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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 816218, 11 pages
http://dx.doi.org/10.1155/2012/816218
Research Article

On Maximal and Minimal Solutions for Set-Valued Differential Equations with Feedback Control

Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

Received 11 September 2011; Accepted 8 November 2011

Academic Editor: Ibrahim Sadek

Copyright © 2012 Ngo Van Hoa and Nguyen Dinh Phu. 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

In this paper, we present the existence of extremal solutions of set-valued differential equations with feedback control on semilinear Hausdorff space under Hukuhara derivative which is developed under the form , ,  for all  with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem, that is useful for optimization problems.

1. Introduction

Recently, the study of set differential equations was initiated in a metric space and some basic results of interest were obtained. Some interesting results in this direction have been obtained in a series of works of Professor V. Lakshmikantham and other authors (see [15]). Professor V. Lakshmikantham and the other authors considered set differential equations (SDEs) and had some important results on existence, comparison, and stability criteria for SDEs: where , , and.

Based on these results, the authors gave the concept of set-valued control differential equation and studied existence and comparison of its solutions (see [6]). In this paper, we investigate an existence result of Peano’s type and then consider the existence of extremal solution of set-valued control differential equations. For this purpose, one needs to introduce a partial order in , prove the required comparison result for strict inequalities, and then, utilizing it, discuss the existence of extremal solutions.

This paper is organized as follows: in Section 2, we recall some basic concepts and notations which are useful in next sections. In Section 3, we present on the existence of extremal solutions for SSDEs on semilinear Hausdorff space with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem.

2. Preliminaries

We recall some notations and concepts presented in detail in recent series works of Professor V. Lakshmikantham et al. (see [1]). Let denote the collection of all nonempty compact convex subsets of . Given , the Hausdorff distance between and is defined by where denotes the Euclidean norm in and —the zero points set in . It is known that is a complete metric space and is a complete and separable with respect to metric .

We define the magnitude of a nonempty subset of : where is the zero element of which is regarded as a one point set. -norm in is finite when the supremum in (2.2) is attained with .

The Hausdorff metric (1.1) satisfies the properties below:

for all and . If , and , then It is known that is a complete metric space and if the space is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.

Let . The set satisfying is called the Hausdorff difference (the geometric difference) of the sets and and is denoted by the symbol . Given an interval in . We say that the set mapping has a Hukuhara derivative at a point , if exist in the topology of and are equal to .

By embedding as a complete cone in a corresponding Banach space and taking into account result on the differentiation of Bochner integral, we find that ifwhere is integrable in the sense of Bochner, then exists and the equality a.e on holds.

The Hukuhara integral of is given by for any compact set .

Some properties of the Hukuhara integral are in [1]. If is integrable, one has If are integrable, then is integrable and

3. Main Results

We consider the set-valued differential equations (SSDEs) with feedback control under the form where and is a feedback control, state set .

Definition 3.1. The mapping set is called to be a solution of (3.1) on if and only if the following conditions are satisfied:
(i) with Hukuhara derivative by ;(ii);(iii) is integrable on ;(iv)for all , the integral in (3.2) is Hukuhara integral.
In this section, we will use the monotone iterative technique to solve the minimal and maximal solutions of (3.1). To construct the set monotone sequence, we first introduce the following definition.

Definition 3.2. We denote(i)by the subfamily of consisting of sets such that any is a nonnegative (positive) vector of components satisfying for ,(ii)by the subfamily of consisting of sets such that any is a nonpostive (negative) vector of components satisfying for .By Definition 3.2, we notice that is a positive cone in and is the nonempty interior of . is a negative cone in and is the nonempty interior of . We can therefore induce a partial odering in . Thus, if is , that is, with any is satisfying for and is , that is, with any is satisfying for . Now we define the ordering in .

Definition 3.3. For any , if there exists a such that and , then we write . Similarly, if there exists a such that and , then we write .

Theorem 3.4. Assume the following:(H1) is monotone nondecreasing in for every , that is, for fixed , wherever , and is monotone nondecreasing in for every , that is, for fixed , wherever ; (H2) there exist such that and(H3) for any with and some positive number real such that then for provided .

Proof. For any , we define and we note that . By using (2.5), we infer . Let be the supremum of all positive number such that implies on . Thus and . Using (H1)–(H3), we get Equation (2.5), together with (3.5), implies that there exists an such that This contradicts that is the supremum in view of the continuity of function involved and consequently that the inequality holds for . Taking the limit yields the desired result. This proof is complete.

Corollary 3.5. Let such that for all , then implies that for all .

Proof. It is clear from the proof of Theorem 3.4.

Definition 3.6. are said to be the lower solution and upper solution of (3.1) respectively if

Theorem 3.7 (existence of solution). Assume are lower solution and upper solution of (3.1), respectively, and assumptions (H1), (H3) are satisfied, then there exists solution of (3.1).

