Abstract: The existence of solution and positive solution is considered for a fourthorder nonlinear elastic beam equation with derivatives of all orders. In material mechanics, the equation describes deformation of an elastic beam whose two ends are fixed. The first, second and third derivatives express corner, bending moment and shearing stress of the beam, respectively. By choosing suitable equivalent norm in the Banach space -C 3［0,1］ and applying the Leray-Schauder fixed point theorem, several existence results are obtained for the equation. The results show that the equation has at least one solution or positive solution provided the “maximal height” of nonlinear term is appropriate on a bounded set of its domain.

Key words: boundary value problem, solution and positive solution, nonlinear ordinary differential equation