Existence of positive solutions for the cantilever beam equations with fully nonlinear terms

In this paper we discuss the existence of positive solutions of the fully fourth-order boundary value problem {u(4)=f(t,u,u′,u″,u‴),t∈[0,1],u(0)=u′(0)=u″(1)=u‴(1)=0, which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where f:[0,1]×R+3×R−→R+ is continuous. Some inequality conditions on ff guaranteeing the existence of positive solutions are presented. Our conditions allow that f(t,x0,x1,x2,x3) is superlinear or sublinear growth on x0,x1,x2,x3. For the superlinear case, a Nagumo-type condition is presented to restrict the growth of ff on x2x2 and x3x3. Our discussion is based on the fixed point index theory in cones.