Solvability of multi-point boundary value problem at resonance––Part IV

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: (1.2)x(0)=∑i=1m−2αix(ξi),x(1)=∑j=1n−2βjx(ηj),(1.3)x(0)=∑i=1m−2αix(ξi),x′(1)=∑j=1n−2βjx′(ηj),(1.4)x′(0)=∑i=1m−2αix′(ξi),x(1)=∑j=1n−2βjx(ηj),where αi(1⩽i⩽m−2), βj(1⩽j⩽n−2)∈R, 0<ξ1<ξ2<⋯<ξm−2<1, 0<η1<η2<⋯<ηn−2<1. When all the αis have no the same sign and all the βjs have no the same sign, some existence results are given for (1.1) with boundary conditions , , at resonance case. We also give some examples to demonstrate our results.