The Solvability Of The Complete Form Of Simply Supported Beam Equation

In this paper,by using the method of upper and lower solutions,fixed point theorem of completely continuous operators and the fixed point index theory in cones,we deal with the existence and uniqueness of solutions and the existence of positive solutions to the fully fourth-order boundary value problem where f:[0,1]×R×R×R×R?R is a continuous function.The problem models deformations of an elastic beam in equilibrium state,whose two ends are simply supported.The main results of this paper are as follows:1.By the estimate of norm for the linear operator of the corresponding fourth-order linear differential equation,we get the results of the existence and uniqueness of a fourth order nonlinear differential equation under the two-parameter nonresonance conditions.2.By the existence and uniqueness of solutions for corresponding fourth or-der linear differential equation,we obtain the existence and uniqueness of a fourth order nonlinear differential equation without growth restriction by using the Leray-Schauder fixed point theorem of completely continuous operators.3.With the aid of Nagumo condition,we obtain the existence of solutions for the simply supported beam equations with fully nonlinear terms by applying the technic of truncating function and the method of upper and lower solutions.4.By constructing a suitable cone and Nagumo condition,and applying the fixed-point index theory in cone,we obtain the existence of positive solutions for the simply supported beam equations with fully nonlinear terms under the case of superlinear and sublinear conditions.