Positive solutions for boundary value problems of nonlinear fractional differential equations

Abstract

In this paper, we study the existence of positive solutions for the nonlinear fractional differential equation boundary value problem. D 0 + α u ( t ) = λ f ( u ( t ) ) , 0 < t < 1 , u ( 0 ) + u ′ ( 0 ) = 0 , u ( 1 ) + u ′ ( 1 ) = 0 , where 1 < α ⩽ 2 is a real number, D 0 + α is the Caputo fractional derivative, λ > 0 and f : [0, +∞) → (0, +∞) is continuous. By using the properties of the Green function and Guo–Krasnosel’skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established.