Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in Banach spaces
© Ravichandran and Baleanu; licensee Springer 2013
Received: 2 April 2013
Accepted: 29 June 2013
Published: 15 July 2013
In this paper, we investigate the existence results for a class of abstract fractional neutral integro-differential evolution systems involving the Caputo derivative in Banach spaces. The main techniques rely on the fractional calculus, properties of characteristic solution operators, Mönch’s fixed point theorem via measures of noncompactness. Particularly, we do not assume that characteristic solution operators are compact. The application is given to illustrate the theory. The results of this article are generalization and improvement of the recent results on this issue.
MSC:26A33, 34A12, 47H08, 47H10.
Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. For more details on fractional calculus theory and applications, one can see the monographs of Kilbas et al. , Lakshmikantham et al. , Miller and Ross , Podlubny , Baleanu et al. [5–7] and the papers [8–12] as well as the references therein.
Recently, the existence of solutions for fractional semilinear differential or integro-differential equations is one of the theoretical fields investigated by many authors [11, 13–16]. Very recently, Ji et al.  studied the controllability of impulsive differential systems with nonlocal conditions by using Mönch’s fixed point theorem and Wang et al.  established the sufficient conditions for nonlocal controllability for fractional evolution systems and the results were obtained by using fractional calculus and Mönch’s fixed point theorem.
However, to the best of our knowledge, no work has been reported on the existence results for fractional neutral integro-differential systems with infinite delay in an abstract phase space via measures of noncompactness combined with the help of characteristic solution operators.
where is the Caputo fractional derivative of order , A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X, which means that there exists such that , , and are given functions, where is a phase space defined in Section 2. The histories , defined by , , belong to some abstract phase space .
The paper is organized as follows. In Section 2, we recall some basic definitions, notations and preliminary facts. Section 3 is devoted to the existence results for fractional neutral integro-differential evolution systems with infinite delay. The application of our theoretical results is given in Section 4. The last section is devoted to our conclusions.
In this section, we mention notations, definitions, lemmas and preliminary facts needed to establish our main results.
Throughout this paper, we denote by X a Banach space with the norm . Let Y be another Banach space, let denote the space of bounded linear operators from X to Y. We also use norm of f whenever for some p with . Let denote the Banach space of functions which are Bochner integrable normed by . Let , be the Banach space of continuous functions from J into X with the usual supremum norm , for .
is a Banach space with the norm for .
for each and .
- (iv)For every , is bounded on X and there exists such that
is a bounded linear operator for in X.
then it is clear that is a Banach space.
Lemma 2.1 (See )
provided the right-hand side is pointwise defined on , where is the gamma function, which is defined by .
where the function has absolutely continuous derivative up to order .
- (i)If , then
The Caputo derivative of a constant is equal to zero.
If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.
Definition 2.4 (See )
The following results of and are used throughout this paper.
Remark 2.2 (See )
and are strongly continuous.
For and any bounded subsets , and are equicontinuous if with respect to as for each fixed .
- (iv)For any , , we have
Moreover, let us recall some definitions and properties of the measures of noncompactness.
Definition 2.5 (See )
Let be a positive cone of an ordered Banach space . A function Φ defined on the set of all bounded subsets of the Banach space X with values in is called a measure of noncompactness (MNC) on X iff for all bounded subsets , where stands for the closed convex hull of Ω.
- (1)Monotone iff for all bounded subsets , of X we have:
Nonsingular iff for every , ;
Regular iff if and only if Ω is relatively compact in X.
, where ;
for any ;
If the map is Lipschitz continuous with constant k, then for any bounded subset , where Z is a Banach space.
Lemma 2.3 (See )
If is a sequence of Bochner integrable functions from J into X with the estimation for almost all and every , where , then the function belongs to and satisfies .
The following fixed-point theorem, a nonlinear alternative of Mönch’s type, plays a key role in our proof of system (1.1)-(1.2).
Lemma 2.4 (See [, Theorem 2.2])
Let D be a closed convex subset of a Banach space X and . Assume that is a continuous map which satisfies Mönch’s condition, that is, ( is countable, is compact). Then F has a fixed point in D.
3 Existence results
In this section, we present and prove the existence results for problem (1.1)-(1.2). In order to prove the main theorem of this section, we list the following hypotheses.
A generates a strongly continuous semigroup in X;
For all bounded subsets and , as for each fixed .
is measurable for all and is continuous for a.e. and for , is strongly measurable.
There exists a constant and and a nondecreasing continuous function, there is a positive integrable function such that , for all , where Ω satisfies .
- (iii)There exists a constant and such that, for any bounded subset , ,
where and β is the Hausdorff MNC.
is measurable for and is continuous for a.e. .
There exists a constant such that , for all , .
- (iii)There exists such that for any bounded subset ,
For our convenience, let us take , . , and .
Proof In order to prove the existence of mild solutions for system (1.1)-(1.2), transform it into a fixed point problem.
Obviously, the operator Φ has a fixed point is equivalent to has one. So, our aim is to show that has a fixed point. The proof will be given in several steps.
Step 1: We show that there exists some such that . If it is not true, then for each positive number r, there exists a function and some such that .
Dividing both sides of (3.9) by r, and taking , we have ≥1, which is a contradiction.
Hence for some positive number r, .
Step 2: is continuous on .
Let with in as .
which implies that is continuous on .
Using Lemma 2.2, we can verify that the right-hand side of the above inequality tends to zero as . Therefore, is equicontinuous on J.
Step 4: Mönch’s condition holds.
Suppose that is countable and . We show that , where β is the Hausdorff MNC. Without loss of generality, we may suppose that . Now we need to show that is relatively compact in X for each .
which implies that .
Hence, using Lemma 2.4, has a fixed point y in . Then is a mild solution of system (1.1)-(1.2). This completes the proof. □
4 An application
We take , is a constant. F is Lipschitz continuous for the second variable. Then f satisfies hypotheses (H3) and (H4). This completes the example.
In the current paper, we are focused on establishing the existence result for a class of abstract fractional neutral functional integro-differential evolution systems involving the Caputo fractional derivative in Banach spaces. By using fractional calculus, the properties of characteristic solution operators, Mönch’s fixed point theorem via MNC, we have found the existence results. Here, we do not assume that characteristic solution operators are compact. An example is provided to show the effectiveness of the proposed results.
The authors sincerely thank the reviewers for their valuable comments that led to the present improved version of the original manuscript. And the first author wish to thank Dr.Y. Robinson, Director, RVS Technical Campus, Coimbatore - 641 402, Tamilnadu, India for his constant encouragements and support for this research work.
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