Second Order Impulsive Retarded Differential Inclusions with Nonlocal Conditions

Henríquez, Hernán R.; Rabelo, Marcos; Vale, Luciana

doi:https://doi.org/10.1155/2014/131379

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 131379 | https://doi.org/10.1155/2014/131379

Second Order Impulsive Retarded Differential Inclusions with Nonlocal Conditions

Hernán R. Henríquez,¹Marcos Rabelo,²and Luciana Vale²

Academic Editor: Geraldo Botelho

Received27 Aug 2013

Accepted03 Dec 2013

Published23 Jan 2014

Abstract

In this work we establish some existence results for abstract second order Cauchy problems modeled by a retarded differential inclusion involving nonlocal and impulsive conditions. Our results are obtained by using fixed point theory for the measure of noncompactness.

1. Introduction

In this paper we are interested in studying the existence of solutions to evolution systems that can be described by equations that suffer abrupt changes in their trajectories and simultaneously depend on nonlocal initial conditions. More specifically, the aim of this paper is to establish existence results for abstract second order evolution problems with delay whose equations can be written as differential inclusions with nonlocal initial conditions and subjected to impulses.

To describe the problem, throughout this work we denote by a Banach space provided with a norm . We assume that is the infinitesimal generator of a cosine functions of operators on . We study the system on an interval , for some , and we assume that the impulses occur at fixed moments . Moreover, denotes the system delay. Specifically, we will consider abstract second order systems where , , denotes the function defined by for , indicates the gap of a piecewise continuous function at , is an appropriate function, and , , , and are maps that will be specified later.

As a model, we consider a general wave equation described by a second order differential inclusion with impulses and nonlocal initial conditions for , , and . In this system we assume that is a multivalued map, and the inclusion indicated in (5) will be explained in Section 4. Moreover, , , , , and are appropriate functions.

Here we briefly discuss the context in which our work is inserted. We do not intend to make an exhaustive list of references but just mention those most recent and directly related to the topic of this paper. Differential inclusions and impulsive differential inclusions are used to describe many phenomena arising from different fields as physics, chemistry, population dynamics, and so forth. For this reason, last years several researchers have studied various aspects of the theory. We mention here to [1–6] and references in these texts for the motivations of the theory.

In particular, there are phenomena in nature that experiment abrupt changes at fixed moments of time. Such kind of systems are well described by impulsive systems. In the study of ordinary and partial differential equations with impulsive action, interesting questions appear such as local and global existence, stability, controllability, and so forth. For this reason this topic has attracted the attention of many authors in the last time. We only mention here the papers [7–17] which are directly related with the objective of this paper.

The concept of nonlocal initial condition was introduced by Byszewski and Lakshmikantham to extend the classical theory of initial value problems ([18–22]). This notion is more appropriate than the classical theory to describe natural phenomena because it allows us to consider additional information. Thenceforth, the study of differential equations with nonlocal initial conditions has been an active topic of research. The interested reader can consult [23–26] and the references therein for recent developments on issues similar to those addressed in this paper.

On the other hand, it is well known that retarded functional differential equations are used to model important concrete phenomena. For general aspects of the theory of partial differential equations with delay we refer to [27], and for functional differential inclusions we refer to [7, 9, 12–14, 28]. In similar way, there exists an extensive literature concerning abstract second order problems. In the autonomous case, the existence of solutions to the second order abstract Cauchy problem is strongly related with the concept of cosine functions.

In this paper, we combine the theory of cosine functions with the properties of the measure of noncompactness and some properties of function spaces introduced in [9] to establish the existence of solutions to the problems (1)–(4).

This paper has four sections. In Section 2 we develop some properties about the abstract Cauchy problem of second order, the measure of noncompactness, and multivalued analysis which are needed to establish our results. In Section 3 we discuss the existence of mild solutions to problems (1)–(4). Finally, in Section 4 we apply our results to establish the existence of solutions to problems (5)–(9).

