Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

Abstract EN

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 .