Solvability of functional differential equations with multi-point boundary value problems at resonance

In this paper, we discuss the following third order functional differential equations x‴(t)=f(t,x(t),(Fx)(t),x′(t),(Gx′)(t),x″(t),(Hx″)(t)),t∈[0,1], subject to the boundary conditions x(0)=0,x″(0)=0,x′(1)=∑i=1m−2αix′(ηi), where f:[0,1]×R6⟶Rf:[0,1]×R6⟶R, F,G,HF,G,H are three operators, αi(i=1,…,m−2)≥0, 0<η1<η2<⋯<ηm−2<10<η1<η2<⋯<ηm−2<1. Under some appropriate conditions, some existence and multiplicity results are given for the problem at resonance by using a priori estimates and the topological degree theory of Mawhin.