sufficient to show that the set is precompact in . Let be fixed, and let be a real number satisfying . For , we define the operators
Since is compact operator, the set is precompact in , for every , . Moreover, for each , we have
Therefore there are precompact sets arbitrarily close to the set . Thus the set is precompact in . *Step **4. * is continuous.

Let , with in . Then there is a number such that for all and a. e. , so and .

By using the hypotheses (H4), (H5) and condition (36) we have
for each , and since
we have by the dominated convergence theorem that
Therefore,
This implies that is continuous.

From Steps 1–4, we can conclude that the operator is completely continuous and thus satisfies condition (b) in Lemma 2.

Theorem 11. *Assume that the hypotheses (H1)–(H6) hold. Then the problem (1) has at least one mild solution on .*

* Proof. *Let for some . Then for any , the function is a mild solution of the system (22) for which we have proved in Theorem 8 that , , and hence from Lemma 4, we have
which yields that the set is bounded.

Consequently, by virtue of Lemma 2, Theorem 9, and Theorem 10, the equation has a solution . Let , ; then is a fixed point of the operator which is a mild solution of the problem (1).

Theorem 12. *Assume that ( H1), (H2), (H4), (H5), and the following hypotheses are satisfied. *

*There exist integrable functions , , such that(i)*

*, ,*(ii)*, ,*(iii)*, for each , where is continuous nondecreasing function such that for each and .**The condition holds, where , , and are as in (H6) and (24).*

*Then the problem (1) has at least one mild solution on .*

* Proof. *Let be the solution of (22). By using the hypotheses (H1), (H2), and and (23), we obtain
By an application of Lemma 4, we get
Define the function as in the proof of Theorem 8 and, proceeding on the same line, we obtain
Applying Lemma 3 to the above inequality, we obtain
Let
Then , , , and
Since is nondecreasing function, we have
Therefore,
Integrating from 0 to and using the change of variables and the hypothesis , we obtain
This implies that . So there is constant such that , , and hence , , where depends on the functions , , and .

Define the operators , , and as discussed above. Note that the set