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Existence and growth of meromorphic solutions of some nonlinear q-difference equations
Advances in Difference Equations volume 2013, Article number: 33 (2013)
Abstract
In this paper, we investigate the existence of transcendental meromorphic solutions of order zero of some nonlinear q-difference equations and some more general equations. We also give some estimates on the growth of transcendental meromorphic solutions of these equations. Some examples are given to illustrate the sharpness of some of our results.
MSC:30D35, 39B32.
1 Introduction and main results
In this paper, a meromorphic function means being meromorphic in the whole complex plane. We also assume that the readers are familiar with the usual notations of Nevanlinna theory (see, e.g., [1–3]). Especially, we use notations and for the order and the lower order of a meromorphic function f. We also denote by any quantity satisfying for all r outside of a possible exceptional set of finite logarithmic measure. Moreover, the standard definitions of logarithmic measure and logarithmic density can be found in [4].
In last two decades, there has been a renewed interest in the complex analytic properties of complex differences and meromorphic solutions of complex difference equations owing to the introduction of Nevanlinna theory in this field. And the study of complex q-differences and q-difference equations is an important component of the study of complex differences and difference equations.
The original study of complex nonlinear q-difference equations can be derived from the study of the nonautonomous Schröder equation
by Valiron [5], which is closely related to the equations in complex dynamic systems. Indeed, Ritt [6] is an earlier classical paper on the autonomous Schröder equation
And in the important collection [7] of research problems, Rubel posed the question: What can be said about the more general equation (1.1)?
Recently, a number of papers (including [8–23]) focus on complex differences and complex q-differences. These papers also investigate the existence and the growth of meromorphic solutions of complex difference equations and complex q-difference equations.
In particular, Yang-Laine [21] pointed out some similarities between results on the existence and uniqueness of finite order entire solutions of the nonlinear differential equations and differential-difference equations. They obtained the following results.
Theorem A Let , be polynomials. Then a nonlinear difference equation
has no transcendental entire solutions of finite order.
Theorem B Let be an integer, be a linear differential-difference polynomial of f, not vanishing identically, and h be a meromorphic function of finite order. Then the differential-difference equation
possesses at most one admissible transcendental entire solution of finite order such that all coefficients of are small functions of f. If such a solution f exists, then f is of the same order as h.
Some generalizations of Theorems A and B can be found in Peng-Chen [19], and we omit those results here.
In this paper, we consider a similar problem on the existence and the growth of transcendental meromorphic solutions of complex q-difference equations (resp. differential-q-difference equations) instead of complex difference equation (1.2) (resp. differential-difference equation (1.3)). And the involved equations are more general than (1.1). Moreover, the fact that all meromorphic solutions of the Riccati q-difference equation and linear q-difference equation, both with rational coefficients, are of order zero, shows that it is of great importance to investigate meromorphic solutions of order zero of q-difference equations.
Theorem 1.1 Let (≢0), be rational functions, , and n, m be positive integers such that .
-
(i)
If , then the nonlinear q-difference equation
(1.4)
has no transcendental meromorphic solutions of order zero. Furthermore, when , if there exists a transcendental meromorphic solution f of positive order of (1.4), then
-
(ii)
If , then nonlinear q-difference equation (1.4) has no transcendental entire solutions of order zero.
Corollary 1.1 Let (≢0), be rational functions, , and be an integer. Then the nonlinear q-difference equation
has no transcendental meromorphic solutions of order zero. Furthermore, when , if there exists a transcendental meromorphic solution f of positive order of (1.5), then
Remark 1.1 Wittich [24] and Ishizaki [18] had earlier treated equation (1.5) of the case , which is the first order linear q-difference equation.
Remark 1.2 Equation (1.4) may have meromorphic solutions of order zero, when . For example, the function
is a transcendental meromorphic function of order zero (see Ramis [20]), and it satisfies the nonlinear q-difference equation
where .
Remark 1.3 Equations (1.4) and (1.5) may have meromorphic solutions of positive orders, when . For example, the function
satisfies the nonlinear q-difference equation
where , and . And the function
satisfies the nonlinear q-difference equation
where , and . The above two examples show that the estimates on the growth of meromorphic solutions of equations (1.4) and (1.5) are sharp.
In the following, we consider the existence of entire solutions of order zero of a type of differential-q-difference equation, which includes equations (1.4) and (1.5) as its special cases. We define a differential-q-difference polynomial in f, which is a finite sum of products of f, derivatives of f and of their q-shifts, with all meromorphic coefficients of these monomials of growth . Concretely, we denote a differential-q-difference polynomial in f by
where J is a finite set of indices, , are meromorphic functions of growth , and . And we denote the degree of by
In particular, if each monomial of is of the same degree, then we call a homogeneous differential-q-difference polynomial in f.
