Abstract:

The existence of the solution to a class of nonlinear fourth-order elastic-beam equation is investigated. This class of equation,whose nonlinear terms contain a three-order derivative of the unknown function,mechanical-ly describes the deformation of an elastic beam,one end of which is fixed and the other is clamped by sliding clamps. In the investigation,the decomposition technology of boundary-value problems is adopted to transform the elastic equation into a fixed-point equation. Besides,four existence theorems of the solution to this class of equation are presented by constructing a suitable Banach space and by means of Leray-Schauder fixed-point theorem. The re-suits show that this class of equation possesses at least one solution or one positive solution if the“height’’of its nonlinear item is appropriate in a bounded set.

Key words: nonlinear ordinary differential equation, two-point boundary-value problem, solution, positive solu-tion, existence, fixed-point theorem