Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results - Yi-Chou Chen (Journal of Applied Mathematics)
Let Fx, y=asxys+as-1xys-1+⋯+a0x be a real-valued polynomial function in which the degree s of y in Fx, y is greater than or equal to 1. For any polynomial yx, we assume that T:Rx→Rx is a nonlinear operator with Tyx=Fx, yx. In this paper, we will find an eigenfunction yx∈Rx to satisfy the following equation: Fx, yx=ayx for some eigenvalue a∈R and we call the problem Fx, yx=ayx a fixed point like problem. If the number of all eigenfunctions in Fx, yx=ayx is infinitely many, we prove that (i) any coefficients of Fx, y, asx, as-1x,…, a0x, are all constants in R and (ii) yx is an eigenfunction in Fx, yx=ayx if and only if yx∈R.