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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 605614, 9 pages
Existence Results for Singular Boundary Value Problem of Nonlinear Fractional Differential Equation
Department of Mathematics, Shandong University of Science and Technology, Qingdao 266510, China
Received 27 December 2010; Revised 2 March 2011; Accepted 3 March 2011
Academic Editor: Elena Braverman
Copyright © 2011 Yujun Cui. 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.
By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: , , , where , is the Riemann-Liouville fractional derivative.
Many papers and books on fractional calculus differential equation have appeared recently. Most of them are devoted to the solvability of the linear initial fractional equation in terms of a special function [1–4]. Recently, there has been significant development in the existence of solutions and positive solutions to boundary value problems for fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, etc.), see [5, 6] and the references therein.
In this paper, we consider the following boundary value problems of the nonlinear fractional differential equation where is the standard Riemann-Liouville fractional derivative and is singular at . Our assumptions throughout are(H1) is continuous,(H2) is decreasing in , for each fixed ,(H3) and , uniformly on compact subsets of (0,1), and(H4) for all and as defined in (3.1).
The seminal paper by Gatica et al.  in 1989 has had a profound impact on the study of singular boundary value problems for ordinary differential equations (ODEs). They studied singularities of the type in (H1)–(H4) for second order Sturm-Louiville problems, and their key result hinged on an application of a particular fixed point theorem for operators which are decreasing with respect to a cone. Various authors have used these techniques to study singular problems of various types. For example, Henderson and Yin  as well as Eloe and Henderson [9, 10] have studied right focal, focal, conjugate, and multipoint singular boundary value problems for ODEs. However, as far as we know, no paper is concerned with boundary value problem for fractional differential equation by using this theorem. As a result, the goal of this paper is to fill the gap in this area.
Motivated by the above-mentioned papers and , the purpose of this paper is to establish the existence of solutions for the boundary value problem (1.1) by the use of a fixed point theorem used in [7, 11]. The paper has been organized as follows. In Section 2, we give basic definitions and provide some properties of the corresponding Green's function which are needed later. We also state the fixed point theorem from  for mappings that are decreasing with respect to a cone. In Section 3, we formulate two lemmas which establish a priori upper and lower bounds on solutions of (1.1). We then state and prove our main existence theorem.
For fractional differential equation and applications, we refer the reader to [1–3]. Concerning boundary value problems (1.1) with ordinary derivative (not fractional one), we refer the reader to [12, 13].
2. Some Preliminaries and a Fixed Point Theorem
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature.
Definition 2.1 (see ). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .
Definition 2.2 (see ). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined on .
Lemma 2.4 (see ). Assume that with a fractional derivative of order that belongs to . Then for some , .
Lemma 2.5. Given , and , the unique solution of is where
Lemma 2.6. The function defined by (2.6) satisfies the following conditions: (i), ,(ii) for , where .
Proof. Observing the expression of , it is clear that for . For given , is increasing with respect to . Consequently, for .
If , we have If , we have
Let be a Banach space, be a cone in . Every cone in defines a partial ordering in given by if and only if . If and , we write . A cone is said to be normal if there exists a constant such that implies . If is normal, then every order interval is bounded. For the concepts and properties about the cone theory we refer to [14, 15].
Next we state the fixed point theorem due to Gatica et al.  which is instrumental in proving our existence results.
Theorem 2.7 (Gatica-Oliker-Waltman fixed point theorem). Let be a Banach space, be a normal cone, and be such that if with , then . Let be a continuous, decreasing mapping which is compact on any closed order interval contained in , and suppose there exists an such that is defined (where and , are order comparable to . Then has a fixed point in provided that either: (i) and ;(ii) and ; or(iii)The complete sequence of iterates is defined and there exists such that with for all
3. Main Results
In this section, we apply Theorem 2.7 to a sequence of operators that are decreasing with respect to a cone. These obtained fixed points provide a sequence of iterates which converges to a solution of (1.1).
Let the Banach space with the maximum norm , and let . is a norm cone in . For , let where is given in Lemma 2.6. Define by and the integral operator by where is given in (2.6). It suffices to define as above, since the singularity in precludes us from defining on all of . Furthermore, it can easily be verified that is well defined. In fact, note that for there exists such that for all . Since decreases with respect to , we see for . Thus, Similarly, is decreasing with respect to .
Lemma 3.1. is a solution of (1.1) if and only if .
Proof. One direction of the lemma is obviously true. To see the other direction, let . Then , and satisfies (1.1). Moreover, by Lemma 2.6, we have
Thus, there exists some such that , which implies that . That is, .
We now present two lemmas that are required in order to apply Theorem 2.7. The first establishes a priori upper bound on solutions, while the second establishes a priori lower bound on solutions.
Lemma 3.2. If satisfies (H1)–(H4), then there exists an such that for any solution of (1.1).
Proof. For the sake of contradiction, suppose that the conclusion is false. Then there exists a sequence of solutions to (1.1) such that with . Note that for any solution of (1.1), by (3.5), we have Then, assumptions (H2) and (H4) yield, for and all , for some . In particular, , for all , which contradicts .
Lemma 3.3. If satisfies (H1)–(H4), then there exists an such that for any solution of (1.1).
Proof. For the sake of contradiction, suppose uniformly on as . Let . From (H3), we see that uniformly on compact subsets of . Hence, there exists some such that for and , we have . On the other hand, there exists an such that implies , for . So, for and , But this contradicts the assumption that uniformly on as . Hence, there exists an such that .
We now present the main result of the paper.
Theorem 3.4. If satisfies (H1)–(H4), then (1.1) has at least one positive solution.
Proof. For each , defined by
By conditions (H1)–(H4), for ,
Now define a sequence of functions , , by
Then, for each , is continuous and satisfies (H2). Furthermore, for ,
Note that has effectively “removed the singularity” in at , then we define a sequence of operators , , by
From standard arguments involving the Arzela-Ascoli Theorem, we know that each is in fact a compact mapping on . Furthermore, and . By Theorem 2.7, for each , there exists such that for . Hence, for each , satisfies the boundary conditions of the problem. In addition, for each ,
Arguing as in Lemma 3.2 and using (3.11), it is fairly straightforward to show that there exists an such that for all . Similarly, we can follow the argument of Lemma 3.3 and (3.5) to show that there exists an such that
Since is a compact mapping, there is a subsequence of which converges to some . We relabel the subsequence as the original sequence so that as .
To conclude the proof of this theorem, we need to show that To that end, fixed , and let be give. By the integrability condition (H4), there exists such that Further, by (3.11), there exists an such that, for , so that Thus, for and , and for , Thus, for , Since was arbitrary, we conclude that for all . Hence, and for
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The Project Supported by the National Science Foundation of China (10971179) and Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2010SF023).
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