Existence and Nonexistence Results for a Fourth-Order Boundary Value Problem with Sign-Changing Green’s Function

In this paper, we consider a fourth-order three-point boundary value problem. Despite the fact that the corresponding Green’s function changes its sign on the square of its definition, we obtain the existence of at least one positive and decreasing solution under some suitable conditions. The results are based on the classical Krasosel’skii’s fixed point theorem in cones. Then, we impose some sufficient conditions that allow us to deduce nonexistence results. In the end, some examples are given in order to illustrate our main results.