Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

Hong, Shihuang; Gao, Jing; Peng, Yingzi

doi:https://doi.org/10.1155/2014/610365

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 610365 | https://doi.org/10.1155/2014/610365

Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

Shihuang Hong,¹Jing Gao,¹and Yingzi Peng¹

Academic Editor: Shawn X. Wang

Received09 Nov 2013

Accepted30 Mar 2014

Published04 May 2014

Abstract

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

1. Introduction

One of the most convenient generalizations of differential equations is the notion of set differential equations (SDEs). The main objects in this framework are set-valued functions of the form , where is an interval of real axis and is the space of all nonempty compact and convex subsets of . The increasing need of theoretical models for the study of other problems emerging in optimal control theory, dynamic economy, and biological theory motivated the advance in SDEs. In the last decade, the study of SDEs has attracted the attention of many researches [1–28]. In [21–23] the authors extended SDEs to set dynamic equations on time scales in order to unify such problems in the framework of set dynamic equations on a time scale.

Hukuhara derivative (-derivative, for short) of set-valued functions is the starting point for the topic of SDEs [29] and later also for fuzzy differential equations (FDEs) [30]. In [17, 18] the authors were concerned with the interrelation between SDEs and FDEs. However, Hukuhara differentiability concept has some drawbacks as pointed out in [31, 32]. For instance, -derivative depends on the existence of Hukuhara difference but the latter does not always exist. Recently, several generalized -derivatives are proposed to overcome some shortcomings of this approach. Let us mention that the strongly generalized differentiability (-differentiability) was defined by considering lateral -derivatives (four cases) in [32]. In [33] the authors have introduced the concept of -differentiability, which is based on a generalization of the Hukuhara difference between two intervals. Chalco-Cano et al. in [34] defined -differentiability of set-valued functions from the real axis into and studied its relationship to -differentiability. In this paper we shall adopt -differentiability of set-valued functions on time scales, which is corresponding in character with the above-mentioned -differentiability and is an extension of the -derivative on time scales introduced by [21].

A number of processes in physics, biology, and control theory during their evolutionary development are subject to the action of short-time forces in the form of impulses. In most cases the duration of the action of these forces is negligibly small, as a result of which one can assume that the forces act only at certain moments of time. The impulsive differential equations represent a mathematical model of such processes. The theory of impulsive differential equations has attracted the attention of many scientists; see, for instance, [18, 20, 22] and [35–38].

In the present paper, we investigative the existence and stability theory for impulsive set dynamic equations on time scales by introducing the notion of the exponential dichotomy and using fixed point theorems. Our main contribution of the paper lies in showing how the basic properties of exponential dichotomy theory for homogeneous linear set dynamic equations can be used to establish the existence and stability of nonlinear set dynamic equations on time scales, proposing the possibility to get some insight into and better understanding of the subtle difference between discrete and continuous systems and building a unified study framework of the corresponding problems.

The paper contains five sections. In Section 2 several basic definitions and properties of time scales and set-valued analysis are collected; especially, the -derivative of set-valued functions on time scales is defined. Subsequently, in Section 3, the exponential dichotomy of the homogeneous linear set dynamic equation is introduced and the existence and uniqueness of solutions to a class of linear impulsive set dynamic equations under -derivative of set-valued functions and the existence of bounded solutions to its nonlinear counterpart are presented. In Section 4 several preliminary results concerning stability are given by applying a fixed point theorem. In the final section, several examples are given to show the applicability of our main results.

2. Preliminaries

In this section, we recall briefly the necessary background material for a self-contained presentation of our study. We first recall the notion of the time scale built by Hilger and Bohner. For more details, we refer the reader to [39, 40].

A closed nonempty subset of real axis is called a time scale or measure chain. For we define the forward jump operator by , while the backward jump operator is defined by . The function called the graininess function is defined by . In this definition we put (i.e., if has a maximum ) and (i.e., if has a minimum ), where denotes the empty set. is said to be right scattered if and is said to be right dense (rd) if . is said to be left scattered if and is said to be left dense (ld) if . A point is said to be isolated (dense) if it is right scattered (right dense) and left scattered (left sense) at the same time. In this paper we stipulate that the time scale is if has a left scattered maximum .