The terminology and notations are those generally used in functional analysis. In particular, if and are Banach spaces, we denote by the Banach space of the bounded linear operators from into and we abbreviate this notation to whenever .

2. Preliminaries

2.1. The Second Order Abstract Cauchy Problem

In this section we collect the main facts concerning the existence of solutions for second order abstract differential equations. For the theory of cosine functions of operators we refer to [29–34]. We next only mention a few concepts and properties relative to the second order abstract Cauchy problem. Throughout this paper, is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators on the Banach space . We denote by the sine function associated with which is defined by We denote by , some positive constants such that and for . The function is continuous for the norm of operators and for every . The notation stands for the space formed by the vectors for which is a function of class on . We know from Kisyński [35] that endowed with the norm is a Banach space.

The operator valued function is a strongly continuous group of bounded linear operators on the space , generated by the operator defined on . It follows from this property that is a bounded linear operator and that is a strongly continuous operator valued map. We denote by a positive constant such that for all . In addition ([34]) which implies that Furthermore, if is a locally integrable function, then defines an -valued continuous function.

The existence of solutions of the second order abstract Cauchy problem where is an integrable function has been discussed in [30, 32–34, 36]. Similarly, the existence of solutions for the semilinear second order abstract Cauchy problem has been treated in [37]. We only mention here that the function given by is called mild solution of (15), and that when the function is continuously differentiable and

2.2. Measure of Noncompactness and Multivalued Maps

In this subsection we recall some facts concerning multivalued analysis, which will be used later. Let be a metric space. Throughout this paper denotes the collection of all nonempty subsets of and denotes the collection of all bounded nonempty subsets of .

Some of our results are based on the concept of measure of noncompactness. For this reason, we next recall a few properties of this concept. For general information the reader can see [5, 9, 38, 39]. In this paper, we use the notion of Hausdorff measure of noncompactness.

Definition 1. Let be a bounded subset of a metric space . The Hausdorff measure of noncompactness of is defined by

Remark 2. Let be bounded sets. The Hausdorff measure of noncompactness has the following properties. (a)If , then .(b).(c) if and only if is totally bounded.(d).

In what follows, we assume that is a normed space. For a bounded set , we denote by the closed convex hull of the set .

Remark 3. Let be bounded sets. The following properties hold. (a)For , .(b), where .(c).

Henceforth we use the notations and to denote the following sets:(s1),(s2).

We refer the reader to the already mentioned references to abstract concepts of measure of noncompactness and for many examples of measure of noncompactness.

Definition 4. Let be metric space. We said that a multivalued map is said to be (i)upper semicontinuous (u.s.c. for short) if is an open subset of for all open set ;(ii)closed if its graph is a closed subset of ;(iii)compact if its range is relatively compact in ;(iv)quasicompact if is relatively compact in for any compact subset .

Definition 5. A multivalued map is said to be a condensing map with respect to (abbreviated, -condensing) if for every bounded set , , .

The next result is essential for the development of the rest of our work. We point out that if is u.s.c., then is closed. This allows us to establish the following version of the fixed point theorem [5, Corollary 3.3.1].

Theorem 6. Let be a convex closed subset of , and let be a u.s.c. -condensing multivalued map. Then is a nonempty compact set.

2.3. Function Spaces

Let be any of the intervals or . The space is formed by all piecewise continuous functions satisfying the following conditions:(c1)the function is continuous on , and(c2)there exist and and for all .We consider endowed with the norm of the uniform convergence It is well known that is a Banach space. Furthermore, let , , be the map defined by where we set , , and . For each and , we denote by the range of under the operator ; that is, .

We define the subspace of consisting of all functions which are continuously differentiable at and there exist and for all . It is straightforward to show that the space endowed with the norm is a Banach space.

From now on we denote by , , the space of piecewise continuous functions endowed with the norm In what follows we denote by the Hausdorff measure of noncompactness in and by the Hausdorff measure of noncompactness in a space of continuous (or piecewise continuous) functions with values in . We next collect some properties of measure which are needed to establish our results.

