Research Article | Open Access

Jun-Rui Yue, Jian-Ping Sun, Shuqin Zhang, "Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation", *Discrete Dynamics in Nature and Society*, vol. 2015, Article ID 736108, 6 pages, 2015. https://doi.org/10.1155/2015/736108

# Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

**Academic Editor:**Chris Goodrich

#### Abstract

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.

#### 1. Introduction

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 [1].

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 [6] 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 [12] 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.

In order to obtain our main results, we need the following Guo-Krasnoselskii fixed point theorem [14, 15].

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 .*

#### 2. Preliminaries

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 [1].

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*

*Proof. *In view of the equation in (9) and Lemma 5, we know thatand so, which together with the boundary conditions in (9) implies that Therefore, BVP (9) has a unique solution:

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.

#### References

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - W. Feng, S. Sun, X. Li, and M. Xu, “Positive solutions to fractional boundary value problems with nonlinear boundary conditions,”
*Boundary Value Problems*, vol. 2014, article 225, 2014. View at: Publisher Site | Google Scholar - 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 - Z.-W. Lv, “Positive solutions of m-point boundary value problems for fractional differential equations,”
*Advances in Difference Equations*, vol. 2011, Article ID 571804, 2011. View at: Publisher Site | Google Scholar - W. G. Yang, “Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions,”
*Journal of Applied Mathematics and Computing*, vol. 44, no. 1-2, pp. 39–59, 2014. 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 - Y. D. Ji, Y. P. Guo, J. Q. Qiu, and L. Y. Yang, “Existence of positive solutions for a boundary value problem of nonlinear fractional differential equations,”
*Advances in Difference Equations*, vol. 2015, article 13, 2015. View at: Publisher Site | Google Scholar - C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,”
*Computers and Mathematics with Applications*, vol. 59, no. 3, pp. 1363–1375, 2010. View at: Publisher Site | Google Scholar - W. Z. Xie, J. Xiao, and Z. G. Luo, “Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions,”
*Applied Mathematics Letters*, vol. 41, pp. 46–51, 2015. View at: Publisher Site | Google Scholar - J. F. Xu, Z. L. Wei, and W. Dong, “Uniqueness of positive solutions for a class of fractional boundary value problems,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 590–593, 2012. View at: Publisher Site | Google Scholar - K. Zhao and P. Gong, “Existence of positive solutions for a class of higher-order Caputo fractional differential equation,”
*Qualitative Theory of Dynamical Systems*, vol. 14, no. 1, pp. 157–171, 2015. View at: Publisher Site | Google Scholar - Z. B. Bai and T. T. Qiu, “Existence of positive solution for singular fractional differential equation,”
*Applied Mathematics and Computation*, vol. 215, no. 7, pp. 2761–2767, 2009. View at: Publisher Site | Google Scholar - X. Q. Zhang, L. Wang, and Q. Sun, “Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter,”
*Applied Mathematics and Computation*, vol. 226, pp. 708–718, 2014. View at: Publisher Site | 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.

#### Copyright

Copyright © 2015 Jun-Rui Yue et al. 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.