The poles and growth of solutions of systems of complex difference equations
© Wang et al.; licensee Springer. 2013
Received: 7 January 2013
Accepted: 5 March 2013
Published: 26 March 2013
In view of the Nevanlinna theory, we study the growth and poles of solutions of some classes of systems of complex difference equations and obtain some interesting results such as the lower bounds for Nevanlinna lower order, a counting function of poles and maximum modulus for solutions of such systems. They extend some results concerning functional equations to the systems of functional equations in the fields of complex equations.
1 Introduction and main results
The purpose of this paper is to study some properties of the poles and growth of meromorphic solutions of the systems of complex difference equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions are used (see [1–3]). In this paper, a meromorphic function means being meromorphic in the whole complex plane ℂ; for a meromorphic function f, denotes any quantity satisfying for all r outside a possible exceptional set E of finite logarithmic measure , and a meromorphic function is called a small function with respect to f if .
Theorem 1.1 (see [, Theorem 2.5])
has a non-trivial entire solution.
Theorem 1.2 (see [, Corollary 6])
has a non-trivial meromorphic solution in the complex plane.
where the coefficients are meromorphic functions, and obtained the following results.
Theorem 1.3 (see )
If difference equation (1) (or (2)) with polynomial coefficients , admits a transcendental meromorphic solution of finite order, then .
In 2001, Heittokangas et al.  further investigated some complex difference equations which are similar to (1) and (2) and obtained the following results which are improvements of Theorems 1.1 and 1.2.
Theorem 1.4 (see [, Proposition 8 and Proposition 9])
with rational coefficients , admit a transcendental meromorphic solution of finite order, then .
In the same paper, some results of the lower bound for the characteristic functions, poles and maximum modulus of transcendental meromorphic solutions of some complex difference equations are obtained as follows.
Theorem 1.5 (see [, Theorem 10])
- (1)If y is entire or has finitely many poles, then there exist constants and such that
- (2)If y has infinitely many poles, then there exist constants and such that
holds for all .
Theorem 1.6 (see [, Theorem 11])
for all , where and is a constant.
Recently, a number of papers have focused on difference equations, difference product and q-difference in the complex plane ℂ, and considerable attention has been paid to the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory [4, 6–25].
In 2012, Gao [14–16] also investigated the growth and existence of meromorphic solutions of some systems of complex difference equations and obtained some existence theorems and estimates on the proximity function and the counting function of solutions of some systems of complex difference equations.
Remark 1.1 Since system (4) is a particular case of system (5), from the conclusions of Theorem 1.8, we can get the following result.
Theorem 1.10 Suppose that all coefficients in (6) are of growth , and that all the other assumptions of Theorem 1.9 hold. Then ().
2 Some lemmas
Lemma 2.1 (Valiron-Mohon’ko )
where and .
Lemma 2.2 (see )
where is an index set consisting of s elements, and .
Lemma 2.3 (see )
Let , be monotone increasing functions such that outside of an exceptional set of finite logarithmic measure. Then for any , there exists such that for all .
Lemma 2.4 (see [, Lemma 1])
for all , for some constant κ.
3 The proof of Theorem 1.7
where the coefficients , () are polynomials.
Next, two cases will be considered as follows.
where (>0) () are some constants.
From Case 1 and Case 2, this completes the proof of Theorem 1.7.
4 The proof of Theorem 1.8
If is a pole or a zero of the coefficients of , then this process will be terminated and we can choose another pole of in the way we did above.
- (ii)If is neither a pole nor a zero of the coefficients of , thus the right-hand side of (18) has a pole of multiplicity at , then there exists at least one index such that is a pole of of multiplicity . Replacing z by in the second equation of (5), we have
where and .
Thus, from (21) and (22), this completes the proof of Theorem 1.8.
5 The proof of Theorem 1.9
Thus, this completes the proof of Theorem 1.9.
6 The Proof of Theorem 1.10
outside of a possible exceptional set of finite logarithmic measure.
By using the same argument as in [, Theorem 1.11], we can get that () easily.
Thus, this completes the proof of Theorem 1.10.
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20122BAB201016, No. 20122BAB201044) and the National Science and Technology Support Plan (2012BAH25F02).
