Infinitely many eigenfunctions for polynomial problems: Exact results

Let f (x,y) = a s (x) y s + a s 1 (x) y s 1 +⋯+ a o 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. © 2015 yi-chou chen.