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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 828721, 7 pages

http://dx.doi.org/10.1155/2014/828721
Research Article

Existence and Uniqueness of Positive Solutions for a Fractional Switched System

1Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, China

2Department of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455000, China

Received 25 January 2014; Accepted 11 March 2014; Published 13 April 2014

Academic Editor: Xinan Hao

Copyright © 2014 Zhi-Wei Lv and Bao-Feng Chen. 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 discuss the existence and uniqueness of positive solutions for the following fractional switched system: ( ); , where is the Caputo fractional derivative with , is a piecewise constant function depending on , and , ], . Our results are based on a fixed point theorem of a sum operator and contraction mapping principle. Furthermore, two examples are also given to illustrate the results.

1. Introduction

Fractional differential equations arise in various areas of science and engineering. Due to their applications, fractional differential equations have gained considerable attention (cf., e.g., [115] and references therein). Moreover, the theory of boundary value problems with integral boundary conditions has various applications in applied fields. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and population dynamics can be reduced to the nonlocal problems with integral boundary conditions. In [2], Cabada and Wang considered the following m-point boundary value problem for fractional differential equation where , is the Caputo fractional derivative, and is a continuous function.

On the other hand, a switched system consists of a family of subsystems described by differential or difference equations, which has many applications in traffic control, switching power converters, network control, multiagent consensus, and so forth (see [1618]). When we consider a switched system, we always suppose that the solution is unique. So it is important to study the uniqueness of solution for a switched system. In [1], Li and Liu investigated the uniqueness of positive solution for the following switched system: where is a piecewise constant function depending on , and , , .

In this paper, we discuss the existence and uniqueness of positive solutions for the following fractional switched system: where is the Caputo fractional derivative with , is a piecewise constant function depending on , and , , .

The paper is organized as follows. In Section 2, we present some background materials and preliminaries. Section 3 deals with some existence results. In Section 4, two examples are given to illustrate the results.

2. Background Materials and Preliminaries

Definition 1 (see [3, 4]). The fractional integral of order with the lower limit for a function is defined as where is the gamma function.

Definition 2 (see [3, 4]). For a function , the Caputo derivative of fractional order is defined as

In the following, let us recall some basic information on cone (see more from [19, 20]). Let be a real Banach space and let be a cone in which defined a partial ordering in by if and only if . is said to be normal if there exists a positive constant such that implies . is called solid if its interior is nonempty. If and , we write . We say that an operator is increasing if implies .

For all , the notation means that there exist and such that . Clearly, is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .

Definition 3. Let or and let be a real number with . An operator is said to be concave if it satisfies

Definition 4. An operator is said to be homogeneous if it satisfies An operator is said to be subhomogeneous if it satisfies

From [2], we have the following result.

Lemma 5. Assume that and , . Then the problem (3) has a solution if and only if is a solution of the integral equation where

Lemma 6. in Lemma 5 has the following property:(i) . (ii)

Proof. From [2], we know that is obvious. For , , we have This means that holds.

Theorem 7 (see [19]). Let be a normal cone in a real Banach space , an increasing concave operator, and an increasing subhomogeneous operator. Assume that (i)there is such that and ;(ii)there exists a constant such that , .Then the operator equation has a unique solution in . Moreover, constructing successively the sequence , , for any initial value , we have as .

3. Main Results

In this section, we will deal with the existence and uniqueness of positive solutions for problem (3). Let It is obvious that We consider the Banach space endowed with the norm defined by . Letting , then is a cone in . Define an operator as Then has a solution if and only if the operator has a fixed point.

Theorem 8. Let , . Suppose that the following conditions are satisfied: where Then the problem (3) has a unique solution on .

Proof. It follows from Lemma 6 that . For , , we set , , and , where

Step  1. We show that .

For and , , which implies that . Thus, . Therefore,

Step  2. We show that is a contraction mapping.

For and for each , , we have This, together with , , yields that where Thus, This means that is a contraction mapping.

It follows from Banach’s contraction mapping that has a unique fixed point in . Therefore, the problem (3) has a unique solution.

Corollary 9. Let , . Suppose that the following conditions are satisfied: where Then the following fractional switched system has a unique solution on .

Theorem 10. Assume that; and , are increasing in for , , ; for , and there exists a constant such that , ; there exists a constant such that , , .Then problem (3) has a unique solution in , where . Moreover, for any initial value , constructing successively the sequence we have as .

Proof. Define the two operators From Lemma 6, we have and . It is obvious that is the solution of problem (3) if and only if . It follows from that and are two increasing operators. Thus, for , we have and .

Step 1. We show that is a - operator and is a subhomogeneous operator.

In fact, for , from , we have which yields that Thus, is a operator. By a closely similar way, we can see that is a subhomogeneous operator.

Step 2. We show that and .

From Lemma 6 and , we have, for , , For , let It follows from that .

Thus, Letting and , then and . Therefore, which implies that Similarly, we have .

Step 3. There exists a constant such that , .

For and , , by , we have This means that Therefore, the conditions of Theorem 7 are satisfied. By means of Theorem 7, we obtain that the operator equation has a unique solution in . Moreover, for any initial value , constructing successively the sequence we have as .

In Theorem 10, if we let be a null operator, we have the following conclusion.

Corollary 11. Assume that; and is increasing in for , , ; there exists a constant such that , .

Then the following fractional switched system has a unique solution in , where . Moreover, for any initial value , constructing successively the sequence we have as .

4. Examples

Example 1. Consider the following boundary value problem: where , is a finite switching signal, Thus, By computation, we deduce that On the other hand, Hence, by Theorem 8, BVP (41) has a unique positive solution on .

Example 2. Consider the following boundary value problem: where , is a finite switching signal, Let and . It is obvious that and are increasing with respect to the second argument, . On the other hand, for , we have Moreover, for , we have where Hence all the conditions of Theorem 10 are satisfied. Thus, BVP (46) has a unique positive solution in , where , .

Conflict of Interests

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

Acknowledgments

This research was supported by Henan Province College Youth Backbone Teacher Funds (2011GGJS-213) and the National Natural Science Foundation of China (11271336).

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