Proof. For any , we define and . Let be the supremum of all positive number such that implies on . Thus and , by putting the above we infer that Similarly, and . By using Theorem 3.4, we have . Since are lower and upper solutions of (3.1), we have that where is solution of (3.1). Now, we wish to show that on . If it is not true, then there exists a such that and on . This implies that and . Equation (2.5), together with , implies that there exists an such that This contradicts that , hence we have that . Similarly, we can show that and hence relation holds for all . Now as , we conclude that . The proof is complete.

Definition 3.8. Let , are said to be minimal and maximal solutions of (3.1), respectively, if they both are solution of (3.1) and satisfy for every solution of (3.1) with for all , where are the lower and upper solutions of (3.1) respectively with for all .

Theorem 3.9. Assume that(M1) equation (3.1) has the lower solution and upper solution with and for all ;(M2) hypotheses (H1), (H3) satisfy;(M3) is map bounded sets into bounded sets in .Then there exists monotone sequence and in such that , as in , where , are the minimal and maximal solutions of (3.1), respectively.

Proof. Let us construct the set of integrodifferential sequences by for , we prescribe and , for all . From (2.5), (3.1) and using Definition 3.1 we get First, we claim that the iterations are such that Now we show that . Consequently, we have to show that (i) , (ii) and (iii) . By using Definition 3.6 and (3.11), (3.12), then (i) is proved. Indeed, by is a lower solution of (3.1) and following Definition 3.6 we get , addition Hence and using Corollary 3.5 we infer for all . Similarly, we use Definition 3.6 and (3.11), (3.12), then (ii) is proved. Using (M1), we get addition and Corollary 3.5, then (iii) is proved.
By using inductive method, we assume on , then we have to claim that , by means of the monotone property of we obtain From , and by virtue of Corollary 3.5 we get and for all . Again, by means of the monotone property of and our assumption, we have for all . Using again Corollary 3.5, we get . Consequently, Combining (3.11) and is continuous multiplication, it follows that , are continuous for .
Now using the corresponding of (3.11) and the properties of the Hausdorff metric and the Hukuhara integral, together with the assumption (M3), we prove the equicontinuity of the sequences and below. Consider for any , we have Hence and are uniformly bounded and equicontinuity on . On using Ascoli-Arzela theorem (see [1]) in this setup, we obtain a subsequence which converges uniformly to on . Arguing in a similarly to the , we conclude that converges uniformly to on . Next, we again consider (3.12), (3.18), respectively, and by using the convergence properties we infer that Moreover, by means of (3.18) we easily get on .
Finally, we show that and are the minimal and maximal solutions of (3.1), respectively. Let be any solution of (3.1) such that for all and and we need to prove that on . Suppose that for some , on . By using monotone nondecreasing of , , we get where . Applying Corollary 3.5, then we get on for all . Similarly, we get for all . By using assumption from the principle of mathematical induction, we infer that for all . Taking limit as , then we obtain . The proof is complete.

Corollary 3.10. If addition to the assumptions of Theorem 3.7 assume that satisfies for and , then is the unique solution of (3.1).

Example 3.11. We consider set-valued differential equation with feedback control in : where with is a contraction feedback.
We see that satisfies (M1)–(M3). Now, we show that (3.23) exits as extremal solutions on . We prescribe , are lower and upper solutions of (3.23) for all . We note that and . Next, let us construct the set sequences by for all we verify that monotone sequences of constructions above such that (a)and is a minimal of (3.23);(b)and is a maximal of (3.23).First, we prove (a). Indeed, let , then for each positive integer , we consider Because and , otherwise , hence . By using Corollary 3.5, to get for all and . On the other hand with fixed.
Since the family of functions is equicontinuous and uniformly bounded on , it follows Ascoli-Arzela theorem (see [4]) that there exists a decreasing sequence and uniform limit exits on . Obviously , the uniform continuity of implies that tends uniformly to as , and thus Which in turn yields that the limit is a solution of (3.23) on .
Next we will show that is a required maximal solution of (3.23) on . For this purpose, we observe that and is nondecreasing, hence we get
By using Corollary 3.5, then we get on . The uniqueness of maximal solution show that tends uniformly to is the maximal solution of (3.23) with Finally, we will prove (b). Similarly, let , then for each positive integer , we consider Because and , otherwise , hence . By using Corollary 3.5, to get for all and . On the other with fixed.
Since the family of functions is equicontinuous and uniformly bounded on , it follow by Ascoli-Arzela theorem (see [4]) that there exists a decreasing sequence and uniform limit exits on . Obviously , the uniform continuity of implies that tends uniformly to as , and thus which in turn yields that the limit is a solution of (1) on .
Next we will show that is a required minimal solution of (3.23) on . For this purpose, we observe that and is nondecreasing, hence we get

By using Corollary 3.5, then we get on . The uniqueness of minimal solution show that tends uniformly to is the minimal solution of (3.23) with

Based on (3.25) combining (3.28), (3.29) and (3.32), we will solve the minimal and maximal solutions of (3.23). Its graphical representation can be seen in Figure 1.

816218.fig.001
Figure 1: The minimal and maximal solutions of (3.23).

Acknowledgments

The authors gratefully acknowledge the referees for their careful reading and many valuable remarks which improved the presentation of the paper.

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