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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 391971, 14 pages
Existence of Positive Solutions for a Class of Delay Fractional Differential Equations with Generalization to N-Term
1Faculty of Basic Science, Babol University of Technology, Babol 47148-71167, Iran
2Department of Mathematics and Computer Science, Faculty of Arts and Sciences Cankaya University, 06530 Balgat, Ankara, Turkey
3Institute of Space Sciences, P.O. Box, MG-23, RO 76900 Magurele-Bucharest, Romania
Received 25 July 2010; Accepted 17 March 2011
Academic Editor: Yoshikazu Giga
Copyright © 2011 Azizollah Babakhani and Dumitru Baleanu. 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.
We established the existence of a positive solution of nonlinear fractional differential equations , with finite delay , , where , that is, is singular at and . The operator of involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.
Fractional differential equations have gained considerable importance due to their varied applications in viscoelasticity, electroanalytical chemistry, and many other physical problems [1–7]. So far there have been several fundamental works done on the fractional derivative and fractional differential equations [1–4, 6]. These works are in the introduction of 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 of some analytic methods for solving fractional differential equations, namely, the Green’s function method, the Mellin transform method, and the power series.
The existence of positive solutions for fractional differential equations with delay has been carried out by various researchers [8–17]. In  the authors have investigated the following type of fractional differential equations: where and is the standard Riemann-Liouville fractional derivative, , , and is a given continuous function, .
As a pursuit of this in this paper, we discuss the existence of positive solutions for initial nonlinear fractional differential equations with finite delay, with initial conditions , , and , where , , for all , , , is a given continuous function so that (i.e., is singular at ), where is the space of continuous functions from to and defined by for each . In the initial conditions of (1.1), we also assume that The paper is organized as follows.
In Section 2, we provide some definitions about the fractional derivatives and the fractional integrals as well as we list their basic properties. Required topics of functional analysis such as Banach fixed point theorem and Leray-Schauder Theorem were also introduced. Section 3 deals with existence of a positive solution theorem and it gives an explainable example. The unique positive solution theorem with an explainable example has been discussed in Section 4.
Let be a real Banach space with a cone . introduces a partial order ≤ in as if .
Definition 2.1. For , the order interval is defined as
Theorem 2.2 (Leray-Schauder Theorem ). Let be a Banach space with closed and convex. Assume is relatively open subset of with and is a continuous, compact map. Then, either (i) has fixed point in or (ii) there exists and with .
Theorem 2.3 (Banach fixed point theorem ). Let be a closed subspace of a Banach space . Let be a contraction mapping with Lipschitz constant from to itself. Then, has a unique fixed point in . Moreover, if is an arbitrary point in and is defined by , then and .
The complete Gamma function plays an important role in the theory of fractional integral and derivatives. A comprehensive definition of is provided by the Euler limit as
If , then has the following familiar integral representation: In this paper, the Beta function is used. We notice that is closely related to the Gamma function. If , then it has the integral representation
The definitions of Riemann-Liouville fractional derivative/integral and their properties are given bellow.
Definition 2.4. Let and ; then the expression is called a left-sided factional integrals of order .
Definition 2.5. Let be a positive integer number and . Then the left-sided fractional derivative of a function is defined as
We denote as and as . Further, and are referred as and , respectively.
If the fractional derivative is integrable; then where , and . Further, if , then and (1.4) reduces to
3. Existence Theorem
Lemma 3.1. Let be a continuous function and . If there exits such that and by letting be a continuous function on , then is continuous , where , .
Proof. Let us consider , . For all and for given all , Hence we conclude that . We notice that a similar result is conclusion for .
In the following, we want to show that (1.1) is equivalent to an integral equation.
Theorem 3.2. Suppose that is a given continuous function so that (i.e., is singular at ), where is the space of continuous functions from to and defined by for each . If there exists such that and is a continuous function on , then the fractional differential equation is equivalent to the integral equation where
Proof. From (3.2), we have
By using (2.7), we conclude that
and for ,
Note that, by Lemma 3.1, exists and , as is continuous and . In view of (3.5), (3.6), and (3.7), (3.2) is equivalent to the following integral equation:
The proof is completed. Therefore, by Theorem 3.2, the other expression of (1.1) is given as follows:
where and are mentioned in above.
