Solvability for some fourth order two-point boundary value problems

Abstract

Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green's function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.