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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 536317, 16 pages
http://dx.doi.org/10.1155/2010/536317
Research Article

Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

Department of Basic Science, Babol University of Technology, Babol 47148-71167, Iran

Received 2 November 2009; Revised 20 January 2010; Accepted 31 March 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Azizollah Babakhani. 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 investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

1. Introduction

Fractional differential equations have gained considerable importance due to their varied applications [15] in viscoelasticity, electroanalytical chemistry, and many physical problems [1]. So far there have been several fundamental works on the fractional derivative and fractional differential equations, written by Oldham and Spanier [3], Miller and Ross [2], Podlubny [1], and others. These works are an introduction to the theory of the fractional derivative and fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and the uniqueness, some analytic methods for solving fractional differential equations, namely, Green`s function method, the Mellin transform method, and the power series.

Control systems subject to delays have been extensively studied [6] and the delay differential equations are large and important class of dynamic systems. They often arise in either natural or technological control problems. Equations with discontinuity often appear in various control theory models [7, 8]. Time delay, always existing in real systems, usually results in oscillations around the discontinuity surface. Shustin [9] has studied various aspects of oscillations for the system of differential equations with the delay in dynamics of oscillations in a multidimension, where and are positive constants.

Roy et al. developed a control methodology for linear time-invariant plants that uses multiple delayed observations in feedback [10]. Fractional differential systems have proved to be useful in control processing for the last two decades [11, 12].

As explained in the above text regarding the ordinary differential equations in control theory and others, we can modify the applications of ordinary differential equations to ordinary fractional differential equations.

The existence and uniqueness of solution for the system of fractional differential equation have been studied in the papers [13, 14]. As a pursuit in this paper, we discuss the existence and uniqueness of positive solutions for system of nonlinear fractional differential equations in two-dimensional with the delay where and are the Riemann-Liouville fractional derivatives,: →  are given continuous functions so that ) =  (i.e., and are singular at and ,, where

The paper is organized as follows. In Section 2, we give definitions of the fractional derivative and fractional integral with some basic properties. Required topics of functional analysis were also introduced. Section 3 deals with existence of positive solution theorem and gives an illustrative example. The unique positive solution theorem with an example has been discussed in Section 4.

2. Basic Definitions and Preliminaries

We recall some standard definitions and results [14, 15, 16]. In the following and are real Banach spaces and is an operator (not necessarily linear) with domain in and range in .

Definition 2.1. A subset of is called a cone if the following conditions hold well:
(i) the set is closed,
(ii) if    implies that ,
(iii) if implies that which is the zero element of .

Note:
Every cone induces a semiordering in namely, if

Definition 2.2. For the order interval is defined as [17]

Definition 2.3. A subset is called order bounded if is contained in some order interval.

Definition 2.4. A cone is called normal if there exists a positive constant such that and implies that

We state the two fixed point results which will be needed in this paper. Our first result is a nonlinear alternative of Leray-Schauder type in a cone whereas our second is Krasnoselskii`s fixed point theorem.

Theorem 2.5 (Leray-Schauder Theorem). Let be a Banach space with closed and convex. Assume that is relatively open subset of with and is a continuous, compact map. Then either
has fixed point in or
there exist and with .

Theorem 2.6 (Krasnoselskii`s fixed point theorem [16]). Let be a Banach space and let be a cone in . Assume that and are open subsets of with and and let be continuous and completely continuous. In addition suppose that either
for and for or
for and for .
Then, has a fixed point in .

In this paper the Beta function is used also. is closely related to the Gamma function [1]. If then it has the integral representation It may be written in terms of the Gamma function as .

Definitions of Riemann-Liouville fractional derivative/integral and their properties are given bellow [14].

Definition 2.7. Let  :  and then the expression is called a left-sided fractional integral of order .

Definition 2.8. Let be a positive integer number and . The left-sided fractional derivative of a function is defined as We denote as and as . Further and are referred to as and respectively.

If the fractional derivative is integrable then where and [1]. Further, if then and then (2.5) implies that

3. Existence Theorem

In this section we discuss the system of nonlinear fractional differential equation (1.2) which has at least one positive solution.

Lemma 3.1 (see [18]). Let be a continuous function and . If there exits such that and is a continuous function on then is continuous .

In the following theorem we want to prove that (1.2) is equivalent to a system of integral equations.

Theorem 3.2. Suppose that  :  are given continuous functions with . If there exist such that and are continuous functions on , then (1.2) is equivalent to the system of integral equations

Proof. It is sufficient to prove that the first equation of system equation (1.2) is equivalent to the first equation of system equation (3.1). So, Using (2.6) we get . Hence (3.2) implies that .
Conversely, first we note that  =  exists by Lemma 3.1. =  as is continuous and . Therefore, the system of fractional integral equation (3.1) is a solution of (1.2).
Let  :  be a function defined by for each with where and belong to . We introduce the notation defined by We can decompose as which implies that . Hence, by Theorem 3.2, (1.2) is equivalent to the system of integral equations where . Set . For each , let be the seminorm in defined by is a Banach space with norm . Let be a cone of For each , we define the operator  :  by where

Lemma 3.3. Let and be nonnegative continuous functions, where . Then, the operator with the following conditions is maps-bounded set into bounded sets in .
   if there exist are continuous on .

