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International Journal of Partial Differential Equations

Volume 2014 (2014), Article ID 640931, 8 pages

http://dx.doi.org/10.1155/2014/640931
Research Article

Existence of the Mild Solution for Impulsive Semilinear Differential Equation

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India

Received 26 December 2013; Revised 15 April 2014; Accepted 18 April 2014; Published 18 May 2014

Academic Editor: Yuri N. Skiba

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.

Abstract

We study the existence of solutions of impulsive semilinear differential equation in a Banach space in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.

1. Introduction

In a few decades, impulsive differential equations have received much attention of researchers mainly due to its demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, medicine and so on. The real world processes and phenomena which are subjected during their development to short-term external inuences can be modeled as impulsive differential equation. Their duration is negligible compared to the total duration of the entire process or phenomena. Impulsive differential equations are an appropriate model to hereditary phenomena for which a delay argument arises in the modelling equations. To further study on impulsive differential equations, we refer to books [1, 2] and papers [311].

In this paper, our purpose is to establish the existence of a solution to the following differential equations with non instantaneous impulses where is a closed and bounded linear operator with dense domain . We assume that is the infinitesimal generator of a strongly continuous semigroup in a Banach space . Here, , and , for all are suitable functions to be specified later.

In [4], authors have introduced a new class of abstract impulsive differential equations in which impulses are not instantaneous and established the existence of solutions to the problem (1)–(3) with the assumption that operator generates a -semigroup of bounded linear operators. In this system of (1)–(3), the impulses begin all of a sudden at the points and their proceeding continues on a finite time interval [5]. To concern the hemodynamical harmony of an individual we think about the following simplified situation. One can recommend a few intravenous sedates (insulin) on account of a decompensation (e.g., high or low level of glucose). Since the presentation of the medications in the bloodstream and the ensuing retention for the form are progressive and continuous processes, we can depict this circumstance as an impulsive activity which begins abruptly and stays animated on a finite time interval.

In [12], the generalization of the condensing operator as convex-power condensing operator has been introduced by Sun and Zhang and a new fixed point theorem for convex-power condensing operator has been established. The new fixed point theorem for convex-power condensing operator, defined by Sun and Zhang, is the generalization of the famous Schauder’s fixed point theorem and Sadovskii’s fixed point theorem. Sun and Zhang [12] have considered the following problem: in a Banach space and established existence theorems for positive mild solutions and global mild solutions to the problem (4) with noncompact semigroup. The nonlinear function fulfills the suitable conditions on the measure of noncompactness as for any bounded set , where is the Kuratowski measure of noncompactness and is a positive constant. For more details about measure of noncompactness, we refer to [1220].

In the present work, our aim is to obtain results concerning the existence of mild solutions to problem (1)–(3) by using convex-power condensing operator and fixed point theorem for convex-power condensing operator Sun and Zhang [12].

The organization of the paper is as follows. We provide some basic definitions, Lemmas and theorems in Section 2 as “preliminaries.” We prove the existence of a mild solution for system (1) in Section 3 as “existence of mild solution.” In the last section, we present an example to illustrate the application of the abstract results.

2. Preliminaries

In this section, we give some definitions, notations, theorems, and lemmas which will be used in later sections.

Let be a real Banach space. The symbol stands for the Banach space of all the continuous functions from into equipped with the norm and stands for the space of -valued Bochner integrable functions on endowed with the norm , .

The operator is the infinitesimal generator of a uniformly continuous semigroup and denotes the domain of , which is densely defined, endowed with the graph norm. A semigroup is said to be equicontinuous if is equicontinuous at arbitrary , , for any bounded subset . Throughout the paper we assume that(H1)the operator generates the equicontinuous semigroup and there exists a positive number such that .

For the study of impulsive differential equation, we define the space which contains all the functions such that is continuous at , and exists for all . Clearly, is a Banach space endowed with norm . For a function and , we define the function such that For , we have and we have following Accoli-Arzelà type criteria.

Lemma 1 (see [4]). A set is relatively compact in if and only if each set    is relatively compact in    .

Now we present the following definition of mild solution.

Definition 2. A piecewise continuous function is said to be a mild solution of the (1)–(3) if , , for all , , and for all , and for all and every .

Next, we give the definition of the Hausdorff measure of noncompactness (MNC).

Definition 3 (see [14]). The Hausdorff measure of noncompactness of the set in Banach space is the greatest lower bound of those for which the set has in the space a finite -net; that is, for every bounded subset in a Banach space .

Definition 4 (see [14]). The Kuratowski measure of noncompactness defined on each bounded subset of as

The relation between Kuratowski measure of noncompactness and the Hausdorff measure of noncompactness is given by To set the structure for our primary existence results, we review some essential properties about the Kuratowski and Hausdorff measure of noncompactness.