Lemma 7. Let be a strongly continuous operator valued map. Let be a bounded set. Then .

Lemma 8. Let be a bounded set. Then .

Lemma 9 (see [39]). Let be a bounded set. Then for all . Furthermore, if is equicontinuous on , then is continuous on , and

Lemma 10. Let be a bounded set. Then there exists a countable set such that .

A set is said to be uniformly integrable if there exists a positive function such that a.e. for and all .

Lemma 11. Let be a strongly continuous operator valued map and be the map defined by Let . Assume that there is a compact set and a positive function such that for all and . Then

Proof. It is clear that is uniformly integrable. Applying Lemma 10 and [5, Theorem 4.2.2], we can affirm that Since the set is equicontinuous, using Lemma 11, we obtain the assertion.

We also need to consider the product space provided with the norm The following property is immediate.

Lemma 12. Let be a bounded set. (a)Assume that , where and are bounded sets. Then .(b)Let Then .

3. Existence Results

In this section we establish some results of existence of mild solutions of problems (1)–(4). Initially we will establish the general framework of conditions under which we will study this problem. Throughout this section, denotes the Hausdorff measure of noncompactness in . We assume that . Moreover, in what follows we assume that is a multivalued map from into that satisfies the following properties.(F1)The function admits a strongly measurable selection for each , , and .(F2)For each , the function is u.s.c.(F3)For each , there is a function such that for all , , and such that (F4)There exists a positive integrable function on such that for all bounded sets , , and such that .

Remark 13. Let and . Then the function , , is continuous. Hence, the function , , is strongly measurable. Combining this assertion with conditions (F1) and (F2) and applying [5, Theorem 1.3.5] we infer that the function , admits a Bochner integrable selection. As a consequence, the set and is convex.

Next we introduce the conditions on the function . We assume that is a map from into such that the values for all and that the following conditions are fulfilled.(g1)The function is continuous and takes bounded sets in into bounded subsets of . Moreover, the map is continuous and takes bounded sets in into bounded subsets of .(g2)There is a continuous function and a constant such that for all bounded set .(g3)For each bounded set the set is equicontinuous.

Next we establish the conditions on maps , , .

We assume that and satisfy the following conditions.(I1)The maps , , are continuous and takes bounded sets into bounded sets.(I2)There are positive constants , , , , such that for all bounded subsets of , and such that .

Remark 14. Let be a bounded set. Then for all , is a bounded subset of and for all .

Motivated by expressions (16) and (17) (see also [12]), we introduce the following concept of mild solution to problems (1)–(4).

Definition 15. A function is said to be a mild solution of (1)–(4) if conditions (2)–(4) are satisfied, and the integral equation is verified for and all .

To establish our results, we need to study two integral operators defined on the set for functions and . Initially we mention some properties of . A first result establishes that is closed. Specifically we have the following property ([5, Lemma 5.1.1]).

Lemma 16. Let and be sequences that converge to and , respectively. Suppose that , , is a sequence that converges weakly to . Then .

On the other hand, since the values of are convex compact sets, and, as already mentioned, the graph of is closed, we can assert that for functions and , the set is compact in . In addition, as a consequence of (F3), the set is uniformly integrable over ; that is to say, there exists a positive function such that a.e. for and all .

We introduce now the operators given by It is clear that , are bounded linear operators. Using , we can construct the multivalued maps given by Since and are strongly continuous operator valued functions, the assertion in [5, Lemma 4.2.1] remains valid for and . Hence, combining our previous remarks with [5, Lemma 4.2.1, Corollary 5.1.2] we can establish the following property.

Lemma 17. Let be a multivalued map satisfying conditions (F1)–(F4). Then and are u.s.c. maps with convex compact values.

We next define the solution map for problems (1)–(4) as follows. Assume that and let . We define to be the set formed by all functions given by for . It follows from our hypotheses that . Hence, . Furthermore, it is clear that is a mild solution of problems (1)–(4) if and only if is a fixed point of .