Proof. By assumptions of Lemma 3.3 and by using Lemma 3.1, it is clear that . There exist positive constants such that and as and are continuous on . Hence, Similarly,
Let be bounded, that is, for each there exist positive constants such that . In view of (3.10), (3.11) we have Therefore, where Hence, is bounded.

Lemma 3.4. Suppose that are continuous functions and further two conditions and are satisfied. Then, the operator is continuous and completely continuous.

Proof. In the following we want to prove that is continuous. It is sufficient to prove that and are continuous. Let with and . If with and , then and . By the continuity of and on , we conclude that and are uniformly continuous on Therefore, are continuous and hence, is continuous.
Finally, we want to prove that the operator is equicontinuous. Let be bounded. Suppose that, and such that . For given , there exists , so that if , then : Similarly, for given , there exists , so that if , then

Case 1. If and , then Set Hence, and

Case 2. If , then are finite. Set Hence, and . Thus, for any given for all and for all with where , for the Euclidian distance on we have where . Therefore, is equicontinuous and the Ascoli-Arzela theorem implies that is compact and hence is completely continuous.

Theorem 3.5. If and on are nonnegative continuous functions, then the system of nonlinear fractional differential equation (1.2) with the conditions and has at least one positive solution satisfying , where is observed in the proof of the theorem.

Proof. By Lemma 3.3, the operator is continuous and completely continuous. We prove that there exits an open set , with for and . Let be any solution of . In view of Theorem 3.5, we have Hence, There exist positive constants such that and , as and are continuous on Hence, If we consider then any solution satisfies Set Theorem 2.5 guarantees that has fixed point . Theorem 3.2 gives that (1.2) has a positive solution satisfying .

Example 3.6. Consider the system of nonlinear fractional differential equation where Here, and with If we select and then and are continuous on
By Theorem 3.2, the system of nonlinear fractional differential equation (3.26) is equivalent to the system of integral equation Therefore, by Theorem 3.5, (3.26) has at least one positive solution satisfying where Note that, for each we have and

Theorem 3.7. Suppose that  :  are nonnegative continuous with . If there exist such that and and are continuous on then (1.2) has at least one positive solution with the following conditions:


where ,, and

Proof. For proving this theorem we provide the conditions required in Theorem 2.6. Set For each and we have and Condition implies that Hence . Similarly . Thus, for . By using condition (H4) and the above-mentioned proof, we have for . Therefore, by Theorem 2.6, (ii), and Theorem 3.2, the proof is completed.

Example 3.8. Consider the system of nonlinear fractional differential equation where are continuous functions from to so that Here  =  =  =  =  =  =   such that  =  =  =  Also and are continuous, where  =  =  and  = . In the following calculations, we review conditions and by using (3.34): Hence,  =  =  and the above calculations satisfy condition On the other hand, Hence and the above calculations satisfy condition Then, (3.33) has at least one positive solution.

4. Unique Existence of Solution

In this section we give conditions on and which render unique positive solution to (1.2).

Theorem 4.1. Let  :  be continuous and . If there exist such that and are continuous functions on then (1.2) has unique positive solution with the following conditions:
where and are positive constants.

Proof. As pointed out in the proceeding section, (1.2) is equivalent to the integral equation (3.1) and the solution of (3.1) is equivalent to the fixed point of operator . Thus for Similarly, Hence, for each , we have Hence, by application of the Banach fixed point theorem, has unique fixed point in , which is the unique positive solution of (1.2).

Example 4.2. Consider the system of nonlinear fractional differential equation where Here,  =  =  =  and  =  such that  =  = . We select  =  and  = . Hence and are continuous on : where . Equation (4.4) satisfies the conditions required in Theorem 4.1 and this theorem implies that the nonlinear equation (4.4) has unique solution in . Using Theorem 3.2, (4.4) is equivalent to the system of integral equations and the solution of (4.4) is where

5. Conclusions

In this paper the existence of positive solutions for system of nonlinear fractional differential equations in two dimensions with the delay comprising of standard Riemann-Liouville fractional derivatives has been discussed in Banach space. The unique solution under Lipschitz condition is also derived. For getting our results in this paper Leray-Schauder`s theorem, Krasnoselskii`s fixed point theorem, and Banach contraction principle had played important roles. Our research work in this paper can be generalized to system of nonlinear fractional differential equations in multiple dimensions with the finite delay or infinite delay. The present work provides insights in the equations encountered in fractional order dynamic systems and the control systems which further may be explored.

Acknowledgments

The authors thank the referees’ efforts for their remarks and Professor Dumitru Bleanu for his regular contact.

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