- Hayman WK: Meromorphic Functions. The Clarendon Press, Oxford; 1964.MATHGoogle Scholar
- Yang L: Value Distribution Theory. Springer, Berlin; 1993.MATHGoogle Scholar
- Yi HX, Yang CC: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003. (Chinese original: Science Press, Beijing, 1995.)MATHGoogle Scholar
- Shimomura S: Entire solutions of a polynomial difference equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1981, 28: 253–266.MathSciNetMATHGoogle Scholar
- Yanagihara N: Meromorphic solutions of some difference equations. Funkc. Ekvacioj 1980, 23: 309–326.MathSciNetMATHGoogle Scholar
- Ablowitz MJ, Halburd RG, Herbst B: On the extension of the Painlevé property to difference equations. Nonlinearity 2000, 13: 889–905. 10.1088/0951-7715/13/3/321MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Finite-order meromorphic solutions and the discrete Painleve equations. Proc. Lond. Math. Soc. 2007, 94: 443–474.MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG R, Korhonen RJ: Meromorphic solutions of difference equations, integrability and the discrete Painlev’e equations. J. Phys. A, Math. Theor. 2007, 40: R1-R38. 10.1088/1751-8113/40/6/R01View ArticleMATHGoogle Scholar
- Heittokangas J, Korhonen RJ, Laine I, Rieppo J, Tohge K: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 2001, 1(1):27–39.MathSciNetView ArticleMATHGoogle Scholar
- Barnett DC, Halburd RG, Korhonen RJ, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb., Sect. A, Math. 2007, 137: 457–474.MathSciNetView ArticleMATHGoogle Scholar
- Bergweiler W, Ishizaki K, Yanagihara N: Growth of meromorphic solutions of some functional equations I. Aequ. Math. 2002, 63(1/2):140–151.MathSciNetView ArticleMATHGoogle Scholar
- Chen ZX: Growth and zeros of meromorphic solution of some linear difference equations. J. Math. Anal. Appl. 2011, 373: 235–241. 10.1016/j.jmaa.2010.06.049MathSciNetView ArticleMATHGoogle Scholar
- Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1MathSciNetView ArticleMATHGoogle Scholar
- Gao LY: On meromorphic solutions of a type of system of composite functional equations. Acta Sci. Math. 2012, 32B(2):800–806.MathSciNetMATHGoogle Scholar
- Gao LY: Systems of complex difference equations of Malmquist type. Acta Math. Sin. 2012, 55: 293–300.MathSciNetMATHGoogle Scholar
- Gao LY: Estimates of N -function and m -function of meromorphic solutions of systems of complex difference equations. Acta Sci. Math. 2012, 32B(4):1495–1502.MathSciNetMATHGoogle Scholar
- Gundersen GG, Heittokangas J, Laine I, Rieppo J, Yang D: Meromorphic solutions of generalized Schröder equations. Aequ. Math. 2002, 63: 110–135. 10.1007/s00010-002-8010-zMathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463–478.MathSciNetMATHGoogle Scholar
- Laine I, Rieppo J, Silvennoinen H: Remarks on complex difference equations. Comput. Methods Funct. Theory 2005, 5(1):77–88. 10.1007/BF03321087MathSciNetView ArticleMATHGoogle Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleMATHGoogle Scholar
- Korhonen R: A new Clunie type theorem for difference polynomials. J. Differ. Equ. Appl. 2011, 17: 387–400. 10.1080/10236190902962244MathSciNetView ArticleMATHGoogle Scholar
- Silvennoinen H: Meromorphic solutions of some composite functional equations. Ann. Acad. Sci. Fenn., Math. Diss. 2003, 13: 14–20.MathSciNetMATHGoogle Scholar
- Wang J: Growth and poles of meromorphic solutions of some difference equations. J. Math. Anal. Appl. 2011, 379: 367–377. 10.1016/j.jmaa.2011.01.022MathSciNetView ArticleMATHGoogle Scholar
- Zheng XM, Chen ZX: Some properties of meromorphic solutions of q -difference equations. J. Math. Anal. Appl. 2010, 361: 472–480. 10.1016/j.jmaa.2009.07.009MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.