Positive Solutions for Two-Point Boundary Value Problems for Fourth-Order Differential Equations with Fully Nonlinear Terms

In this paper, we consider the existence of positive solutions for the fully fourth-order boundary value problem {u((4)) (t) = f(t, u(t),u' (t),u '' (t),u''' (t)), 0 <= t <= 1, u(0) = u(1) = u '' (0) = u '' (1) = 0 , where f: [0, 1] x [0, +infinity] x (-infinity, +infinity) x (-infinity, 0) x (-infinity, +infinity) -> [0, +infinity] is continuous. This equation can simulate the deformation of an elastic beam simply supported at both ends in a balanced state. By using the fixed-point index theory and the cone theory, we discuss the existence of positive solutions of the fully fourth-order boundary value problem. We transform the fourth-order differential equation into a second-order differential equation by order reduction method. And then, we examine the spectral radius of linear operators and the equivalent norm on continuous space. After that, we obtain the existence of positive solutions of such BVP.