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

In this paper, we consider the following second order ordinary differential equation<no>(1.1)</no>x′′=f(t,x(t),x′(t))+e(t), t∈(0,1),subject to one of the following boundary value conditions: <no>(1.2)</no>x(0)=∑i=1m−2αix(ξi), x(1)=∑j=1n−2βjx(ηj),<no>(1.3)</no>x(0)=∑i=1m−2αix(ξi), x′(1)=∑j=1n−2βjx′(ηj),<no>(1.4)</no>x′(0)=∑i=1m−2αix′(ξi), x(1)=∑j=1n−2βjx(ηj),where <f>αi (1&les;i&les;m−2)</f>, <f>βj (1&les;j&les;n−2)∈R</f>, <f>0<ξ1<ξ2<⋯<ξm−2<1</f>, <f>0<η1<η2<⋯<ηn−2<1</f>. When all the <f>αi</f>s have no the same sign and all the <f>βj</f>s have no the same sign, some existence results are given for (1.1) with boundary conditions (1.2)–(1.4) at resonance case. We also give some examples to demonstrate our results. [Copyright &y& Elsevier]