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Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in Banach spaces
Advances in Difference Equations volume 2013, Article number: 215 (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.
Our aim of this paper is to close the gap, and motivated by works [11, 17, 18], in this paper we investigate the existence of solutions of fractional neutral integro-differential systems with infinite delay of the form
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 .
In this paper, we assume that is the infinitesimal generator of a strongly continuous semigroup , then there exists a constant . Without loss of generality, we assume that . Then it is possible to define the fractional power for , as a closed linear operator on its domain with inverse (see ). The following are basic properties of .
is a Banach space with the norm for .
for each and .
For every , is bounded on X and there exists such that
is a bounded linear operator for in X.
Now we define the abstract phase space , which has been used in . Assume that is a continuous function with . For any , we define
and equip the space ℬ 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 be a seminorm in defined by
Lemma 2.1 (See )
Assume , then for , . Moreover,
Definition 2.1 The fractional integral of order α with the lower limit zero for a function f is defined as
provided the right-hand side is pointwise defined on , where is the gamma function, which is defined by .
Definition 2.2 The Riemann-Liouville fractional derivative of order , , , is defined as
where the function has absolutely continuous derivative up to order .
Definition 2.3 The Caputo derivative of order α for a function can be written as
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 )
A continuous function is said to be a mild solution of system (1.1)-(1.2) if on and the following integral equation
is satisfied, where and are called characteristic solution operators and given by
Here, is a probability density function defined on , that is
The following results of and are used throughout this paper.
Remark 2.2 (See )
It is not difficult to verify that for ,
The operators and have the following properties:
For any fixed , and are linear and bounded operators, that is, for any ,
and are strongly continuous.
For and any bounded subsets , and are equicontinuous if with respect to as for each fixed .
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.
The MNC Φ is said to be
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.
One of the many examples of MNC is the noncompactness measure of Hausdorff β defined on each bounded subset Ω of X by
, 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 bounded and equicontinuous, then is continuous for and
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 .
(H2) The function satisfies the following:
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 .
There exists a constant and such that, for any bounded subset , ,
where and β is the Hausdorff MNC.
(H3) The function satisfies:
is measurable for and is continuous for a.e. .
There exists a constant such that , for all , .
There exists such that for any bounded subset ,
(H4) The function is continuous and there exists a constant , such that g is valued and
For our convenience, let us take , . , and .
Theorem 3.1 Assume that the hypotheses (H1)-(H4) are satisfied, then system (1.1)-(1.2) has at least one mild solution provided that,
Proof In order to prove the existence of mild solutions for system (1.1)-(1.2), transform it into a fixed point problem.
We consider the operator defined by
For , we define by
then . Let , . It is easy to see that x satisfies (2.1) if and only if y satisfies and
Let . For any ,
thus is a Banach space. Set for some , then is uniformly bounded, and for , from Lemma 2.1, we have
Define the operator by
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 .
On the other hand, from hypotheses the (H2)(i), (ii), (H4), Lemma 2.2(i) and Hölder’s inequality, we obtain
Let us estimate, , . By assumption (H4), we have
By using Lemma 2.2 and Hölder’s inequality, one can deduce that
Using assumptions (H2) and (H3), we have
Combining estimates (3.5)-(3.8) yields
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 .
Then, by using hypotheses ()(i), (ii), ()(i), (ii) and Lebesgue’s dominated convergence theorem, we obtain
Observing (3.6) and (3.7), we have
which implies that is continuous on .
Step 3: is equicontinuous on J. Indeed, let and . Then there is such that
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 .
By Theorem 2.1, we have
which implies, by Lemma 2.3, , where is defined in condition (3.1). Thus, from Mönch’s condition, we get
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
For , consider the following fractional order neutral functional integro-differential of the form
where is continuous functions. To write system (1.1)-(1.3) to the form (4.1)-(4.3), let and be defined as follows: with domain
It is well known that A is an infinitesimal generator of a semigroup in X and is given by , for , is not a compact semigroup on X with , where β is the Hausdorff MNC and there exists an such that . Moreover, is equicontinuous  for and . Define by
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.
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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.
The authors declare that they have no competing interests.
The authors have made equal contributions of this manuscript. All authors read and approved the final version.