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Discussion on the differentiable solutions of the iterated equation ${\sum }_{i=1}^{n}{\lambda }_{i}{f}^{i}\left(x\right)=F\left(x\right)$. (English) Zbl 0717.39005
The author considers the differentiable solutions of the functional equation (1) ${\lambda }_{1}f\left(x\right)+{\lambda }_{2}{f}^{2}\left(x\right)+···+{\lambda }_{n}{f}^{n}\left(x\right)=F\left(x\right),$ $x\in \left[a,b\right]=I,$ where f: $I\to I$, ${f}^{0}\left(x\right)=x$ and ${f}^{k}=f\circ {f}^{k-1}$, ${\lambda }_{i}\in ℝ$. 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 $A=\left\{\phi \in {C}^{1}\left[I,I\right],\phantom{\rule{1.em}{0ex}}\phi \left(a\right)=a,\phantom{\rule{1.em}{0ex}}\phi \left(b\right)=b,\phantom{\rule{1.em}{0ex}}0\le {\phi }^{\text{'}}\le M\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}|{\phi }^{\text{'}}\left({x}_{1}\right)-{\phi }^{\text{'}}\left({x}_{2}\right)|\le L|{x}_{1}-{x}_{2}|,\phantom{\rule{1.em}{0ex}}{x}_{1},{x}_{2}\in I\right\}·$ The proof of the main result relies on the Schauder fixed point theorem.
Reviewer: M.C.Zdun

##### MSC:
 39B12 Iterative and composite functional equations 26A18 Iteration of functions of one real variable