We are now in a position to prove the main result of this section. We introduce the map defined as follows. For , is the set consisting of all functions given by for . It follows from (g1) and (I1) that is well defined.

We use the following notations:

Theorem 18. Assume that , and conditions (F1)–(F4), (g1)–(g3) and (I1)-(I2) are fulfilled. If , then the map is u.s.c. and -condensing.

Proof. It follows from our hypotheses and Lemma 17 that is a u.s.c. multivalued map with convex compact values. It remains to prove that is -condensing. Let be a bounded set such that . It follows from Lemma 10 that there exists a sequence in such that . We can write for some . It follows from Lemma 12 that Here we will estimate separately the values and . To estimate , using (39), we can write for .
For , applying (g2), we get Using now condition (g3) and Lemma 9 we infer that Now we consider functions defined on . From (43) and using Lemma 7, we get Using now conditions (g2), (I2), and Remark 14, we have On the other hand, since , for we have that . This implies that is uniformly integrable and, applying condition (F4), Combining this estimate with Lemma 11 we infer that Substituting in (46), we obtain Combining with (45), and using Lemma 12, it yields We next estimate . Using (40) we can write for .
From (52) and using Lemma 7, we get Using again conditions (g2) and (I2), Lemma 12, and also our previous estimates, we obtain Finally, collecting these estimates, we get This implies that , which in turn implies that is a -condensing map.

Corollary 19. Under the hypotheses of Theorem 18, there exists a mild solution of problems (1)–(4).

Proof. It follows from Theorem 18 and Theorem 6 that there is a fixed point of . It is follows from (17), (39), and (40) that and that is a fixed point of .

The sine functions involved in concrete problems are frequently compact. This allows us to reduce the conditions to obtain the existence of mild solutions to problems (1)–(4). To establish this result some previous properties about sine operators are needed.

Lemma 20. Assume that is a compact operator for all . If is a bounded set, then the set is relatively compact in .

Proof. The set is relatively compact in for all . Moreover, for fixed and such that we can decompose If we restrict us to consider , using that is relatively compact, is bounded, and , we obtain that and when uniformly for . Consequently, the set is equicontinuous, and the Ascoli-Arzelá theorem implies that is relatively compact in .

Lemma 21. Assume that is a compact operator for all . Then the map is compact.

Proof. Let be a bounded set. It follows from [40, Theorem 5] that the set is relatively compact in for every . On the other hand, using again (56) we can write Since is relatively compact in , as uniformly for . Combining with the above estimate, it follows that as uniformly for . Therefore, the set is equicontinuous. The Ascoli-Arzelá theorem shows that is a compact operator.

We define the constants

Corollary 22. Assume that the operator is compact for all . Assume further that and that conditions (F1)–(F4), (g1)–(g3), and (I1)-(I2) hold. If , then there exists a mild solution of problems (1)–(4).

Proof. We repeat the construction carried out in the proof of Theorem 18. The only modification is related with the estimate of for defined on . Using Lemmas 20 and 21 we can see that Combining with (45), for defined on , we obtain Proceeding as in the proof of Theorem 18 and Corollary 19, we get that has a fixed point , which is a mild solution of problems (1)–(4).

We now are concerned with the following particular case of problems (1)–(4): From an intuitive viewpoint this model corresponds to an incomplete second order equation in which the impulses on the path do not lead to changes in the velocity.

We can reduce this problem to a particular case of problems (1)–(4) taking as with and modifying slightly the conditions about , , and . We assume that is a multivalued map from into that satisfies conditions (F1)–(F4) (now we omit the variable in these conditions). Proceeding as in Remark 13, for the function , admits a Bochner integrable selection. As a consequence, the set and is convex.

Next we describe the conditions on the function . We assume that is a map from into that satisfies the following.(g1)The function is continuous and takes bounded sets in into bounded subsets of .(g2) There is a continuous function such that for all bounded sets .(g3)For each bounded set the set is equicontinuous.

