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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 ${\cal P}f = F$ with an operator ${\cal P}: {\cal K}(X) \to {\cal K}(X)$, where ${\cal K}(X) = \{f: X \to X$ Lipschitz$: \Vert f - \text{id}\Vert _{\sup} < \infty\}$. 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 ${\cal P} - I$ and the application of the Banach-Caccioppoli fixed point principle.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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