Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance

Abstract

We consider the fractional differential equation D 0 + q u(t)=f ( t , u ( t ) ) ,0<t<1, satisfying the boundary conditions D 0 + p u(t) | t = 0 = D 0 + p − 1 u(t) | t = 0 =⋯= D 0 + p − n + 1 u(t) | t = 0 =0,u(1)= ∑ i = 1 m − 2 α i u( ξ i ), where D 0 + q is the Riemann-Liouville fractional order derivative. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is a Fredholm map of index zero. As a result, the minimal and maximal nonnegative solutions for the problem are obtained by using a fixed point theorem of increasing operators.MSC:26A33, 34A08.