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A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. (English) Zbl 1237.39022
Let $$X$$ be an arbitrary set, $$Y$$ be a complete ultrametric space, $$f_1,\dots ,f_k:\;X\to X$$, $$\Phi:\;X\times Y^k\to Y$$. The authors find conditions for the solvability of the functional equation $$\Phi (x,\psi (f_1(x)),\dots ,\psi (f_k(x)))=\psi (x)$$, $$x\in X$$ (with respect to $$\psi:\;X\to Y$$) and its generalizations. The result, in the spirit of the Hyers-Ulam stability where a solution is obtained as a limit of approximate solutions, is based on a new fixed point theorem.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 39B82 Stability, separation, extension, and related topics for functional equations 54H25 Fixed-point and coincidence theorems (topological aspects)
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