Let be function defined by for each with ; we denote by the function define by We can decompose as , which implies that , . Hence by Theorem 3.2, (1.1) is equivalent to the following integral equation: Set , and for each , let be the seminorm in defined by is a Banach space with norm . Let be a cone of . and Define the operator by
Theorem 3.3. Suppose that , , , is a continuous function and . If there exits such that and is a continuous function on , then the operator , defined as (3.12), maps bounded set into bounded sets in , continuous and completely continuous.
Proof. For all , since by Lemma 3.1 and the nonnegativeness of , we obtain .
Since is continuous on , there exists a positive constant such that . Hence Let be bounded, that is, there exists a positive constant such that , for all . In view of (3.13), for each , we obtain Thus, is bounded. In the following text, we would like to show that is continuous. Let and , if and , then ; by the continuity of , we know that is uniformly continuous on . Thus, for all there exists a such that for all and with . Obviously, if , then and for each . Hence, we have for all and with . For all , let , and , we choose . By using (3.20), we get Finally, we want to prove that the operator is equicontinuous. Let be bounded, that is, three exits a positive constant such that , for all . Suppose, and . For a given , there exists , so that if , then Set , where
Case 1. If , we choose . Hence, Case 2. If , we choose . Hence, Therefore, is equicontinuous. The Arzela-Ascoli Theorem implies that is compact. Thus, is completely continuous.
Theorem 3.4. With all the details of (1.1)-(1.3), further, if exists in such that , , and is a continuous function on , then (1.1) has at least one positive solution , satisfying , where and is a positive constant which is observed in the proof of the theorem.
Proof. We know that the operator is continuous and it is completely continuous by Theorem 3.3. Here we show that there exists an open set , with for and . Let be any solution of , . According to the Theorem 3.3, since is continuous and it is completely continuous, we have Since there exists a positive constant such that , as is continuous on , therefore Set , where Equation (3.20) implies that Therefore, we conclude that . Then, as a result, any solution satisfies . Let . Theorem 2.2 guarantees that has a fixed point . By using Theorem 3.2, (1.1) under the conditions of (1.3) has a positive solution satisfying .
Example 3.5. Consider the following nonlinear fractional differential equation: where . Here, , , and such that . We select , . Hence is continuous on . We consider the initial conditions By using Theorem 3.2, the nonlinear fractional differential equation (3.20) is equivalent to an integral equation given below where in the view of the procedure proving Theorem 3.2, we have In addition of that we have that , where Therefore, by using the Theorem 3.4, (3.30) has at least one positive solution satisfying where and .
4. Unique Existence of Solution
In this section, we give conditions on and , which render a unique positive solution to (1.1).
Theorem 4.1. Let be continuous and . Suppose that there exists so that , , and is a continuous function on . If further, the following conditions are satisfied (H1) is Lipschitz with respect to the second variable with Lipschitz constant , that is, (H2) where and ; then (1.1) under the conditions of (1.3) has a unique positive solution.
Proof. As it was pointed out in the previous section, (1.1) is equivalent to (3.13). Thus, for we obtain where is given in (3.16). Hence, in view of the Theorem 2.3, will have a unique fixed point in , which is the unique positive solution of (1.1).
Example 4.2. Consider the nonlinear fractional differential equation given below where with initial conditions , , , , , , and . Here . We select ; then and is continuous on . Now, we review the Lipschitz condition with respect to the second variable. Note that from equations (3.11) and (3.12), for , we have and , , where . Without loss of generality, we suppose that , for all . Then, for each , Hence, the condition of (H1) holds with . On other hand, we have Thus, (4.4) satisfies the conditions required by the Theorem 4.1. This Theorem implies that the nonlinear equation (4.4) has a unique solution in . By using the Theorem 3.2, (4.4) is equivalent to the following integral equation: The solution of (4.4) is , where . Therefore, the iterated sequence is given by
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