We continue with a description of the basic known results for Hausdorff metrics, continuity, and differentiability for set-valued mappings on time scales and their corresponding properties within the framework of time scales. We refer readers to [18, 21] for details. The following operations can be naturally defined on it: Here, assume that some product operation is defined on . The set satisfying is known as the geometric difference (Hukuhara difference) of the set and set and is denoted by the symbol . It is worthy to note that the geometric difference of two sets does not always exist but if it does, it is unique. A generalization of geometric difference proposed in [33] aims to guarantee the existence of difference for any two intervals in . In the light of this, a generalized difference called the -difference, “”, can be defined for any ; that is,

It is clear that if the -difference exists, it is unique and it is a generalization of the geometric difference since , whenever exists. In addition, the authors in [33] enumerated the following properties.

Lemma 1. Let be two compact convex sets. Then,(i); ; ;(ii)if exists in the sense (a), then exists in the sense (b) and vice versa;(iii) if and only if and ;(iv) exists if and only if and exist and .

Proof. (i)–(iii) for the proof we refer to [41]. To prove the first part of (iv) let ; that is, or . Then or and this means ; the second part is immediate.

Throughout this paper, we always assume that the -difference of any two elements under consideration in exists. We remark that the assumption may be valid; for instance, in the unidimensional case (with , a class of all closed bounded intervals of the real line) the -difference exists for any two compact intervals.

We define the Hausdorff metric as where and are bounded subsets of .

Notice that with this distance is a complete metric space. On the other hand, the Hausdorff metric is compatible with the operations defined on it as described by the following properties: for any and , Here for .

In order to define the continuity and regularity of set-valued functions on time scales, we first need the notion of selectors of set-valued functions; that is, a function is called a selector of the set-valued function if for all .

Definition 2. A set-valued mapping , where with , is said to have the limit at if there exists an element such that, for any , there exists a such that , for all with . We denote the limit by ; that is, .
Let be well defined. is called continuous at if its limit at exists and equals .
is called regulated provided its right-sided limit exists at any right-dense point in , its left-sided limit exists at any left-dense point in , and its regulated selector exists.
is called right dense continuous, denoted -continuous, provided is continuous at each right-dense point in , its left-sided limits exist at each left-dense points in , and its -continuous selector exists. Similarly we can define -continuity.
is said to be uniformly -continuous on if it is -continuous and for any , there exists such that for each right-dense point and any with .

Lemma 3. Let the set-valued function be regulated, where is a compact interval. Then is bounded; that is, there exists a positive number such that for each .

The following definition we refer to the -differentiability in [34] which can be regarded as an improvement of -differentiability introduced in [21].

Definition 4. Suppose that is a set-valued function. Let and be an element of (provided it exists) with the property that, for given any , there exists a neighborhood of (i.e., for some ) such that for all with . We call the -derivative of at .
We say that is -differentiable at if its -derivative exists at . Moreover, we say is -differentiable on if its -derivative exists at each . The set-valued function is then called the -derivative of on .

We denote the sets of all -continuous set-valued functions and all set-valued functions whose -continuous -derivative exists, respectively, by

It is significant to refer that if we restrict ourselves to single valued mappings, then the previous notions reduce to their classical counterparts, that is, to ordinary -continuity and -differentiability in (in the sense, defined as [40] if is a single valued function). We enumerate the following properties for the g-differentiable set-valued functions.

Proposition 5. Assume that is a set-valued function and ; then we have the following.(I)If is -differentiable at , then is continuous at .(II)If is continuous at and is right scattered, then is -differentiable at . Moreover, we have (III)If is right dense, then is -differentiable at if and only if

Proof. (I) Assume that is -differentiable at . Let . Define Clearly, . Note that and by Definition 4 there exists a neighborhood of such that for all with . Therefore, we have for all with This implies that is continuous at .
(II) Assume that is continuous at and is right scattered. By the continuity, we have This guarantees the existence of . By virtue of Lemma 1(iv), the difference exists and Hence, given , there exists a neighborhood of such that for all with . It follows that for all with . According to Definition 4, (II) is valid as desired.
(III) Assume that is -differentiable at and is right dense. Let be given. Since is -differentiable at , there exists a neighborhood of such that for all with . Since , that is, , we have that for all with . This yields for all with . Therefore, from the arbitrariness of we get
Conversely, suppose that is right dense, the limit exists in Part (III). Then, for any given , there is a neighborhood of such that for all with . This easily infers the desired result.

