Iterative equations in Banach spaces. (English) Zbl 1069.47064

Tabor, Jacek; Żołdak, Marek

J. Math. Anal. Appl. 299, No. 2, 651-662 (2004).

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.

Reviewer: Peter Zabreiko (Minsk)

MSC:

47H10

Fixed-point theorems

Keywords:

Banach contraction principle; equations with iterations and superposition operators; unique solvability

References:

[1]	Abel, N., Oeuvres complètes, vol. II, christiania, (1981), pp. 36-39
[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), Cambridge Univ. Press · Zbl 0376.01002
[6]	Kuczma, M., Functional equations in a single variable, Monografie matematyczne, vol. 46, (1968) · Zbl 0196.16403
[7]	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
[8]	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
[9]	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)
[10]	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