Necessary and sufficient conditions for the existence of solution to three-point BVP

We study the existence of a solution of the three-point boundary value problem equation(P)x″+λx+g(x)=h,x(0)=0,x(η)=x(π). We give sufficient and necessary conditions on gg and hh in order for (P) to have a real solution. Our conditions are of Landesman–Lazer type and depend on the real values of λλ and ηη.