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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.

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