Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

Yi-Chou Chen

Journal of Applied Mathematics, 2015, vol. 2015, issue 1

Abstract: Let F(x, y) = as(x)ys + as−1(x)ys−1 + ⋯+a0(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:Rx→Rx is a nonlinear operator with T(y(x)) = F(x, y(x)). In this paper, we will find an eigenfunction yx∈Rx to satisfy the following equation: F(x, y(x)) = ay(x) for some eigenvalue a∈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), as(x), as−1(x), …, a0(x), are all constants in R and (ii) y(x) is an eigenfunction in F(x, y(x)) = ay(x) if and only if yx∈R.

Date: 2015
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https://doi.org/10.1155/2015/516159

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