In this paper, we mainly apply a new, asymptotic method to investigate the growth of meromorphic solutions of linear higher order difference equations and differential equations. We delete the condition (1.6) of Theorems E and F, yet obtain the same results for Theorems E and F. We also weaken the condition (1.4) of Theorems C and D.
Zongxuan CHEN
,
Ranran ZHANG
,
Shuangting LAN
,
Chuangxin CHEN
. THE GROWTH OF DIFFERENCE EQUATIONS AND DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2021
, 41(6)
: 1911
-1920
.
DOI: 10.1007/s10473-021-0608-0
[1] Bank S, Laine I, On the oscillation theory of f'' + Af=0 where A is entire. Trans Amer Math Soc, 1982, 273:351-363
[2] Bergweiler W, Langley J K. Zeros of differences of meromorphic functions. Math Proc Camb Phil Soc, 2007, 142:133-147
[3] Chen C X, Chen Z X. Some results concerning meromorphic solutions for the Pielou Logistic equation. Bull Malays Math Sci Soc, 2020, 43:1775-1792
[4] Chen Z X. Complex Differences and Difference Equations, Mathematics Monograph Series 29. Beijing:Science Press, 2014
[5] Chen Z X. Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl, 2011, 373:235-241
[6] Chen Z X. On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations. Science China Math, 2011, 54(10):2123-2133
[7] Chen Z X. The growth of solutions of f'' + e-zf' + Q(z) f=0 where the order (Q)=1. Science in China (series A), 2002, 45(3):290-300
[8] Chen Z X, Shon K H. On existence of solutions of difference Riccati equation. Acta Mathematica Scientia, 2019, 39B(1):1-9
[9] Chiang Y M, Gao S A. On a problem in complex oscillation theory of periodic second order linear differential equations and some related perturbation results. Ann Acad Sci Fenn Math, 2002, 27:273-290
[10] Chiang Y M, Feng S J. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane. Ramanujan J, 2008, 16:105-129
[11] Cui N, Chen Z X. The conjecture on unity of meromorphic functions concerning their differences. J Difference Equa Appl, 2016, 22(10):1452-1471
[12] Frank G, Hellerstein S. On the meromorphic solutions of non-homogeneous linear differetial equations with polynomial coefficients. Proc London Math Soc, 1986, 53(3):407-428
[13] Gao S A, Chen Z X, Chen T W. Complex Oscillation Theory of Linear Differential Equations. Wuhan:Huazhong University of Science and Technology Press, 1998(Chinese)
[14] Gundersen G. Finite order solutions of second order linear differential equations. Trans Amer Math Soc, 1988, 305:415-429
[15] Halburd R G, Korhonen R. Meromorphic solution of difference equation, integrability and the discrete Painlevé equations. J Phys A:Math Theor, 2007, 40:1-38
[16] Hayman W K. Meromorphic Functions. Oxford:Clarendon Press, 1964
[17] He Y Z, Xiao X Z. Algebroid Functions and Ordinary Differential Equations. Beijing:Science Press, 1988(Chinese)
[18] Ishizaki K, Wen Z T. Binomial series and complex difference equations. J Math Anal Appl, 2021, 497(1):Article 124844
[19] Ishizaki K, Yanagihara N. Wiman-Valiron method for difference equations. Nagoya Math J, 2004, 175:75-102
[20] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin:W de Gruyter, 1993
[21] Lan S T, Chen Z X. On fixed points of meromorphic functions f(z) and f(z+c), Δc f(z). Acta Mathematica Scientia, 2019, 39B(5):1277-1289
[22] Yang L. Value Distribution Theory. Beijing:Science Press, 1993
[23] Zhang R R, Chen Z X. Fixed points of meromorphic functions and of their differences, divided differences and shifts. Acta Mathematica Sinica, English Series, 2016, 32(10):1189-1202