Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique

Abstract

In this paper, we consider the existence of solution to the following fractional boundary value problem {ddx(p0Dx−β(u′(x))+qxD1−β(u′(x)))+f(x,u(x))=0,x∈(0,1),u(0)=u(1)=0, where the constants β∈(0,1), 0Dx−β and xD1−β denote left and right Riemann–Liouville fractional integrals of order β respectively, 0<p=1−q<1 and f:[0,1]×R→R is continuous. Due to the general assumption on the constants p and q, the problem does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem under suitable assumptions. The results extend the results in [F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011) 1181–1199].