Integral Characteristic Function of a Nonlinear Sturm-Liouville Problem

We propose a new approach to the study of Sturm-Liouville problems nonlinear in the spectral parameter. The approach is based on the introduction of a transcendental function related to the problem under study, called the integral characteristic function, which determines the eigenvalues of the Sturm-Liouville problem in question. The study of this function permits one to prove the solvability of the problem, find the eigenvalue asymptotics, obtain comparison theorems, and introduce a natural numbering of the eigenvalues and zeros of the eigenfunctions. We use this approach to study a nonlinear Sturm-Liouville problem on an interval with boundary conditions of the first kind.