Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz continuous operator and let fix$(f)$ denote the set of fixed points of $f$.
Define the operator $g = a f + (1-a) f^2$, for some $0 < a < 1$, where $f^2 = f \circ f$ denotes the composition.
Prove or disprove that, and find conditions under which, fix$(f) = $ fix$( g )$, for all $0 < a < 1$.
Please also indicate a book reference for nonlinear transformations and their fixed points.
Some comments/trials:
- It holds that if $x \in $ fix$(f)$, then $x \in$ fix$(g)$.
- The claim holds for linear operators, for $f = x^3$, for $f=x^2-1$, (other polynomials), $f = \sin$, $f = \sin^2$, (other polynomials of trigonometrics), $f = $exp -2, ...
- The claim is false for $f=-x^3$.