Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

摘要

LetFx, y=asxys+as-1xys-1+⋯+a0xbe a real-valued polynomial function in which the degreesofyinFx, yis greater than or equal to 1. For any polynomialyx, we assume thatT:Rx→Rxis a nonlinear operator withTyx=Fx, yx. In this paper, we will find an eigenfunctionyx∈Rxto satisfy the following equation:Fx, yx=ayxfor some eigenvaluea∈Rand we call the problemFx, yx=ayxa fixed point like problem. If the number of all eigenfunctions inFx, yx=ayxis infinitely many, we prove that (i) any coefficients ofFx, y, asx, as-1x,…, a0x, are all constants inRand (ii)yxis an eigenfunction inFx, yx=ayxif and only ifyx∈R.