The spectral properties of some nonlinear operators of Sturm-Liouville type

For certain quasilinear second-order ordinary differential equations, we study the eigenvalue problem of Sturm-Liouville type on a closed interval with conditions of the first kind. To determine the discrete eigenvalues we use an additional (local) condition on one of the boundaries of the interval. The problem is (equivalently) reduced to a transcendental equation with respect to the spectral parameter. The analysis of this equation makes it possible to prove the existence of infinitely many (isolated) eigenvalues, indicate their asymptotics, find conditions under which the eigenfunctions are periodic, calculate the period, and give an explicit formula for the zeros of the eigenfunction. Several comparison theorems are obtained. We also study a problem to which perturbation theory cannot be applied.
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