Proposition 6. Assume that set-valued functions are -differentiable; then one has the following.(d1) The sum defined by for each and the difference defined by are -differentiable at . Moreover, (d2) For any constant , is -differentiable at with (d3) The product function defined by for is -differentiable at with

Proof. (d1) Since and are -differentiable at , for any , there exist neighborhoods and of such that for all with and for all with . Let . Then we have for all with Therefore is -differentiable at and as desired. The proof of the second formula is similar.
(d2) We assume that . Otherwise, the desired result trivially holds. The differentiability of at guarantees that there exists the neighborhood of such that, for any given , we have for all with . This implies that for all with . Therefore, is -differentiable at and as desired.
(d3) Let . Define . Then and there exist neighborhoods , , and of such that for all with and for all with . From Proposition 5(I) it follows that for all with . Let and . Then Thus, . The second product rule follows from this last equation by interchanging the set-valued functions and .

From [40] it follows that if a single valued function is -differentiable and , then we define the Cauchy integral by In this case, we say to be -integrable on interval . In particular, by we mean that is -integrable on provided this limit exists.

Similarly, we can introduce the integral of the set-valued functions. By we mean the set of all -integrable selectors of on .

Definition 7 (see [21]). A set-valued function is called -integrable on if has at least a -integrable selector on . In this case, we define the -integral of on , denoted by , as the set

Lemma 8 (see [21]). Assume that , and are -integrable and have -continuous selectors; then we have the following.(i).(ii).(iii) with .(iv). Specially, for .(v)If , then, for any , we have (vi)Let . If , then where stands for the -derivative of with respect to the first variable .(vii)If implies that , then is -integrable and (viii)If and imply that and , respectively, then is -integrable and

Proof. We only prove (vi). Let . By assumption there exists a neighborhood of such that for all with Since is continuous at , there exists a neighborhood of such that for Now define and let satisfy , and . Then Here, we also have used Lemma 8 (viii).

Definition 9. A set-valued function is called predifferentiable with (region of differentiation) , provided is countable and contains no right-scattered elements of , and is -differentiable at each .

Lemma 10. (i) Let be an interval. If , then defined by is -differentiable and one has (ii) If is -continuous and , then

Proof. We only prove (i). Let . By virtue of Proposition 6(d1) exists and provided is -differentiable. Thus, it is sufficient to check the -differentiability of and .
It is evident that is regulated provided is -continuous. As similar argument to Theorems 8.12 and 8.13 in [40], we can show that the set-valued function is predifferentiable with region of differentiation such that for all . If , then is a right-dense point of . On the other hand, for nay , let be a neighborhood of such that for all with . So we have This implies that exists and on . This proof is complete.

Finally, we recall the concept of the matrix-valued functions introduced by [40]. An -matrix-valued function is said to be -differentiable on provided each entry of is -differentiable on . In this case we put

An -matrix-valued function on is called regressive provided Here, stands for -identity matrix. The sets of regressive and -continuous matrix-valued functions will be denoted by . The set consists of all positively regressive and -continuous functions satisfying for . From now on, unless otherwise mentioned, the matrix-valued functions involved in equations are always assumed to belong to .

For , the “circle plus” and “circle minus” of matrix-valued functions are referred to as, respectively,

A matrix exponential function is defined as a unique matrix-valued solution of the following initial value problem: where is an -matrix-valued function and . Denote

Assume is a constant matrix. If , then , while if and is invertible, then , where stands for the integral number set and . In [40] the elements of have been proved to possess the following properties.

