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Journal of Applied Mathematics
Volume 2015 (2015), Article ID 516159, 6 pages
http://dx.doi.org/10.1155/2015/516159
Research Article

Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

Department of General Education, National Army Academy, Taoyuan 320, Taiwan

Received 1 September 2014; Accepted 28 October 2014

Academic Editor: Gue Myung Lee

Copyright © 2015 Yi-Chou Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a real-valued polynomial function in which the degree of in is greater than or equal to 1. For any polynomial , we assume that is a nonlinear operator with . In this paper, we will find an eigenfunction to satisfy the following equation: for some eigenvalue and we call the problem a fixed point like problem. If the number of all eigenfunctions in is infinitely many, we prove that (i) any coefficients of , are all constants in and (ii) is an eigenfunction in if and only if .