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Solutions of equivariance for a polynomial-like iterative equation. (English) Zbl 0983.39010
Let $\Gamma$ be a topologically finitely generated Lie group acting on $\bbfR$. Let $I=[-1,1]$ and $M>1$. Denote by $F_\Gamma (I,M)$ the family of all continuous maps of the interval $I$ into $\bbfR$ such that $f(-1)= -1$, $f(1)=1$, $0\le f(y)-f (x)\le M(y-x)$ for $-1\le x\le y\le 1$ and $f (\gamma x)=\gamma f(x)$ for $\gamma\in \Gamma$ and $x\in I\cap\gamma^{-1}I$. Define ${\cal I}_\Gamma= \bigcup_{M>1}F_\Gamma(I,M)$. Let $\lambda_1,\dots, \lambda_n \in[0,\infty)$ and $\sum^n_{i=1} \lambda_i=1$ and $F\in{\cal I}_\Gamma$. Using the fixed point theorems of Banach and Schauder, the author discusses the existence, uniqueness and stability of solutions $f$ to the iterative functional equation $\lambda_1f(x)+ \cdots+\lambda_n f^n(x)= F(x)$, $x\in I$ in the class of functions ${\cal I}_\Gamma$. Similar problem in $\bbfR^n$ space, where $\Gamma$ is the orthogonal group $O(n)$ in $\bbfR^n$ is considered.

39B12Iterative and composite functional equations
39B82Stability, separation, extension, and related topics
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