A two-parametric three-point problem

We study the existence of a solution x=x(t)x=x(t), t∈Rt∈R, of the three-point problem equation(P)x″+λx+g(t,x)=h,x(0)=0,Bηx=0. We give sufficient and necessary conditions on gg and hh in order for (P) to have a solution. Our conditions depend on the values of λλ and ηη.