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International Journal of Differential Equations
Volume 2014 (2014), Article ID 780636, 10 pages
Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand Pin-247667, India
Received 24 December 2013; Revised 6 March 2014; Accepted 3 April 2014; Published 4 May 2014
Academic Editor: Kanishka Perera
Copyright © 2014 Alka Chadha and Dwijendra N. Pandey. 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.
We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space . The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.
In recent years, fractional calculus has becomes an active area of research due to its demonstrated applications in widespread fields of science and engineering such as mechanics, electrical engineering, medicine, biology, ecology, and many others. The memory and hereditary properties of various materials and processes can be described by a differential equation with fractional order. The fractional differential equation also describes the efficiency of nonlinear oscillations of the earthquake. The details on the theory and its applications can be found in [1–4] and references given therein.
On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, and medicine. In addition, the improvement of the hypothesis of the functional differential equation with infinite delay relies on a choice of phase space. There are various phase spaces which have been studied. Hale and Kato in  introduced a common phase space . For more details on phase space, we refer to books by Hale and Kato , Hino et al.  and papers [7–10]. For the study of impulsive differential equation, we refer to papers [7, 8, 11–18] and references given therein.
The purpose of this work is to establish the existence of mild solution for impulsive fractional differential equation with infinite delay: where , , is a closed and densely defined linear operator and infinitesimal generator of a solution (resolvent) operator on Banach space , and denotes the fractional derivative in Caputo sense and denotes the Riemann-Liouville fractional integral operator. The history defined by for belongs to some abstract phase space defined axiomatically and ; are fixed numbers and denotes the jump of the function at the point , given by . The functions are appropriate functions and satisfy some conditions to be specified later.
In , authors have considered the following impulsive fractional differential equation in a Banach space of the form where is the infinitesimal generator of a -semigroup on a Banach space, is continuous, and , are the element of . Authors have established some existence and uniqueness results for system (2) under the different assumptions on initial conditions.
In this work, we adopt the idea of Wang et al.  and establish the existence of a mild solution for the problem (1) by using the measure of noncompactness and solution operator. The tool of measure of noncompactness has been used in linear operator theory, theory of differential and integral equations, the fixed point theory, and many others. For an initial study of theory of the measure of noncompactness, we refer to book of Banaś and Goebel  and Akhmerov et al.  and papers [9, 21–25] and references given therein.
This paper is organized as follows: In Section 2 we recall some basic definitions, lemmas, and theorems. We will prove the existence of a mild solution for the system (1) in Section 3. In the last section, we shall discuss an example to illustrate the application of the abstract results.
Now we provide some basic definitions, notations, theorems, lemmas, and preliminary facts which will be used throughout this paper.
Let be a Banach space and let be the Banach space of continuous functions from to equipped with the norm and denotes the Banach space of all Bochner-measurable functions from to with the norm
Assume that , that is, is invertible. Then, this permits us to define the positive fractional power as closed linear operator with domain for . Moreover, is dense in with the norm It is easy to see that which is dense in is a Banach space. Henceforth, we use as notation of . Also, we have that for and, therefore, the embedding is continuous. Then, we define , for each . The space , standing for the dual space of , is a Banach space with the norm for . For more details on the fractional powers of closed linear operators, we refer to book by Pazy .
To consider the mild solution for the impulsive problem, we propose that the set = . Clearly, is a Banach space endowing the norm . For a function and , we define the function such that For and , we have and following Accoli-Arzel type criteria.
Lemma 1. A set is relatively compact in if and only if each set is relatively compact in .
For the differential equation with infinite delay, Hale and Kato  proposed the phase space satisfying certain fundamental axioms.
Definition 2 (see ). A phase space is a linear space which contains all the functions mapping into Banach space with a seminorm . The fundamental axioms assumed on are the following,(A)If , is a continuous function on such that and , then for every , the following conditions hold:(i),(ii),(iii). Where is a positive constant, , is a locally bounded, is continuous, and , , are independent of .(A1)For the function in , is a -valued continuous function for .(B)The space is complete.
Now, we state some basic definitions and properties of fractional calculus.
Mittag-Leffler. The definition of one parameter Mittag-Leffler function is given as and two-parameter Mittag-Leffler function is defined as where is a contour which starts and ends at and encircles the disc counter clockwise. The Laplace transform of the Mittag-leffler is defined as For more details we refer to .
Laplace transform of integer order derivatives is defined as
Definition 3. The Riemann-Liouville fractional integral operator is defined as where and is the order of the fractional integration.
Definition 4. The Riemann-Liouville fractional derivative is given as where , , and . Here, the notation stands for the Sobolev space defined as Note that and .
Definition 5. The Caputo fractional derivative is given as where and the following holds
Definition 6 (see ). A family of bounded linear operators in is called a resolvent (or solution operator) generating by if the following conditions are fulfilled:(1) is strongly continuous on and ;(2)for and , and ;(3) is the solution of the equation where denotes the space of all bounded linear operators from into endowed with the norm of operators.
Definition 7 (see ). A solution operator is said to be analytic if admits analytic extension to a sector for some . Furthermore, An analytic resolvent is said to be of analyticity type if, for and , there exists such that for ; here means the real part of .
In this work, we assume that solution operator is analytic; that is, satisfy the following property.The map is continuous from to endowed with the uniform operator norm .
Without loss of generality, we have that there exist a positive constant such that , for .
Definition 8 (see ). The Hausdorff measure of noncompactness is defined as for bounded set , where is a Banach space.
