## Abstract and Applied Analysis

### The Fixed Points of Solutions of Some $q$-Difference Equations

#### Abstract

The purpose of this paper is to investigate the fixed points of solutions $f(z)$ of some $q$-difference equations and obtain some results about the exponents of convergence of fixed points of $f(z)$ and $f({q}^{j}z)\text{\hspace\{0.17em\}\hspace\{0.17em\}}(j\in {\mathbb{N}}_{+})$, $q$-differences ${\mathrm{\Delta }}_{q}f(z)=f(qz)-f(z)$, and $q$-divided differences ${\mathrm{\Delta }}_{q}f(z)/f(z)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 262570, 7 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049069

Digital Object Identifier
doi:10.1155/2014/262570

Mathematical Reviews number (MathSciNet)
MR3272192

#### Citation

Zheng, Xiu-Min; Xu, Hong-Yan; Xu, Jun-Feng. The Fixed Points of Solutions of Some $q$ -Difference Equations. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 262570, 7 pages. doi:10.1155/2014/262570. https://projecteuclid.org/euclid.aaa/1425049069

#### References

• W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
• L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.
• C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic, Dordrecht, The Netherlands; Chinese Original: Science Press, Beijing, China, 2003.
• W. Bergweiler and J. K. Langley, “Zeros of differences of meromorphic functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 142, no. 1, pp. 133–147, 2007.
• Z. Chen, Z. Huang, and X. Zheng, “On properties of difference polynomials,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 627–633, 2011.
• Z. Chen and K. H. Shon, “On zeros and fixed points of differ-ences of meromorphic functions,” Journal of Mathematical Ana-lysis and Applications, vol. 344, no. 1, pp. 373–383, 2008.
• Z. X. Chen and K. H. Shon, “Properties of differences of mero-morphic functions,” Czechoslovak Mathematical Journal, vol. 61, no. 136, pp. 213–224, 2011.
• Y. Chiang and S. Feng, “On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008.
• I. Laine and C. Yang, “Value distribution of difference polynomials,” Proceedings of Japan Academy A, vol. 83, no. 8, pp. 148–151, 2007.
• A. Fletcher, J. K. Langley, and J. Meyer, “Slowly growing meromorphic functions and the zeros of differences,” Mathematical Proceedings of the Royal Irish Academy, vol. 109, no. 2, pp. 147–154, 2009.
• G. G. Gundersen, J. Heittokangas, I. Laine, and D. Yang, “Mero-morphic solutions of generalized Schröder equations,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 110–135, 2002.
• H. Wang, H.-Y. Xu, and B.-X. Liu, “The poles and growth of solutions of systems of complex difference equations,” Advances in Difference Equations, vol. 2013, article 75, 11 pages, 2013.
• H. Y. Xu, “On the value distribution and uniqueness of difference polynomials of meromorphic functions,” Advances in Difference Equations, vol. 2013, article 90, 2013.
• H. Xu, B. Liu, and K. Tang, “Some properties of meromorphic solutions of systems of complex $q$-shift difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 680956, 6 pages, 2013.
• H. Y. Xu, J. Tu, and X. M. Zheng, “Some properties of solutions of complex q-shift difference equations,” Annales Polonici Mathematici, vol. 108, no. 3, pp. 289–304, 2013.
• J. Xu and X. Zhang, “The zeros of q-shift difference polynomials of meromorphic functions,” Advances in Difference Equations, vol. 2012, article 200, 2012.
• G. Zhang, “Growth of meromorphic solutions of some q-difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 943209, 6 pages, 2013.
• S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
• A. A. Mohon'ko and V. D. Mohon'ko, “Estimates of the Nevan-linna characteristics of certain classes of meromorphic functions, and their applications to differential equations,” Sibirskii Matematicheskii Zhurnal, vol. 15, pp. 1305–1322, 1974 (Russian).
• D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, “Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations,” Proceedings of the Royal Society of Edinburgh A; Mathematics, vol. 137, no. 3, pp. 457–474, 2007.
• W. Bergweiler, K. Ishizaki, and N. Yanagihara, “Meromorphic solutions of some functional equationsčommentComment on ref. [3a?]: We split this reference to [3a,3b?]. Please check it.,” Methods and Applications of Analysis, vol. 5, no. 3, pp. 248–258, 1998.
• W. Bergweiler, K. Ishizaki, and N. Yanagihara, “Meromorphic solutions of some functional equations (correction),” Methods and Applications of Analysis, vol. 6, pp. 617–618, 1999.
• J. Zhang and R. Korhonen, “On the Nevanlinna characteristic of $f(qz)$ and its applications,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 537–544, 2010. \endinput