Solvability of sequential fractional order multi-point boundary value problems at resonance

Abstract

In this paper, we study the following sequential fractional differential equationa(t)CD0+αu(t)′=f(t,u(t),u′(t),CD0+αu(t)),t∈J,u(0)=0,CD0+αu(0)=0,u(1)=∑j=1m-1σju(ξj),where CD0+α is the Caputo fractional derivative, 1<α⩽2,σj∈R,ξj∈(0,1),j=1,2,…,m-1,m∈N,m>1,J=[0,1] and ∑j=1m-1σjξj=1. By the coincidence degree continuation theorem, we get some existence results about the boundary value problem at resonance.