## Abstract and Applied Analysis

### Differentiable Solutions of Equations Involving Iterated Functional Series

Wei Song

#### Abstract

The nonmonotonic differentiable solutions of equations involving iterated functional series are investigated. Conditions for the existence, uniqueness, and stability of such solutions are given. These extend earlier results due to Murugan and Subrahmanyam.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 636843, 13 pages.

Dates
First available in Project Euclid: 2 March 2010

https://projecteuclid.org/euclid.aaa/1267538480

Digital Object Identifier
doi:10.1155/2008/636843

Mathematical Reviews number (MathSciNet)
MR2485405

Zentralblatt MATH identifier
1167.39309

#### Citation

Song, Wei. Differentiable Solutions of Equations Involving Iterated Functional Series. Abstr. Appl. Anal. 2008 (2008), Article ID 636843, 13 pages. doi:10.1155/2008/636843. https://projecteuclid.org/euclid.aaa/1267538480

#### References

• M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, vol. 32 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.
• K. Baron and W. Jarczyk, Recent results on functional equations in a single variable, perspectives and open problems,'' Aequationes Mathematicae, vol. 61, no. 1-2, pp. 1--48, 2001.
• S. Nabeya, On the functional equation $f(p+qx+rf(x))=a+bx+cf(x)$,'' Aequationes Mathematicae, vol. 11, pp. 199--211, 1974.
• J. Z. Zhang, L. Yang, and W. N. Zhang, Iterative Equations and Embedding Flow, Scientific and Technological Education, Shanghai, China, 1998.
• N. Abel, Oeuvres Complètes, vol. 2, Christiana, Christiana, Ten, USA, 1881.
• J. M. Dubbey, The Mathematical Work of Charles Babbage, Cambridge University Press, Cambridge, UK, 1978.
• J. G. Dhombres, Itération linéaire d'ordre deux,'' Publicationes Mathematicae Debrecen, vol. 24, no. 3-4, pp. 277--287, 1977.
• A. Mukherjea and J. S. Ratti, On a functional equation involving iterates of a bijection on the unit interval,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 8, pp. 899--908, 1983.
• W. N. Zhang, Discussion on the iterated equation $\sum _i=1^n\lambda _if^i(x)=F(x)$,'' Chinese Science Bulletin, vol. 32, no. 21, pp. 1444--1451, 1987.
• W. N. Zhang, Stability of the solution of the iterated equation $\sum _i=1^n\lambda _if^i(x)=F(x)$,'' Acta Mathematica Scientia, vol. 8, no. 4, pp. 421--424, 1988.
• W. N. Zhang, Discussion on the differentiable solutions of the iterated equation $\sum _i=1^n\lambda _if^i(x)=F(x)$,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 15, no. 4, pp. 387--398, 1990.
• W. N. Zhang, An application of Hardy-Boedewadt's theorem to iterated functional equations,'' Acta Mathematica Scientia, vol. 15, no. 3, pp. 356--360, 1995.
• J. G. Si, The existence and uniqueness of solutions to a class of iterative systems,'' Pure and Applied Mathematics, vol. 6, no. 2, pp. 38--42, 1990.
• J. G. Si, The $C^2$-solutions to the iterated equation $\sum _i=1^n\lambda _if^i(x)=F(x)$,'' Acta Mathematica Sinica, vol. 36, no. 3, pp. 348--357, 1993.
• M. Kulczycki and J. Tabor, Iterative functional equations in the class of Lipschitz functions,'' Aequationes Mathematicae, vol. 64, no. 1-2, pp. 24--33, 2002.
• W. Zhang, K. Nikodem, and B. Xu, Convex solutions of polynomial-like iterative equations,'' Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 29--40, 2006.
• B. Xu and W. Zhang, Decreasing solutions and convex solutions of the polynomial-like iterative equation,'' Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 483--497, 2007.
• J. Zhang, L. Yang, and W. Zhang, Some advances on functional equations,'' Advances in Mathematics, vol. 24, no. 5, pp. 385--405, 1995.
• X. Wang and J. G. Si, Differentiable solutions of an iterative functional equation,'' Aequationes Mathematicae, vol. 61, no. 1-2, pp. 79--96, 2001.
• V. Murugan and P. V. Subrahmanyam, Existence of solutions for equations involving iterated functional series,'' Fixed Point Theory and Applications, vol. 2005, no. 2, pp. 219--232, 2005.
• V. Murugan and P. V. Subrahmanyam, Special solutions of a general iterative functional equation,'' Aequationes Mathematicae, vol. 72, no. 3, pp. 269--287, 2006.
• X. Li and S. Deng, Differentiability for the high dimensional polynomial-like iterative equation,'' Acta Mathematica Scientia, vol. 25, no. 1, pp. 130--136, 2005.
• X. P. Li, The $C^1$ solution of the high dimensional iterative equation with variable coefficients,'' College Mathematics, vol. 22, no. 3, pp. 67--71, 2006.
• J. Mai and X. Liu, Existence, uniqueness and stability of $C^m$ solutions of iterative functional equations,'' Science in China Series A, vol. 43, no. 9, pp. 897--913, 2000. \endthebibliography