Existence result for a singular nonlinear boundary value problem at resonance

We study the existence of solutions of the second-order boundary value problems u″(t)+π2u(t)+a(t)g(u(t))=h(t),a.e. t∈(0,1),u′∈ACloc(0,1),u(0)=u(1)=0,u(0)=u(1)=0, where g:R→Rg:R→R is continuous, a,h∈{z∈Lloc1(0,1)∣∫01t|z(t)|dt<∞}. The proof of the main result is based upon the Lyapunov–Schmidt procedure and the connectivity properties of the solution set of parametrized families of compact vector fields.