Positive solutions for boundary value problem of nonlinear fractional differential equation

Abstract

In this paper, we investigate the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem: D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 , where 1 < α ⩽ 2 is a real number, D 0 + α is the standard Riemann–Liouville differentiation, and f : [ 0 , 1 ] × [ 0 , ∞ ) → [ 0 , ∞ ) is continuous. By means of some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. The proofs are based upon the reduction of problem considered to the equivalent Fredholm integral equation of second kind.