Next we establish the conditions on maps , . We assume that satisfy the following conditions.(I1)The maps , are continuous and takes bounded sets into bounded sets.(I2) There are positive constants , , , such that for all bounded sets and such that .

We now establish our concept of mild solution.

Definition 23. A function is said to be a mild solution of (61)–(63) if conditions (62)-(63) are satisfied, and the integral equation is verified for and all .

We next define the solution map associated with our concept of mild solution for problems (61)–(63) as follows. Let . We define to be the set formed by all functions given by for . It follows from our hypotheses that . Hence, . Furthermore, it is clear that is a mild solution of problems (61)–(63) if and only if is a fixed point of .

We define We are now in a position to prove the following result.

Theorem 24. Assume that conditions (F1)–(F4), (g1)–(g3), and (I1)-(I2) hold. If , then the map is u.s.c. and -condensing.

Proof. We proceed as in the proof of Theorem 18. We only include here a sketch of the proof. To prove that is -condensing. Let be a bounded set such that . It follows from Lemma 10 that there exists a sequence in such that . We can write for some .
To estimate , using (68) we can write for .
From (45), we have
Now we consider functions defined on . From (70) and using Lemma 7, we get
Using now conditions (g2), (I2), and Remark 14, we have
On the other hand, since , for , we have that . This implies that is uniformly integrable and, applying condition (F4),
Combining this estimate with Lemma 11 we infer that
Substituting this estimate in (72), we obtain Collecting these assertions, we get which implies that , which in turn shows that is -condensing and completes the proof.

The following assertions are immediate consequences of Theorem 24.

Corollary 25. Under the hypotheses of Theorem 24, there exists a mild solution of problems (61)–(63).

Corollary 26. Assume that the operator is compact for all . Assume further that conditions (F1)–(F4), (g1)–(g3), and (I1)-(I2) hold. If then there exists a mild solution of problems (61)–(63).

4. Applications

In this section we apply our abstract results to study the existence of solutions to the impulsive retarded wave equation described by (5)–(9). To model this problem in abstract form, in what follows we consider the space and is the map defined by with domain . It is well known that is the infinitesimal generator of a strongly continuous cosine function on . Furthermore, has a discrete spectrum and the eigenvalues are , , with corresponding eigenvectors . Furthermore, the set is an orthonormal basis of and the following properties hold.(a)For , .(b)For , Consequently, for all and is a compact operator for every .(c)The space (see [30] for details) and . In particular, we observe that the inclusion is compact. Moreover, the function is -periodic. Using this property and (13) we can show that In fact, using the periodicity of is sufficient to establish the property for . It is an immediate consequence of the definition of the norm in that , for . For , we can write with , and using (13) we have Since for all , we obtain Similarly, for , we can write with , and using again (13), we can write which implies In view of that , this completes the proof of the assertion.

In what follows we assume that and that , where we have identified for and .

Initially we construct the multivalued function . We assume that is a bounded multivalued map that satisfies the following conditions.(f1)There exist positive constants , , and such that for all , where denotes the Hausdorff metric and denotes the Euclidean norm in .(f2)There exists a positive function such that for all .In a metric space , we denote . We will use the following property of the Hausdorff metric.

Lemma 27. Let be a metric space, and let be bounded sets. Then, for every ,

We have the following consequences.

Proposition 28. Under the previous conditions, the following properties hold. (i)For each and , the function is measurable.(ii)For each and , the function is upper semicontinuous.(iii)For each and the set is closed convex in .

Proof. Consider the following.(i)It is an immediate consequence of the fact that the multivalued map is -continuous.(ii)We know that is bounded and, therefore, relatively compact. We will show that the is closed. Assume that and , , as . Using Lemma 27 we can write Since is -continuous, it follows that . In view of that the set is closed, we conclude that . Applying [3, Proposition 1.2] we obtain that is upper semicontinuous.(iii)For and the map , , is measurable with closed values. It follows from [3, Proposition 3.2] that there exists a measurable selection such that . Using that is bounded it follows that . This shows that . Since the values of are convex, it follows that is also convex.
To establish that is closed, we consider a sequence , such that , , for the norm in . By passing to a subsequence if necessary, we can assume that , , a.e. . Since is closed, it follows that , which in turn implies that .

