%0 Journal Article
%T Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with -Laplacian Operator
%A Xiangshan Kong
%A Haitao Li
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/898206
%X This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with -Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective. 1. Introduction Fractional differential equation has recently attracted many scholarsĄ¯ interest due to its wide applications [1¨C3] in engineering, technology, biology, chemical process, and so on. The first issue for the theory of fractional differential equations is the existence of solutions to kinds of boundary value problems (BVPs), which has been studied recently by many scholars, and lots of excellent results have been obtained [4¨C17] by means of fixed point theorems, upper and lower solutions technique, and so forth. As an important branch of BVPs, -Laplacian equation was firstly introduced in [18] to model the following turbulent flow in a porous medium: where , , , and . Then, it was investigated in both integer-order BVPs [19, 20] and fractional BVPs [21¨C25]. In [21], Chen and Liu considered the antiperiodic boundary value problem of fractional differential equation with -Laplacian operator and obtained the existence of one solution by using SchaeferĄ¯s fixed point theorem under certain nonlinear growth conditions. Han et al. [22] investigated a class of fractional boundary value problem with -Laplacian operator and boundary parameter and presented several existence results for a positive solution in terms of the boundary parameter. It is noted that although there exist several results on the existence of one solution to fractional -Laplacian BVPs, there are, to our best knowledge, relatively few results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to fractional -Laplacian BVPs. In this paper, we study the following two-point boundary value problem of nonlinear fractional differential equations with -Laplacian operator: where , , , , , , , , , , and is the Caputo derivative. We first convert BVP (2) into an equivalent operator equation by using the property of the Caputo derivative and then present some sufficient conditions
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/898206/