Theorem 1.2 Let n, m be integers such that , (≢0) be a homogeneous differential-q-difference polynomial in f of degree m, with all meromorphic coefficients of growth , and be a rational function. Then the differential-q-difference equation
has no transcendental entire solutions of order zero.
Corollary 1.2 Let be an integer, (≢0) be a linear differential-q-difference polynomial in f, with all meromorphic coefficients of growth , and be a rational function. Then differential-q-difference equation (1.6) has no transcendental entire solutions of order zero.
Corollary 1.3 Let , be rational functions, not all vanishing identically, , and let n (≥3), m be positive integers. Then the nonlinear q-difference equation
has no transcendental entire solutions of order zero. Furthermore, if , then nonlinear q-difference equation (1.7) has no transcendental meromorphic solutions of order zero, and any transcendental meromorphic solution f of positive order of (1.7) satisfies .
Remark 1.4 The results concerning the existence of meromorphic solutions of order zero in Theorem 1.1(i) and Corollary 1.1 are not only special cases of Theorem 1.2 and Corollary 1.2 respectively, but also more precise.
Remark 1.5 Clearly, equations (1.4)-(1.7) can have rational solutions.
2 Preliminary lemmas
Lemma 2.1 (See [22])
Let f be a non-constant zero-order meromorphic function and . Then
on a set of lower logarithmic density 1.
The following two lemmas can be seen as special cases of [[23], Theorem 3]. In fact, the present versions we give here are more precise than the original one. To facilitate the readers, we give the corresponding proofs here.
Lemma 2.2 Suppose that f is a transcendental meromorphic solution of the equation
where , and all meromorphic coefficients are of growth . If , then for sufficiently large r,
where K (>0) is a constant. Thus, the lower order of f satisfies .
Proof We will use the observation (see [[10], p.249]) as
Noting that , by (2.1), (2.2) and Valiron-Mohon’ko theorem (see [[2], Theorem 2.2.5 and Corollary 2.2.7]), we have that for any given ε (),
outside of a possible exceptional set of finite logarithmic measure. We apply [[2], Lemma 1.1.1] to deal with the exceptional set here. It follows by (2.3) that for any given , there exists an such that
holds for all . Hence,
Inductively, for any , we have by (2.4) that
For sufficiently large s, there exists a such that
We have by (2.5)-(2.6) that
Letting and , we have by (2.7) that
where (>0) is a constant. Thus, we get . □
Lemma 2.3 Suppose that f is a transcendental meromorphic solution of the equation
where , and all meromorphic coefficients are of growth . If , then for sufficiently large r,
where K (>0) is a constant. Thus, the lower order of f satisfies .
Proof Note that (2.3) holds for both (2.1) and (2.8), then the proof of Lemma 2.3 is similar to that of Lemma 2.2. □
Lemma 2.4 (See [9])
Let f be a non-constant zero-order meromorphic function and . Then
on a set of logarithmic density 1.
Lemma 2.5 (See [14])
Suppose that f is a transcendental meromorphic solution of an equation of the form (1.1) with and meromorphic coefficients of growth . Then we have that
where .
3 Proofs of Theorem 1.1 and Corollary 1.1
3.1 Proof of Theorem 1.1
-
(i)
Suppose that is a transcendental meromorphic solution of order zero of (1.4). It follows by (1.4) and Lemma 2.1 that
(3.1)
on a set of lower logarithmic density 1. It is clear that (3.1) is a contradiction since . Thus, (1.4) has no transcendental meromorphic solutions of order zero if .
Furthermore, when , we can transform (1.4) into
By (3.2) and Lemma 2.2, we have
-
(ii)
Suppose that is a transcendental entire solution of order zero of (1.4). By (3.2) and Lemmas 2.1, 2.4, we have
(3.3)
on a set of logarithmic density 1. It is clear that (3.3) is a contradiction since . Thus, (1.4) has no transcendental meromorphic solutions of order zero if .
3.2 Proof of Corollary 1.1
Since , we immediately know by Theorem 1.1(i) that (1.5) has no transcendental meromorphic solutions of order zero. And when , if there is a transcendental meromorphic solution f of positive order of (1.5), then we have
On the other hand, we have by (1.5) and Lemma 2.5 that . Thus, f has a regular order
4 Proofs of Theorem 1.2 and Corollaries 1.2, 1.3
4.1 Proof of Theorem 1.2
Suppose that is a transcendental entire solution of order zero of (1.6). It follows by (1.6) that
Noting that is a homogeneous differential-q-difference polynomial in f of degree m, by the logarithmic derivative lemma, Lemma 2.4 and (4.1), we have
on a set of logarithmic density 1. It is clear that (4.2) is a contradiction since . Thus, (1.6) has no transcendental entire solution of order zero if .
