Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

Abstract

Let $F\left(x, y\right)=a_s\left(x\right)y^s+a_{s-1}\left(x\right)y^{s-1}+\cdots +a_0\left(x\right)$ be a real-valued polynomial function in which the degree $s$ of $y$ in $F\left(x, y\right)$ is greater than or equal to 1. For any polynomial $y\left(x\right)$, we assume that $T:\mathbb{R}\left[x\right]\to \mathbb{R}\left[x\right]$ is a nonlinear operator with $T\left(y\left(x\right)\right)=F\left(x, y\left(x\right)\right)$. In this paper, we will find an eigenfunction $y\left(x\right)\in \mathbb{R}\left[x\right]$ to satisfy the following equation: $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ for some eigenvalue $a\in \mathbb{R}$ and we call the problem $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ a fixed point like problem. If the number of all eigenfunctions in $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ is infinitely many, we prove that (i) any coefficients of $F\left(x, y\right), a_s\left(x\right), a_{s-1}\left(x\right),\dots , a_0\left(x\right)$, are all constants in $\mathbb{R}$ and (ii) $y\left(x\right)$ is an eigenfunction in $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ if and only if $y\left(x\right)\in \mathbb{R}$.