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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 $$\mathbb{R}$$. Let $$I=[-1,1]$$ and $$M>1$$. Denote by $$F_\Gamma (I,M)$$ the family of all continuous maps of the interval $$I$$ into $$\mathbb{R}$$ such that $$f(-1)= -1$$, $$f(1)=1$$, $$0\leq f(y)-f (x)\leq M(y-x)$$ for $$-1\leq x\leq y\leq 1$$ and $$f (\gamma x)=\gamma f(x)$$ for $$\gamma\in \Gamma$$ and $$x\in I\cap\gamma^{-1}I$$. Define $${\mathcal I}_\Gamma= \bigcup_{M>1}F_\Gamma(I,M)$$. Let $$\lambda_1,\dots, \lambda_n \in[0,\infty)$$ and $$\sum^n_{i=1} \lambda_i=1$$ and $$F\in{\mathcal I}_\Gamma$$. 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 $${\mathcal I}_\Gamma$$. Similar problem in $$\mathbb{R}^n$$ space, where $$\Gamma$$ is the orthogonal group $$O(n)$$ in $$\mathbb{R}^n$$ is considered.

##### MSC:
 39B12 Iteration theory, iterative and composite equations 39B82 Stability, separation, extension, and related topics for functional equations
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