Solvability of multi-point boundary value problem at resonance (II)

摘要

In this paper, we consider the following second order ordinary differential equation:(1.1)x″=f(t,x(t),x′(t))+e(t),t∈(0,1),subject to one of the following boundary value conditions:(1.10)x(0)=αx(ξ),x(1)=∑j=1n−2βjx(ηj),(1.11)x(0)=αx(ξ),x′(1)=∑j=1n−2βjx′(ηj),(1.12)x′(0)=αx′(ξ),x(1)=∑j=1n−2βjx(ηj),(1.13)x′(0)=αx′(ξ),x′(1)=∑j=1n−2βjx′(ηj),where α,βj(1⩽j⩽n−2)∈R, 0<η1<η2<⋯<ηn−2<1, 0<ξ<1. When all the βj’s have no same sign, some existence results are given for (1.1) with boundary conditions (1.10)–(1.13) at resonance case. We also give some examples to demonstrate our results.