Existence of solutions for a class of fractional boundary value problems via critical point theory

Authors:
Feng Jiao;Yong Zhou
Affiliations:
School of Mathematics and Information Sciences, Guangzhou University, Guangdong 510006, PR China;School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

In this paper, by the critical point theory, a new approach is provided to study the existence of solutions to the following fractional boundary value problem: {ddt(12"0D"t^-^@b(u^'(t))+12"tD"T^-^@b(u^'(t)))+@?F(t,u(t))=0,a.e. t@?[0,T],u(0)=u(T)=0, where "0D"t^-^@b and "tD"T^-^@b are the left and right Riemann-Liouville fractional integrals of order 0@?@bR is a given function and @?F(t,x) is the gradient of F at x. Our interest in this problem arises from the fractional advection-dispersion equation (see Section 2). The variational structure is established and various criteria on the existence of solutions are obtained.