Lemma 5. For any bounded set , where is a Banach space. Then, we have following results: (i) if and only if   is precompact;(ii) , where and denote 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.

Definition 6. A continuous and bounded map is called -condensing if, for any noncompact bounded subset , where is a Banach space.

To avoid confusion, we denote by the Hausdorff measure of noncompactness on set , and .

Lemma 7 (see [14], Darbo-Sadovskii). Let be bounded, closed, and convex. If the continuous map is a -contraction, then the map has a fixed point in .

In [12], authors have presented the generalization of the condensing operator and suggested a new fixed point theorem for such operators. Firstly, we present some recognition. Let be a bounded, closed, and convex set and let be a continuous map from into itself with . For every , we set

Definition 8 (see [12]). Let be bounded, closed, and convex. A bounded and continuous operator is called a convex-power condensing operator if for any bounded nonprecompact subset , there exist and an integer such that

A -convex-power condensing operator is -condensing for . Therefore, it is clear that convex-power condensing operator is a generalization of the condensing operator.

Lemma 9 (see [12]). Let be bounded, closed, and convex set. If the continuous map is -convex-power condensing, then there exists a fixed point of map in .

Lemma 10 (see [11, 14]). If is bounded, then , , where . In addition, if is equicontinuous on , then is continuous on and

Lemma 11 (see [11]). If is bounded, then , . Besides, suppose the following conditions are satisfied: (1) is equicontinuous on and each , , ,(2) is equicontinuous at , .Then, we have .

Lemma 12 (see [14]). If is bounded and equicontinuous, then is continuous and where .

Lemma 13. Let be a sequence of functions in . Suppose that there exist satisfying for almost all and every . Then, we have

Lemma 14. We assume that (H1) holds. Then the set , for a.e. is equicontinuous for all .

Proof. Let be a positive constant such that . For , we have It is obvious for . Let be arbitary integer with . For , we have Since , is equicontinuous; therefore, as , uniformly for . Second and third terms of (18) tend to zero when since is arbitrary small.

Then from (17), (18), and (19) and the absolute continuity of integrals, we obtain that , for a.e. is equicontinuous for all .

3. Main Results

In this section, the existence of the mild solution of (1)–(3) under some specified conditions on is established by using the measure of noncompactness and fixed point theorem for convex-power condensing operator.

Now, we made the following hypothesis which will be useful for proving our results.(HG)The functions    are compact and continuous.(HF) is a nonlinear function such that satisfies the Carathèodary condition; that is, we have the following.(i) is continuous for a.e. .(ii)For each , is strongly measurable. For any , there exists a function such that for (iii)There exists a constant such that for any bounded set , (HG1)The functions are continuous and there exist positive constants such that for all , , .(Hk) (H) .

Theorem 15. Suppose that (H1), (HG), (HF), and (Hk) are satisfied. Then, there exists at least one mild solution on for the problem (1)–(3).

Proof. We define the operator as , , for all and for and for , where , To prove the result, we show that the operator has a fixed point. Firstly, we show that the is continuous on . Let be a sequence in such that in . For , we have By the continuity of and    , we have Therefore from (26), (27), (28), and Lebesgue dominated convergence theorem, we get which implies that is continuous on . For , we get From the (27), we get thus, is continuous on . Hence, we conclude that is continuous on .

Secondly, we claim that , where . For each and , we get For , , we have which implies that , for all . On the other hand, by the property of , we get for .   By the assumption (Hk), we have .

Therefore, we conclude that ; that is, has values in .

Now we show the equicontinuity of on . Since is compact, therefore it is obvious that is equicontinuous on . Assume . Let be a constant such that . For , we get Using the semigroup property, we have By the strong continuity of and Lemma 14, we conclude that is equicontinuous on .

For we have Since we have is compact and is strongly continuous, which implies that is equicontinuous on . Hence, is equicontinuous on each .

Set , where and denote the convex hull and closure of the convex hull, respectively. It can be shown easily that maps into itself and is equicontinuous on each , , , . Next we prove that is a convex-power condensing operator. We take and show that there exists a positive integer such that for every nonprecompact bounded subset . From (6), (15), and compactness of , for , where , we have and similarly for , we have For , we get by the fact that are compact.

Further for , we have Proceeding with this iterative method, we get for and similarly for , we get . Thus, we obtain for all .

We have that is equicontinuous on by Lemma 14. Therefore, from Lemma 11, we get since we have that as , which infers that there exists a substantial enough positive integer such that which means that is a convex-power condensing operator. Therefore, from Lemma 10, we get that has at least one fixed point in which is just a mild solution to the problem (1)–(3). This completes the proof of the theorem.