4.2 Proof of Corollary 1.2
Since is a linear differential-q-difference polynomial in f, the degree of is . Thus, by , we immediately know by Theorem 1.2 that (1.6) has no transcendental entire solutions of order zero.
4.3 Proof of Corollary 1.3
The first part result of Corollary 1.3 is a special case of Corollary 1.2. The second part result of Corollary 1.3 can be proved similarly to Theorem 1.1(i) by replacing Lemma 2.2 with Lemma 2.3.
References
Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
Yang L: Value Distribution Theory and New Research. Beijing Press, Beijing; 1982. (in Chinese)
Hayman WK: On the characteristic of functions meromorphic in the plane and of their integrals. Proc. Lond. Math. Soc. 1965, s3-14A(1):93-128. 10.1112/plms/s3-14A.1.93
Valiron G: Fonctions Analytiques. Press Univ. de France, Paris; 1954.
Ritt J: Transcendental transcendency of certain functions of Poincaré. Math. Ann. 1925/26, 95(1):671-682.
Rubel L: Some research problems about algebraic differential equations. Trans. Am. Math. Soc. 1983, 280(1):43-52. 10.1090/S0002-9947-1983-0712248-1
Ablowitz MJ, Halburd RG, Herbst B: On the extension of the Painlevé property to difference equations. Nonlinearity 2000, 13(3):889-905. 10.1088/0951-7715/13/3/321
Barnett D, Halburd R, Korhonen R, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb. A 2007, 137(3):457-474.
Bergweiler W, Ishizaki K, Yanagihara N: Meromorphic solutions of some functional equations. Methods Appl. Anal. 1998, 5(3):248-259. (Correction: Methods Appl. Anal. 6(4), 617-618 (1999))
Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142(1):133-147. 10.1017/S0305004106009777
Chen ZX, Shon KH: Value distribution of meromorphic solutions of certain difference Painlevé equations. J. Math. Anal. Appl. 2010, 364(2):556-566. 10.1016/j.jmaa.2009.10.021
Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 2009, 361(7):3767-3791. 10.1090/S0002-9947-09-04663-7
Gundersen G, Heittokangas J, Laine I, Rieppo J, Yang DG: Meromorphic solutions of generalized Schröder equations. Aequ. Math. 2002, 63(1-2):110-135. 10.1007/s00010-002-8010-z
Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314(2):477-487. 10.1016/j.jmaa.2005.04.010
Halburd RG, Korhonen RJ: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 2007, 94(2):443-474.
Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31(2):463-478.
Ishizaki K: Hypertranscendency of meromorphic solutions of a linear functional equation. Aequ. Math. 1998, 56(3):271-283. 10.1007/s000100050062
Peng CW, Chen ZX: Property of meromorphic solutions of certain nonlinear differential and difference equations. J. South China Norm. Univ. 2012, 44(1):24-28. (in Chinese)
Ramis JP: About the growth of entire functions solutions of linear algebraic q -difference equations. Ann. Fac. Sci. Toulouse 1992, 1(1):155-160.
Yang CC, Laine I: On analogies between nonlinear difference and differential equations. Proc. Jpn. Acad., Ser. A, Math. Sci. 2010, 86(1):10-14. 10.3792/pjaa.86.10
Zhang JL, Korhonen RJ:On the Nevanlinna characteristic of and its applications. J. Math. Anal. Appl. 2010, 369: 537-544. 10.1016/j.jmaa.2010.03.038
Zheng XM, Chen ZX: Some properties of meromorphic solutions of q -difference equations. J. Math. Anal. Appl. 2010, 361(2):472-480. 10.1016/j.jmaa.2009.07.009
Wittich H: Bemerkung zu einer Funktionalgleichungen von H. Poincaré. Arch. Math. 1950, 2: 90-95. (in German)
Acknowledgements
The authors thank the referees and the editors for their valuable comments to improve the readability of our paper. And this work was supported by the National Natural Science Foundation of China (11126145 and 11171119) and the Natural Science Foundation of Jiangxi Province in China (20114BAB211003 and 20122BAB211005).
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Zheng, XM., Tu, J. Existence and growth of meromorphic solutions of some nonlinear q-difference equations. Adv Differ Equ 2013, 33 (2013). https://doi.org/10.1186/1687-1847-2013-33
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DOI: https://doi.org/10.1186/1687-1847-2013-33
Keywords
- q-difference equation
- differential-q-difference equation
- meromorphic solution
- order zero