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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


Let F x ,   y = a s x y s + a s - 1 x y s - 1 + + 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 : R x R x is a nonlinear operator with T y x = F x ,   y x . In this paper, we will find an eigenfunction y x R x to satisfy the following equation: F x ,   y x = a y x for some eigenvalue a R and we call the problem F x ,   y x = a y x a fixed point like problem. If the number of all eigenfunctions in F x ,   y x = a y x is infinitely many, we prove that (i) any coefficients of F x ,   y ,   a s x ,   a s - 1 x , ,   a 0 x , are all constants in R and (ii) y x is an eigenfunction in F x ,   y x = a y x if and only if y x R .


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


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

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

Rights: Copyright © 2015 Hindawi


Vol.2015 • No. SI5 • 2015
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