Lemma 11. If , then(i);(ii), where stands for the conjugate transpose of the matrix ;(iii);(iv) if and commute;(v) for ;(vi) for .

3. Solvability of ISDE

We emphasize that and means that, at each point , is continuous if is a right-dense point and has the limit if is a left-dense point. Moreover, let represent the right limit of at if is right dense and if is right scattered for . Let and exists for each and , and and is -differentiable in each interval . It is clear that is a complete metric space if it is endowed with the distance .

Consider the impulsive set dynamic equation (ISDE) where is a continuous linear operator; that is, for any , and , one has , and is a sequence of points such that and . By a solution of ISDE (51) we mean that a set-valued function satisfies (51).

To explore the existence of solutions to ISDE, we introduce the exponential dichotomy of set dynamic equations. We define the product of a matrix and a subset in as follows: for any -matrix-valued function on and subset .

By an analogue of the proofs of Theorems 5.24 and 5.27 in [40], respectively, together with the product rule of differential and Lemma 8, it is easy to prove the following results.

Lemma 12. Let , -conditions, , and . Then the initial problem with having a unique solution given by

Lemma 13. Let , , , and . Then the initial problem has a unique solution given by

In what follows, by means of we denote the unique solution of the linear homogeneous set equation with initial point fixed. We call the fundamental matrix of (57).

Definition 14. Equation (57) is said to admit an exponential dichotomy on if there exist positive constants and a continuous projection (matrix) (i.e., ) on such that where is the fundamental matrix of (57) and is the norm of the -matrix , say, for example, .

We need the following hypotheses.(H1) with any . (In general, for and ).(H2)There exists a constant such that for , where is given as in Definition 14.

Let , with for and , where is the zero element of . It is easy to see that is a complete metric space.

Theorem 15. Let be a -matrix-valued function, the linear homogenous set dynamic equation (57) with admits the exponential dichotomy on with positive constants , and the projection and the conditions (H1) and (H2) hold. If and , then the linear ISDE has a unique bounded solution on satisfying with with and .

Proof. To show that the operator is continuous and , we shall first estimate the of the addends in (61) for . Let be a positive constant such that for all . By Definition 14 and Lemma 8(vii) we have and analogously (noting that ) Assuming for some positive integer , in virtue of (H2), we have Substituting the above two inequalities into (63) and (64), respectively, we have From (61) and (62) and (66) it follows that the set-valued function is bounded on . If , from the exponential dichotomy it follows that is bounded, say, for . Thus, with . This implies that is bounded on , too. Consequently, is bounded on .
The continuity of for () and the existence of the limit values () are immediately verified by Lemma 8(vii); that is, .
To verify that , by differentiating (61) for () and taking into account Propositions 2.6(d1), (d2) and Lemmas 8(vii), (vi) and Lemma 11, we obtain In the case of , observing we have
This implies .
It remains to prove that satisfies (59). We have proved that the first equation of (59) is met for . Consider system (51) with and if any given; it is not hard to see that this system has a unique solution . According to the impulsive condition we obtain Taking into account this and combining (H1), we obtain This shows that the second equation of (59) is met. The third equation of (59) is straightforwardly met. Conclusively, is a desired solution.
Finally, under the assumption of the exponential dichotomy, the zero solution of the linear homogeneous set dynamic equation is the unique solution that is bounded in (see, e.g., [42, Lemma 4.13]). Let , both be the solutions of ISDE (59). From the first equation of (59) it follows that By Proposition 6(d1), this yields , which further shows that is a solution of . Hence, ; that is, for . Since ( is left continuous at , we have for . This guarantees the uniqueness of solutions and the proof is complete.

Corollary 16. Assume that the conditions of Theorem 15 are valid. If , then ISDE has a unique bounded solution on satisfying with and . Especially, ISDE (73) except for the initial condition has a unique solution

Proof. This is an immediate result by taking (), that is, , in Theorem 15.

Remark 17. If we consider the complete metric space consisting of the functions which belong to and satisfy , then, under the assumptions of Corollary 16 without the condition (H2), the result of Corollary 16 is still valid.

