%0 Journal Article
%T Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with -Laplacian Operator and Identities on the Some Special Polynomials
%A Erdo£¿an £¿en
%A Mehmet Acikgoz
%A Jong Jin Seo
%A Serkan Araci
%A Kamil Oru£¿o£¿lu
%J Journal of Function Spaces
%D 2013
%R 10.1155/2013/753171
%X We consider the following boundary-value problem of nonlinear fractional differential equation with -Laplacian operator , , , , , where , are real numbers, are the standard Caputo fractional derivatives, , , , , , are parameters, and are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters and are obtained. The uniqueness of positive solution on the parameters and is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative. 1. Introduction In 1695, L¡¯H£¿pital asked Leibniz: what if the order of the derivative is ? To which Leibniz considered in a useful means, thus it follows that will be equal to , an obvious paradox. In recent years, fractional calculus has been studied by many mathematicians from Leibniz¡¯s time to the present. Also, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [1¨C9]. In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis [10¨C21], upper and lower solutions method [22¨C24], fixed point index [25, 26], coincidence theory [27], Banach contraction mapping principle [28], and so forth. Chai [11] investigated the existence and multiplicity of positive solutions for a class of boundary-value problem of fractional differential equation with -Laplacian operator where ,£¿ , , is a positive constant number, and are the standard Riemann-Liouville derivatives. By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained. Although the fractional differential equation boundary-value problems have been studied by several authors, very little is known in the literature on the existence and nonexistence of positive solutions of fractional differential equation boundary-value problems with -Laplacian operator when a parameter is involved in the boundary conditions. We also mention that, there is very little known about the uniqueness
%U http://www.hindawi.com/journals/jfs/2013/753171/