The Solutions For A Class Of Nonlinear Cantilever Beam Equations

In this paper, by using monotone iterative method of upper and lower solu-tions, 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 positive solutions to the two-point boundary value problem of fourth-order ordinary differential equation where f:[0,1]× R2 → R is a continuous function. The problem describes the static deformation of an tilted cantilever beam whose one end was fixed and other was freed.The main results of this paper are as follows:1. With the aid of the existence and uniqueness of solutions for corresponding fourth-order linear differential equation, we build a new maximum principle by the method of positive operator perturbation, and obtain the existence and uniqueness of solutions under some weak conditions by the method of monotone iterative of upper and lower solutions for the tilted cantilever beam equation.2. By the demonstration of spectral radius for the linear operator of the cor-responding fourth-order linear differential equation, we obtain the existence of so-lutions or positive solutions for the tilted cantilever beam equation by using the fixed point theorem of completely continuous operators under the linear growth conditions.3. Under the conditions concerning the first eigenvalue of the corresponding linear differential equation, the results of the existence of positive solutions for the tilted cantilever beam equation are obtained in the case of superlineax and sublinear by constructing a suitable cone and applying the fixed-point index theory in cone.