We are in a position to discuss the existence of solutions to nonlinear ISDE (51). We need the following well-known fixed point theorem which is the foundational tool to prove our main results.

Lemma 18. Let from into itself be a continuous and compact mapping. If the set is bounded; that is, there exists a positive constant such that for all , then the operator has a fixed point in .

Proof. Let with . Then, by [18, Proposition ], there exists a compact, convex set such that is an open subset of and . Suppose that has no fixed point on (otherwise we are finished) and for all and . Consider Now since . In addition, the continuity of implies that is closed. Note that , and therefore by Urysohn’s lemma there exists a continuous function with and . Let Now it is immediate that is a continuous, compact map since are continuous and is compact. By Tychonoff’s fixed point theorem, has a fixed point . Note that and hence . As a result, has a fixed point in .
Now consider . The above result guarantees that has a fixed point for . In view of the compactness of there exists a subsequence of such that ; that is, . From the continuity of it follows that . Obviously, .

Theorem 19. Suppose that (57) admits an exponential dichotomy on with positive constants , a projection , and the -matrix function which satisfy condition (H2), and satisfies the following.(i)For each , the set-valued function satisfies the hypothesis (H1).(ii)There exists a function satisfying such that (iii)There exists a function which is -integrable on such that Then nonlinear ISDE has a bounded solution on denoted by

Proof. For any fixed , consider ISDE(81) with instead of , from Theorem 15 it follows that the linear ISDE has a unique solution such that , where , is analogously given by (61) with replaced by , and Let with and . It is clear that is a map from into and any of its fixed point (i.e., there exists such that ) is a solution of nonlinear ISDE (81). To this end, we have to prove that and have a fixed point in , respectively. We first observe that assumption (ii) guarantees to be a contractive mapping, and therefore, by Banach fixed point theorem, has a unique fixed point . Let be a set-valued function whose value is for and for . Then is a unique fixed point of in .
We next prove has a fixed point in . As an analogue of the arguments of the proof of Theorem 15, we easily see that is bounded and continuous on . We shall verify that is equicontinuous. In fact, for any with and , in virtue of the properties of Hausdorff distance and Lemma 8, combining our hypotheses, we have where Let , and from assumption (iii), together with the continuity of and -integrable of , it follows that is equicontinuous with respect to . In virtue of Ascoli-Arzela theorem we obtain that is a continuous compact operator.
Now we prove that the set defined by (76) is bounded. If it is contrary, there exist such that and for . On the other hand, for any , from (61) and (66), Definition 14, and assumption (iii) we have in view of the arbitrariness of , we obtain Let us take to be large enough, say, larger than the right hand side of the above inequality; we have a contradiction. Consequently, the set is bounded. Lemma 18 guarantees that has a fixed point . Set a set-valued function such that its value is for and for . Then is a fixed point of in .
We finally prove that has a fixed point in . Let be fixed and define the mapping Then from the fact that has a unique fixed point satisfying for and for , it follows that . Similarly, we can check that meets all conditions of Lemma 18; therefore, has a fixed point satisfying for . This further implies that ; that is, has a fixed point in . This completes the proof.

Corollary 20. Assume that all conditions of Theorem 19 are satisfied, except for (H2). In addition, instead of we consider the complete metric space consisting of the functions and their -derivatives exist; then nonlinear ISDE (51) has at least bounded solution.

4. Some Stability Criteria

In this section, we assume that for and . Moreover, by we mean that and where the function satisfies and with the constants given as in Definition 14 and a graininess function. For the sake of convenience, we assume the projection .

Under the assumptions of , employing the procedure used in the proof of Theorem 19, we can obtain that the set dynamic equation has a solution for . Additionally define for . We now obtain a solution of ISDE (81) on which is left continuous on and defined by

On the other hand, with the additional assumption that , from (82) we obtain a bounded solution of ISDE (81) Since , subtracting from , we have Thus, we have the following attractive result.

Theorem 21. Suppose that (57) admits an exponential dichotomy on with positive constants , the projection , and the -matrix function which satisfy the condition (H2), and . In addition, if (i)for each , the set-valued function satisfies the hypothesis (H1),(ii)there exists such that for ,then .

