Discussion on the differentiable solutions of the iterated equation \(\sum_{i=1}^{n}\lambda_ if^ i(x)=F(x)\). (English) Zbl 0717.39005

Zhang, Weinian

Nonlinear Anal., Theory Methods Appl. 15, No. 4, 387-398 (1990).

The author considers the differentiable solutions of the functional equation (1) \(\lambda_ 1f(x)+\lambda_ 2f^ 2(x)+...+\lambda_ nf^ n(x)=F(x),\) \(x\in [a,b]=I,\) where f: \(I\to I\), \(f^ 0(x)=x\) and \(f^ k=f\circ f^{k-1}\), \(\lambda_ i\in {\mathbb{R}}\). Under suitable assumptions on \(\lambda_ i\), F and constants M and L he proves the existence, uniqueness and stability of the solutions of equation (1) in the following class of functions \(A=\{\phi \in C^ 1[I,I],\quad \phi (a)=a,\quad \phi (b)=b,\quad 0\leq \phi '\leq M\text{ and } | \phi '(x_ 1)-\phi '(x_ 2)| \leq L| x_ 1-x_ 2|,\quad x_ 1,x_ 2\in I\}.\) The proof of the main result relies on the Schauder fixed point theorem.

Reviewer: M.C.Zdun

MSC:

39B12	Iteration theory, iterative and composite equations
26A18	Iteration of real functions in one variable

Keywords:

contraction; iterated functional equation; differentiable solutions; stability; Schauder fixed point theorem

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