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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.

##### MSC:
 47H10 Fixed-point theorems
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##### References:
  Abel, N., Oeuvres complètes, vol. II, christiania, (1981), pp. 36-39  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  Baron, K., Recent results on functional equations in a single variable, perspectives and open problems, Aequationes math., 61, 1-48, (2001) · Zbl 0972.39011  Kulczycki, M.; Tabor, J., Iterative functional equations in the class of Lipschitz functions, Aequationes math., 64, 24-33, (2002) · Zbl 1009.39021  Dubbey, J., The mathematical work of charles babbage, (1978), Cambridge Univ. Press · Zbl 0376.01002  Kuczma, M., Functional equations in a single variable, Monografie matematyczne, vol. 46, (1968) · Zbl 0196.16403  Kuczma, M.; Choczewski, B.; Ger, R., Iterative functional equations, Encyclopedia of mathematics and its application, vol. 32, (1990), Cambridge Univ. Press Cambridge · Zbl 0703.39005  Zhang, W., Discussion on the iterated equation $$\sum_{i = 1}^n \lambda_i f^i(x) = F(x)$$, Chinese sci. bull., 32, 1444-1451, (1987) · Zbl 0639.39006  Zhang, W., Stability of the solution of the iterated equation $$\sum_{i = 1}^n \lambda_i f^i = F(x)$$, Acta math. sci., 8, 421-442, (1988)  Zhang, W., Discussion on the differentiable solutions of the iterated equation $$\sum_{i = 1}^n \lambda_i f^i(x) = F(x)$$, Nonlinear anal., 15, 387-398, (1990) · Zbl 0717.39005
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