In the next result, the existence of the solution for problem (1)–(3) under Lipschitz conditions of is established by using Darbo-Sadovskii’s fixed point theorem.

Theorem 16. Suppose that assumptions (H1), (HF), (HG1), and (H) hold. Then, the impulsive problem (1)–(3) has at least one mild solution on .

Proof. Firstly, we decompose the map such that , where , , , are defined as For and , we have and for therefore, for all we conclude that Since . By Lemma 5 (vii), we have that for any bounded set For operator , we have From (51)–(53), we have From the assumption (H), we get that , which implies that map is -condensing in . Therefore, by Darbo-Sadovskii’s fixed point theorem, the solution map has a fixed point in which is a mild solution of the nonlocal problem (1)–(3). This completes the proof of the theorem.

4. Example

Let us consider the following impulsive problem: where are fixed numbers, , , and , for all .

To convert problem (55) into the abstract form (1)–(3), where , that is, , we introduced the functions and such that and . We consider and the operator defined as with the the domain It is well known (Pazy [21]) that is the infinitesimal generator of an analytic semigroup , and every analytic semigroup is equicontinuous. This means that satisfies the assumption (H1).

Next we have that is a mild solution of the problem (55).

Case 1. We take and here are continuous functions and . Now, we show that satisfies the assumption (HF). Now, we have that for and for Therefore, for any bounded sets , we obtain where . For , we have Clearly, , are compact and satisfy the assumption (HG). Then by Theorem 15, problem (55) has at least a mild solution.

Case 2. Take and , ; here, is a bounded and continuous function and .

For , and , we have where . For any bounded set , we have where . Thus, (HF) holds. For , we have

Thus, (HG1) holds. Then by Theorem 16, problem (55) has at least a mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi.

References

  1. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989.
  2. L. Zhu and G. Li, “Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 5133–5140, 2011. View at Publisher · View at Google Scholar
  3. J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
  4. M. Benchohra, J. Henderson, and S. Ntouyas, IImpulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, NY, USA, 2006.
  5. Y.-K. Chang, V. Kavitha, and M. M. Arjunan, “Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 32–43, 2010. View at Publisher · View at Google Scholar
  6. E. Hernández and D. O'Regan, “On a new class of abstract impulsive differential equations,” Proceedings of the American Mathematical Society, vol. 141, no. 5, pp. 1641–1649, 2013. View at Publisher · View at Google Scholar
  7. M. Pierri, D. O'Regan, and V. Rolnik, “Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 6743–6749, 2013. View at Publisher · View at Google Scholar
  8. Z. Yan, “Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2252–2262, 2011. View at Publisher · View at Google Scholar
  9. E. Hernández M., M. Rabello, and H. R. Henríquez, “Existence of solutions for impulsive partial neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1135–1158, 2007. View at Publisher · View at Google Scholar
  10. J. H. Liu, “Nonlinear impulsive evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 77–85, 1999.
  11. R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publishing, Malabar, Fla, USA, 1987.
  12. J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593–610, 2006. View at Publisher · View at Google Scholar
  13. X. Xue, “Nonlinear differential equations with nonlocal conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 4, pp. 575–586, 2005. View at Publisher · View at Google Scholar
  14. X. Xue, “Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2593–2601, 2009. View at Publisher · View at Google Scholar
  15. R. P. Agarwal, M. Benchohra, and D. Seba, “On the application of measure of noncompactness to the existence of solutions for fractional differential equations,” Results in Mathematics, vol. 55, no. 3-4, pp. 221–230, 2009. View at Publisher · View at Google Scholar
  16. Z. Fan and G. Li, “Existence results for semilinear differential equations with nonlocal and impulsive conditions,” Journal of Functional Analysis, vol. 258, no. 5, pp. 1709–1727, 2010. View at Publisher · View at Google Scholar
  17. J. X. Sun and X. Y. Zhang, “A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 3, pp. 439–446, 2005. View at MathSciNet
  18. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.
  19. H.-B. Shi, W.-T. Li, and H.-R. Sun, “Existence of mild solutions for abstract mixed type semilinear evolution equations,” Turkish Journal of Mathematics, vol. 35, no. 3, pp. 457–472, 2011.
  20. Y. X. Li, “Existence of solutions to initial value problems for abstract semilinear evolution equations,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 6, pp. 1089–1094, 2005.
  21. K. Li, J. Peng, and J. Gao, “Nonlocal fractional semilinear differential equations in separable Banach spaces,” Electronic Journal of Differential Equations, vol. 2013, no. 7, pp. 1–7, 2013.