The Nonlinear Perturbation Of Two Operator

In this paper, the two nonlinear eigenvalue problems are considered: one is the nonlinear eigenvalue problem of Sturm-Liouville under periodic boundary condition, and the other is about perturbed Fuchs operator.The Fuchs operator is a nonlinear eigenvalue problem which in a finite interval and both ends are singular types . We prove the result of this linear problem by using symmetric integral kernel. With this Through constructing a compact mapping ,we connect nonlinear eigenvalue problem with linear eigenvalue problem. By the Schauder fix point theorem, then we can proof that the perturbed operator still have countable eigenvalues, and the existence of its eigenfunction.