ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C.YANG

doi:0.1016/S0252-9602(15)30087-4

Abstract

Abstract:

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations f(z)ⁿ+P_n-1(f)=0, where n≥2 and P_n-1(f) is a difference polynomial of degree at most n-1 in f with small functions as coefficients.Moreover, we give two examples to show that one conjecture proposed by Yang and Laine [2] does not hold in general if the hyper-order of f(z) is no less than 1.

Key words: growth, meromorphic solutions, difference equations, conjectures

CLC Number:

30D35

Yueyang ZHANG, Zongsheng GAO, Jilong ZHANG. ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C.YANG[J].Acta mathematica scientia,Series B, 2016, 36(1): 195-202.

Trendmd

References

[1] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Function. Newtherlands:Kluwer Academic pulishers, 2003
[2] Yang C C, Laine I. On analogies between nonlinear difference and differential equations. Pro Japan Acad, 2010, 86(A):10-14
[3] Li N, Yang L. Some results related to complex linear and nolinear difference equations. J Diff Equ Appl, 2014, 20(2):237-250
[4] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin:W de Gruyter, 1993
[5] Halburd R G, Korhonen R J. Difference analogue of the lemma on the logarithmic derivative with applications to difference equation. J Math Anal Appl, 2006, 314(2):477-487
[6] Chiang Y-M, Feng S-J. On the Nevanlinna charactericstic of f(z+η) and difference equations in the complex plane. Ramanujan J, 2008, 16(1):105-129
[7] Halburd R G, Korhonen R J. Holomorphic curves with shift-invatiant hyperplane preimages. arXiv:0903. 3236v2
[8] Zhang J L, Gao Z S, Li S. Distribution of zeros and shared values of difference operator. Ann Polon Math, 2011, 102:213-221
[9] Wen Z T, Heittokangas J, Laine I. Exponential polynomials as solutions of certain nonlinear difference equations. Act Math Sinica, English Series, 2012, 66(2):1295-1306
[10] Chen Z X. On growth,zeros and poles of meromorphic solutions of linear and nonlinear difference equations. Arch Math, 2011, 10:2123-2133
[11] Mitsuru Ozawa. On the existence of prime periodic entire functions. Kodai Math Sem Rep, 1978, 29(3):308-321
[12] Heittokangas J, Korhonen R, Laine I, Rieppo J, Tohge K. Complex difference equations of Malmquist type. Comput Methods Funct Theory, 2001, 1(1):27-39
[13] Laine I, Yang C C. Clunie theorems for difference and q-difference polynomials. J Lond Math Soc, 2007, 76(3):556-566
[14] Halburd R G, Korhonen R J. Finite order solutions and the discrete Painlevé equations. Proc Lond Math Soc, 2007, 94:443-474
[15] Zhang J L, Korhonen R. On the Nevanlinna characteristic of f(qz) and its applications. J Math Anal Appl, 2010, 369(2):537-544
[16] Li S, Gao Z S. Resolts on a question of Zhang and Yang. Acta Math Sci, 2012, 32B(2):717-723
[17] Gao L Y. Estimates of n-functions and m-function of meromrophic sohctions of systems of complex difference equations. Acta Math Sci, 2012, 32B(4):1495-1502