For we define and is given by .

Proposition 29. Under the previous conditions, is a measurable and upper semicontinuous map with convex compact values.

Proof. Initially we show that is closed. Let be a sequence in that converges to as . Since is sequentially weakly compact, there is and a subsequence such that as in the weak topology. Since is a closed convex set, it follows that . In view of that where denotes the characteristic function of the interval , we obtain that and .
Since the functions in are uniformly bounded, converge to zero as uniformly for . From [41, Theorem IV.8.20] we conclude that is relatively compact.
On the other hand, proceeding as in the proof of Proposition 28(ii), we get that is upper semicontinuous. Finally, as a consequence of a remark in [5, page 21], we can affirm that is measurable.

The following consequence is essential for our construction.

Corollary 30. Under the above conditions, there exists a measurable selection for .

Proof. It follows from [3, Proposition 3.2].

We now consider the map defined by for . Since , it follows from our construction that satisfies condition (F1). Moreover, proceeding as in the proof of Proposition 28(ii) we conclude that is upper semicontinuous, which shows that satisfies condition (F2). On the other hand, if , then there exists such that with . Therefore, there exists such that Hence which shows that satisfies the condition (F3).

Proposition 31. Let , , and . Then

Proof. To abbreviate the text we introduce the notations and . Since , there is such that . Moreover, in view of that is convex compact in , there is the nearest point of in . Consequently, there is such that . Therefore, which completes the proof.

Corollary 32. Under the above conditions, let , , be bounded sets, and let be a set uniformly bounded. Then where denotes the Hausdorff measure of noncompactness for the norm of uniform convergence.

Proof. Let . We abbreviate the notation by writing and . There exist , , and having the following property: given , , and there are , , and such that , and . Hence, if , using Proposition 31, we obtain that Since is a compact set in , and was chosen arbitrarily, this completes the proof of the assertion.

Corollary 32 shows that satisfies the condition (F4).

On the other hand, we define by for and . We assume that is a function of class such that . It is clear that and . This implies that for all . Moreover, is a continuous map that takes bounded sets into bounded sets, and is also continuous and takes bounded sets into bounded sets in . This shows that satisfies condition (g1).

It is clear that is a continuous function from into for each . Let be a bounded set. It is not difficult to see that the set is equicontinuous. Moreover, which shows that is bounded. The Ascoli-Arzelá theorem implies that is relatively compact in . Therefore, is also relatively compact in . Hence , and we can take . This shows that satisfies the first part of condition (g2). Furthermore, it follows from (c) that From the definition of we obtain Arguing as above, we can affirm that the first term on the right hand side of (103) defines a relatively compact set in . Since in the space , using [42, Theorem 3.1], we conclude that which shows that also satisfies the second part of condition (g2) with .

On the other hand, converges to zero as uniformly for . This shows that the set is equicontinuous. Consequently, we can affirm that satisfies condition (g3).

We define by for , and . We assume that for , and that are functions such that . Proceeding as above, it is easy to see that and are continuous maps that take bounded sets into bounded sets. This shows that condition (I1) is verified. In addition, using that the map , , is a bounded linear functional with norm , we deduce that Using this argument together with condition (c), we get which shows that condition (I2) is also verified.

We complete our model by defining . It is not difficult to see that under the conditions specified previously, systems (5)–(9) are described by the abstract models (1)–(4). The constants introduced in Section 3 are the following:

Combining with Corollary 22, we have established the following result.

Theorem 33. Assume that , and Then there exists a mild solution of systems (5)–(9).

Conflict of Interests

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

Acknowledgment

Hernán R. Henríquez was partially supported by CONICYT under Grants FONDECYT 1130144 and DICYT-USACH.

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Copyright © 2014 Hernán R. Henríquez et al. 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.

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