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

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.

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][2][3][4][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 = 0, ℎ = 0, by using the Leray-Schauder alternative. Li et al. [9] investigated the controllability problem when = 0 and = ∫ 0 ( , ) 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 2 International Journal of Differential Equations controllability results with the impulse effect and infinite delay can be found in [13][14][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].
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 B ℎ 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.

Preliminaries
We give some preliminaries from [21,22] that will be used in our subsequent discussion. Assume that ℎ : is bounded and measurable} (2) and equip the space B with the norm Let us define B ℎ = { : (−∞, 0] → such that for any > 0, If B ℎ is endowed with the norm then it is clear that (B ℎ , ‖ ⋅ ‖ B ℎ ) is a Banach space. Now we consider the space Set ‖ ⋅ ‖ to be a seminorm in B ℎ defined by Let : ( ) → be the infinitesimal generator of a compact analytic semigroup of bounded linear operators ( ), ≥ 0 on a Banach space with the norm ‖ ⋅ ‖, and let 0 ∈ ( ); then it is possible to define the fractional power (− ) , for 0 < ≤ 1, 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.
(i) If 0 < < ≤ 1, then ⊂ and the imbedding is compact whenever the resolvent operator of is compact.

Existence Results
In this section we state and prove our main results. We list the following hypotheses for our convenience.
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 ∈ (0, 1). By Definition 6, the mild solution of the system (22) is given by (22),

Theorem 8. If hypotheses (H1)-(H6) are satisfied and letting ( ) be a mild solution of the system
Proof. Using the hypotheses (H1)-(H3) in (23), we get From inequality (25) and Lemma 4, we have Define the function ( ) = sup{‖ ‖ B ℎ : 0 ≤ ≤ }, ∈ ; then ( ) is nondecreasing on , and we get International Journal of Differential Equations 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 ‖ ‖ B ℎ ≤ , ∈ . Now we define the operator Ψ : ℎ → ℎ by Let B ℎ = { ∈ B ℎ : 0 = 0 ∈ B ℎ }; then for any ∈ B ℎ we have International Journal of Differential Equations 7 thus (B ℎ , ‖ ⋅ ‖ ) is a Banach space. Define = { ∈ B ℎ : ‖ ‖ ≤ } for some > 0; then ⊆ B ℎ is uniformly bounded, and for ∈ , from Lemma 4, we have Define the operator Ψ : In the view of Krasnoselski-Schaefer type fixed point theorem, we decompose Ψ as Ψ 1 +Ψ 2 , where Ψ 1 and Ψ 2 are defined on B ℎ , respectively, by (39) Observe that the operator Ψ having a fixed point is equivalent to Ψ having one. Next, our aim is to prove that the operator Ψ 1 is a contraction, while Ψ 2 is a completely continuous operator.
International Journal of Differential Equations This implies that Since 0 < 1, Ψ 1 is contraction on B ℎ . Proof. We give the proof in the following steps.
we have by the dominated convergence theorem that Therefore, This implies that Ψ 2 is continuous. From Steps 1-4, we can conclude that the operator Ψ 2 is completely continuous and thus satisfies condition (b) in Lemma 2.
Then the problem (1) has at least one mild solution on (−∞, ].
Proof. Proceeding as in the proof of Theorem 12 with suitable modification, we can complete the proof. Hence we omit the details.