Multiple positive solutions for nonlinear fractional boundary value problems.

This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem DC0+αu(t)=f(t, u(t), u'(t)), 0<t<1, u(1)=u'(1)=u''(0)=0, where 2<α≤3 is a real number, DC0+α is the Caputo fractional derivative, and f:[0,1]×[0, +∞)×R→[0, +∞) is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.