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Iterative equations in Banach spaces. (English) Zbl 1069.47064
This article deals with equations of type \[ \sum_i A_if^i(x) = F(x) \quad \text{and} \quad \sum_i A_i f(\phi_i(x)) = F(x), \] \(f^i(\cdot) \;(i = 1, 2, \dots\) are iterations of \(f(\cdot)\)). Both types are special cases of the equation \({\mathcal P}f = F\) with an operator \({\mathcal P}: {\mathcal K}(X) \to {\mathcal K}(X)\), where \({\mathcal K}(X) = \{f: X \to X\) Lipschitz\(: \| f - \text{id}\| _{\sup} < \infty\}\). The authors present some theorems about the unique solvability of such equations; their proofs are based on the accurate calculation of the Lipschitz constant for the operator \({\mathcal P} - I\) and the application of the Banach-Caccioppoli fixed point principle.

47H10 Fixed-point theorems
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