Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation
We consider the following boundary value problem of nonlinear fractional differential equation: , , , , where is a real number, denotes the standard Caputo fractional derivative, and is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.
Fractional calculus has demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering, for example, fluid flow, rheology, electrical networks, chemical physics, control theory of dynamical systems, optics, and signal processing .
Since the discussion of many problems can be summed up in the study of boundary value problems (BVPs for short) for nonlinear fractional differential equations, recently, the existence or uniqueness of solutions or positive solutions of BVPs for nonlinear fractional differential equations has received much attention from many authors; see [2–13] and the references therein.
In particular, Zhang  studied the existence and multiplicity of positive solutions to the following BVP of nonlinear fractional differential equation: where is a real number and denotes the standard Caputo fractional derivative. The main tools used were the Guo-Krasnoselskii and Leggett-Williams fixed point theorems.
In 2009, by using the nonlinear alternative of Leray-Schauder type and Guo-Krasnoselskii fixed point theorem, Bai and Qiu  discussed the existence of a positive solution to the following singular BVP of nonlinear fractional differential equation: where is a real number, denotes the standard Caputo fractional derivative, and is singular at .
Motivated greatly by the above-mentioned works, in this paper, we consider the following BVP of nonlinear fractional differential equation:where is a real number, denotes the standard Caputo fractional derivative, and is continuous. First, Green’s function for the associated linear BVP is constructed. It is necessary to point out that is singular at . Next, some useful properties of are studied. Finally, existence results of at least one positive solution for BVP (3) are obtained.
Theorem 1. Let be a Banach space and let be a cone in . Assume that and are bounded open subsets of such that , and let be a completely continuous operator such that either(1) or(2) Then has a fixed point in .
For the convenience of the reader, we present here some necessary definitions and results from fractional calculus theory. These definitions and results can be found in the recent literature .
In this section, we always assume that and denotes the integer part of .
Definition 2. The Riemann-Liouville fractional integral of order is defined by
Definition 3. The Riemann-Liouville fractional derivative of order is defined by
Definition 4. The Caputo fractional derivative of order is defined via the above Riemann-Liouville fractional derivative by where
Lemma 5. Let be given by (7) and or . Then where .
3. Main Results
In the remainder of this paper, we always assume that is a real number and is continuous.
Lemma 6. Let be a given function. Then the BVPhas a unique solution: where
Now, for , we denote
Remark 7. From the definition of function , it is not difficult to get that for and .
Lemma 8. Green’s function defined by (11) has the following properties. Consider(1) and for ,(2)
Proof. Since (1) is obvious, we only need to prove (2).
First, for any , if , then and if , then Next, for any , if , then and if , then
Let be equipped with norm and Then it is easy to check that is a Banach space and is a cone in .
Now, we define an operator on by Since is singular at , we need to prove that operator is well defined. In fact, for any fixed , we have If we let then which together with Lemma 8 implies that Since is convergent, we obtain that is uniformly convergent on . This shows that operator is well defined. Furthermore, we know that if , then and if is a fixed point of in , then is a nonnegative solution of BVP (3).
Lemma 9. is completely continuous.
Proof. First, for any , in view of Lemma 8, we have So, which together with Lemma 8 implies that which shows that Therefore, .
Next, we prove that is continuous. Suppose that and . Then there exists such that for any . If we let then for any and , in view of Lemma 8, we have By applying Lebesgue dominated convergence theorem, we get which indicates that is continuous.
Finally, we show that is compact. Assume that is a bounded set. Then there exists a constant such that for any . In what follows, we will prove that is relatively compact. LetOn the one hand, for any , there exists such that , and so, it follows from (34) and Lemma 8 that which together with the fact that is convergent implies that is uniformly bounded. On the other hand, for any , since is convergent, we may choose such thatsince is uniformly continuous on , there exists such that for any with ,For any , there exists such that , and so, for any with , it follows from (36), (37), and Lemma 8 that which indicates that is equicontinuous. By Arzela-Ascoli theorem, we know that is relatively compact. Therefore, is completely continuous.
For convenience, we denote
Theorem 10. Assume that there exist two different positive constants and such that the following conditions are fulfilled:Then BVP (3) has a positive solution with .
Proof. Without loss of generality, we assume that . Let Then for any , we get , which together with (40) and Lemma 8 implies that This shows thatFor any , we get , which together with (41) and Lemma 8 implies that This indicates thatTherefore, it follows from Theorem 1, Lemma 9, (44), and (46) that the operator has a fixed point , which is a desired positive solution of BVP (3).
Example 11. We consider the BVPSince , a direct calculation shows that So,Let . If we choose and , then it is easy to verify that So, all the hypotheses of Theorem 10 are fulfilled. Therefore, it follows from Theorem 10 that BVP (47) has a positive solution satisfying
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
D. Q. Jiang and C. J. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 2, pp. 710–719, 2010.View at: Publisher Site | Google Scholar
S. Q. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, article 36, 12 pages, 2006.View at: Google Scholar
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.
M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964.