Abstract and Applied Analysis

Fixed Points of Meromorphic Solutions for Some Difference Equations

Abstract

We investigate fixed points of meromorphic solutions $y\left(z\right)$ for the Pielou logistic equation and obtain some estimates of exponents of convergence of fixed points of $y\left(z\right)$ and its shifts $y\left(z+n\right)$, differences $\mathrm{\Delta }y\left(z\right)=y\left(z+1\right)-y\left(z\right)$, and divided differences $\mathrm{\Delta }y\left(z\right)/y\left(z\right)$.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 496096, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511905

Digital Object Identifier
doi:10.1155/2013/496096

Mathematical Reviews number (MathSciNet)
MR3064237

Zentralblatt MATH identifier
1272.30043

Citation

Chen, Zong-Xuan; Shon, Kwang Ho. Fixed Points of Meromorphic Solutions for Some Difference Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 496096, 7 pages. doi:10.1155/2013/496096. https://projecteuclid.org/euclid.aaa/1393511905

References

• W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.
• I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993.
• L. Yang, Value Distribution Theory, Science Press, Beijing, China, 1993.
• M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,” Nonlinearity, vol. 13, no. 3, pp. 889–905, 2000.
• 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.
• B. Q. Chen, Z. X. Chen, and S. Li, “Uniqueness theorems on entire functions and their difference operators or shifts,” Abstract and Applied Analysis, vol. 2012, Article ID 906893, 8 pages, 2012.
• Z. Chen, “On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations,” Science China. Mathematics, vol. 54, no. 10, pp. 2123–2133, 2011.
• Z.-X. Chen and K. H. Shon, “On zeros and fixed points of differences of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 373–383, 2008.
• Z.-X. Chen and K. H. Shon, “Properties of differences of meromorphic functions,” Czechoslovak Mathematical Journal, vol. 61, no. 1, pp. 213–224, 2011.
• Y.-M. Chiang and S.-J. 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.
• Y.-M. Chiang and S.-J. Feng, “On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions,” Transactions of the American Mathematical Society, vol. 361, no. 7, pp. 3767–3791, 2009.
• R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006.
• R. G. Halburd and R. J. Korhonen, “Existence of finite-order meromorphic solutions as a detector of integrability in difference equations,” Physica D, vol. 218, no. 2, pp. 191–203, 2006.
• R. G. Halburd and R. J. Korhonen, “Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations,” Journal of Physics A, vol. 40, no. 6, pp. R1–R38, 2007.
• J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, “Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 352–363, 2009.
• J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, “Complex difference equations of Malmquist type,” Computational Methods and Function Theory, vol. 1, no. 1, pp. 27–39, 2001.
• I. Laine and C.-C. Yang, “Clunie theorems for difference and $q$-difference polynomials,” Journal of the London Mathematical Society. Second Series, vol. 76, no. 3, pp. 556–566, 2007.
• S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
• W. Bergweiler, “Proof of a conjecture of Gross concerning fix-points,” Mathematische Zeitschrift, vol. 204, no. 3, pp. 381–390, 1990.
• C.-T. Chuang and C.-C. Yang, Fix-Points and Factorization of Meromorphic Functions, World Scientific, Singapore, 1990.
• A. A. Gol'dberg and I. V. Ostrovskiĭ, The Distribution of Values of Meromorphic Functions, Nauka, Moscow, Russia, 1970.