Lemma 9 (see ). For any bounded set , where is a Banach space. Then, the following properties are fulfilled:(i) if and only if is pre-compact;(ii), where and denotes the convex hull and closure of , respectively;(iii), when ;(iv), where ;(v);(vi), for any ;(vii)if the map is continuous and satisfies the Lipschitsz condition with constant , then we have that for any bounded subset , where and are Banach space.
Lemma 10 (see ). A bounded and continuous map is a -contraction if there exists a constant such that , for any bounded closed subset , where is a Banach space.
Lemma 11 (see ). Let be closed and convex with and let the continuous map be a -contraction. If the set is bounded, then the map has a fixed point in .
Lemma 12 ((Darbo-Sadovskii) ). Let be bounded, closed, and convex. If the continuous map is a -contraction, then the map has a fixed point in .
In this work, we consider that denotes the Hausdorff measure of noncompactness in , denotes the Hausdorff measure in noncompactness of and denotes the Hausdorff measure of noncompactness in .
Lemma 14 (see ). If is bounded and equicontinuous set, then is continuous and where .
Lemma 15 (see ). (1) If is bounded, then , , where .
(2) If is piecewise equicontinuous on , then is piecewise continuous for and
(3) If is bounded and equicontinuous, then is piecewise continuous for and where .
3. Main Results
In this section, we will establish the existence results of solution for (1) by using solution operator and Hausdorff's measure of noncompactness.
Definition 16. A piecewise continuous function is said to be a mild solution for impulsive problem (1) if , and where
Now we list the following assumptions which are required to establish main results.(Hf)The function satisfies the following conditions:(1)the function is strongly measurable for every and , ;(2) is a continuous function for each ;(3)there exists an integrable function and a nondecreasing continuous function such that (4)there exists an integrable function such that, for any bounded set , we have for almost everywhere , where .(Hg)(1) For , is Lipschitz continuous function for all and there exist positive constants and such that (2)there exists a constant such that for all .(HI)The functions are continuous functions and satisfy the following conditions:(1)There is a constant such that (2)There exist positive constants such that for all .(H1) where (H2)
Now, let be a function given by and on . It is easy to see that , where , .
Theorem 17. Suppose that hypotheses , , , , and are satisfied. Then, there exists a mild solution for the impulsive problem (1).
Proof. Consider the space endowed with supremum norm . Define the operator byand we have that
where . Thus, is well defined and with the values in by our assumptions. By Lebesgue dominated convergence theorem, axioms of phase space and assumptions , , and , it is clear that is continuous map. Furthermore, by uniformly continuity of the map on , we obtain that set is equicontinuous. We prove the result in following steps.
Step 1. The set is bounded.
Let be a solution of for . Therefore, we have that Take , for . Then, we get that for For , we have For , Therefore, for all , we have From , it implies that and consequently, where Let . Thus . Therefore, we get Integrating above inequality we have that It gives that the functions are bounded on interval . Therefore, the functions are bounded and are also bounded on .
Step 2. The map is a -contraction.
Firstly, we introduce decomposition of into , for such that To prove the result, we firstly show that is Lipschitz continuous. For and , we have that For , we have Thus for we have Taking supremum on , we get Hence, it implies that satisfies the Lipschitz condition with Lipschitz constant , where .
Therefore, from Lemma 9 (vii), we have that for any bounded set
Next, we show that is a -contraction. Let be an arbitrary bounded subset . Since is equicontinuous solution operator, therefore is piecewise continuous. From Lemma 9, we have that for any bounded set , Thus for any bounded set By the assumption (H2), we obtain that ; that is, is a contraction. Therefore, has at least one fixed point in by Darbo fixed point theorem. Let be a fixed point of on , then is a mild solution for (1).
Theorem 18. Suppose that (Hf), (Hg), and (HI) are satisfied and Then, there exists a mild solution for the impulsive problem (1).
Proof. Proceeding as in the proof of Theorem 17, we infer that defined by (34) is continuous from into . Next we indicate that there exists such that , where is defined by . To this end, let us assume that assertion is false, then for any there exists and such that . Therefore, for and , For , we have For , we have which implies that . Therefore, we conclude that We divide both the sides of (58) by and letting , we obtain that This contradicts the inequality (54). Hence, there exists a positive constant such that . Moreover, by uniform continuity of the map on , we have that set is equicontinuous. As the proof of Theorem 17, we infer that system (1) has a mild solution.
In this section, we consider an example to illustrate the application of the theory. Here we take the space as the phase space (see, ).
Now we study the following fractional differential equation with infinite delay: where and are prefixed numbers. Let and consider the operator as with the domain Then, Here , is the orthogonal set of eigenvectors of . It is clear that is the infinitesimal generator of an analytic semigroup in which is given by By Theorem 3.1 in , we get that is the infinitesimal generator of solution operator and there exists a positive constant such that for . The functions satisfy the following conditions:(1)the functions , are measurable such that and (2) is continuous and there exist a such that (3)the functions , are continuous and The impulsive system (60) might be reformulated as the abstract impulsive Cauchy problem (1) where It may be verified that , satisfy the assumptions (Hf), (Hg), and (HI); that is, and , are bounded linear operators and the range of is contained in , , , , and for every , where is defined as . Applying Theorem 17, we obtain that problem (60) has a mild solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the UGC (University Grants Commission, India) under Grant no. (6405-11-061).nt.
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