Existence and multiplicity results for some three-point boundary value problems

Using the method of upper and lower solutions and some degree theory arguments, we establish the existence of at least three solutions of some second-order nonlinear differential equations of the type −x″=f(t,x,x′),t∈I=[0,1] subject to nonlinear three-point boundary conditions x(0)=0,x′(1)=g(x(η)),0<η≤1. The growth of ff with respect to x′x′ is allowed to be quadratic.