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