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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.

##### MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
 [1] Abel, N.: Oeuvres complètes, vol. II, christiania. (1981) [2] Baker, J. A.; Zhang, W.: Continuous solutions for a polynomial-like iterative equation with variable coefficients. Ann. polon. Math. 73, 29-36 (2000) · Zbl 0983.39011 [3] Baron, K.: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes math. 61, 1-48 (2001) · Zbl 0972.39011 [4] Kulczycki, M.; Tabor, J.: Iterative functional equations in the class of Lipschitz functions. Aequationes math. 64, 24-33 (2002) · Zbl 1009.39021 [5] Dubbey, J.: The mathematical work of charles babbage. (1978) · Zbl 0376.01002 [6] Kuczma, M.: Functional equations in a single variable. Monografie matematyczne 46 (1968) · Zbl 0196.16403 [7] Kuczma, M.; Choczewski, B.; Ger, R.: Iterative functional equations. Encyclopedia of mathematics and its application 32 (1990) · Zbl 0703.39005 [8] Zhang, W.: Discussion on the iterated equation $\sum$i=1n$\lambda ifi(x)=F(x)$. Chinese sci. Bull. 32, 1444-1451 (1987) · Zbl 0639.39006 [9] Zhang, W.: Stability of the solution of the iterated equation $\sum$i=1n$\lambda ifi=F(x)$. Acta math. Sci. 8, 421-442 (1988) [10] Zhang, W.: Discussion on the differentiable solutions of the iterated equation $\sum$i=1n$\lambda ifi(x)=F(x)$. Nonlinear anal. 15, 387-398 (1990) · Zbl 0717.39005