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2015 Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results
Yi-Chou Chen
J. Appl. Math. 2015(SI5): 1-6 (2015). DOI: 10.1155/2015/516159

## Abstract

Let $F(x, y)={a}_{s}(x){y}^{s}+{a}_{s-1}(x){y}^{s-1}+\cdots +{a}_{0}(x)$ be a real-valued polynomial function in which the degree $s$ of $y$ in $F(x, y)$ is greater than or equal to 1. For any polynomial $y(x)$, we assume that $T:\mathbb{R}[x]\to \mathbb{R}[x]$ is a nonlinear operator with $T(y(x))=F(x, y(x))$. In this paper, we will find an eigenfunction $y(x)\in \mathbb{R}[x]$ to satisfy the following equation: $F(x, y(x))=ay(x)$ for some eigenvalue $a\in \mathbb{R}$ and we call the problem $F(x, y(x))=ay(x)$ a fixed point like problem. If the number of all eigenfunctions in $F(x, y(x))=ay(x)$ is infinitely many, we prove that (i) any coefficients of $F(x, y), {a}_{s}(x), {a}_{s-1}(x),\dots , {a}_{0}(x)$, are all constants in $\mathbb{R}$ and (ii) $y(x)$ is an eigenfunction in $F(x, y(x))=ay(x)$ if and only if $y(x)\in \mathbb{R}$.

## Citation

Yi-Chou Chen. "Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results." J. Appl. Math. 2015 (SI5) 1 - 6, 2015. https://doi.org/10.1155/2015/516159

## Information

Published: 2015
First available in Project Euclid: 15 April 2015

MathSciNet: MR3319187
Digital Object Identifier: 10.1155/2015/516159  