Solvability for fully cantilever beam equations with superlinear nonlinearities

This paper deals with the existence of solution for the fully fourth-order boundary value problem {u((4))(x) = f(x, u(x), u'(x),u ''(1),u'''(x)), x is an element of [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] x R-4 -> R is continuous. Some inequality conditions on f guaranteeing the existence and uniqueness of solutions are presented. The inequality conditions allow f(x,y(0),y(1),y(2),y(3)) to grow superlinearly on y(0), y(1), y(2), and y(3).