On a Third Order Nonlinear Boundary Value Problem at Resonance

In this paper we discuss the existence of solutions to third order boundary value problems of the form x‴ + x′ + g(x, x′) = p(t),x′(0) = x′(π) = x(η) = 0, where η lies in [0, π]. We assume that g is continuous and one-sided bounded (e.g., g ≥ 0) and p is continuous. Solutions are shown to exist if p satisfies a Landesman-Lazer type condition involving g. Several examples are given to illustrate how these conditions depend upon the parameter η.