International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 495138, 17 pages
Research Article

Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with p-Laplacian Operator

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Received 27 July 2009; Accepted 9 March 2010

Academic Editor: Rodica Costin

Copyright © 2010 Jinhua Wang 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.


We consider the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation with p-Laplacian operator D0+γ(ϕp(D0+αu(t)))+f(t,u(t),D0+ρu(t))=0, 0<t<1, u(0)=u(1)=0, u′′(0)=0, D0+αu(t)|t=0=0, where 0<γ<1, 2<α<3, 0<ρ1, D0+α denotes the Caputo derivative, and f:[0,1]×[0,+)×R[0,+) is continuous function, ϕp(s)=|s|p-2s, p>1,  (ϕp)-1=ϕq,  1/p+1/q=1. By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.