Proof. Let for some nature number . Note that , and combining the exponential dichotomy and the hypothesis (ii), one has and therefore, in view of (H2), In view of Gronwall’s inequality ([40, Theorem 6.4]), we obtain where , . From this, it follows that which guarantees that as and the proof is complete.

We are now in a position to formulate the stability criteria for the null solution of ISDE (51). Let us first define the stability of trivial solution.

Definition 22. Let be any solution of ISDE (51). Then the trivial solution is said to be stable if, for each and , there exists a such that implies that .

Theorem 23. Under the assumption of Theorem 21, the trivial solution of ISDE (51) is stable.

Proof. From the above arguments we see that any solution of ISDE (51) can be indicated by We first consider the case of . As an analogy of the proof of Theorem 21, we have obtained where , . For any , choose and we have whenever .
If , by the exponential dichotomy we have and therefore Gronwall’s inequality again implies that which implies that Taking , we have if . Let . Then implies that . This proof is complete.

5. Examples

In this section we present several examples to illustrate the applicability of the results involved in the above sections.

Example 1. Consider the set dynamic equation where , , and , with , a bounded subset, is a -constant-valued matrix, and for .

Conclusion. Equation (106) has a unique bounded solution

Proof. From [40, Theorem 5.35] it is easy to see that the homogeneous equation of (106) admits an exponential dichotomy with the constants , the projection , and the fundamental matrix . Thus, the condition (H1) is naturally satisfied. Let . Now Remark 17 guarantees that the existence of the unique bounded solution to (106) and the expression of are immediately obtained by Corollary 16.

Example 2. Let us consider an interval-valued dynamic equation where , , , and .

Conclusion. If (ii) of Theorem 21 holds, for , and the following hypotheses hold:(I), and ;(II);(III),then the following results are valid.(i)Let be a solution of (90) with (, ). For any , there exists the natural number such that for , where is a solution of (108).(ii)The trivial solution of (108) is stable.

Proof. Clearly, the homogeneous equation admits an exponential dichotomy with constants , the projection , and the fundamental matrix ; that is, the hypothesis (H1) of Theorem 21 is valid. Our assumption guarantees that (H2) is also valid.
For , set and we similarly stipulate . Since , by means of [33] we have This implies that . In virtue of Theorem 21, for any , there exists a such that for any , where is given as in (92). By means of (92), there exists some natural number such that when is given. Thus, . Hence, where . (i) is valid.
(ii) is an immediate result of Theorem 23.

In order to apply our results to fuzzy problems to obtain the existence and stability of bounded solutions to fuzzy dynamic equations on a time scale, we need some terse memories for fuzzy theory for its analogy in referring to [43]. A function is called a fuzzy number on time scale if it satisfies the following properties:(i) is normal; that is, there exists such that ,(ii) is a convex fuzzy set on (i.e., ),(iii) is upper semicontinuous on ,(iv) is compact.

Let denote the space of fuzzy numbers. For , the set is called the -level set of . Obviously is a compact interval of if for all . The notation denotes explicitly the -level set of . we refer to and as the lower and upper branches of , respectively. For any , the metric structure is given by the Hausdorff distance Thus is a complete metric space.

Consider the existence and stability of bounded solutions for the impulsive fuzzy differential equation on where , , , satisfy the hypotheses of Example 2, respectively. The level set of fuzzy number is for all . Let . Then level set of is . Further, we have the following.

Theorem 24. Let . If , , , satisfy the hypotheses of Example 2, then fuzzy dynamic equation (114) has a bounded solution which satisfies the results of Example 2.

Proof. Note that is a metric space with respect to . In virtue of Theorems 21 and 23, it suffices to prove . In fact, for any , we have Let . Then satisfies This implies that and . Consequently, . This proof is complete.

Conflict of Interests

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

Acknowledgment

This research is supported by Natural Science Foundation of Zhejiang Province (LY12A01002).

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Copyright © 2014 Shihuang Hong 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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