Complex oscillation of meromorphic solutions for difference Riccati equation
© Jiang et al.; licensee Springer. 2014
Received: 4 April 2014
Accepted: 5 September 2014
Published: 24 September 2014
The Erratum to this article has been published in Advances in Difference Equations 2015 2015:8
In this paper, we investigate zeros and α-points of meromorphicsolutions for difference Riccati equations, and we obtain someestimates of exponents of convergence of zeros and α-points of and shifts , differences , and divided differences .
MSC: 30D35, 39B12.
1 Introduction and main results
In this paper, we assume that the reader is familiar with the standard notations andbasic results of Nevanlinna’s value distribution theory (see [1, 2]). In addition, we use the notions to denote the order of growth of the meromorphic function, , and to denote the exponents of convergence of zeros and polesof , respectively. We say a meromorphic function is oscillatory if has infinitely many zeros.
The theory of difference equations, the methods used in their solutions, and their wideapplications have advanced beyond their adolescent stage to occupy a central position inapplicable analysis. The theory of oscillation play an important role in the research ondiscrete equations, and it is systematically introduced in . The complex oscillation is the development and deepening of thecorresponding real oscillation, and it can profoundly reveals the essence of theoscillation problem that the property of oscillation is investigated in complex domain.
Recently, as the difference analogs of Nevanlinna’s theory were being investigated [4–6], many results on the complex difference equations have been got rapidly. Manypapers [4, 7–9] mainly deal with the growth of meromorphic solutions of some differenceequations, and several papers [7, 8, 10–15] deal with analytic properties of meromorphic solutions of some nonlineardifference equations. Especially, there has been an increasing interest in studyingdifference Riccati equations in the complex plane [8, 10, 12, 15].
where A is a polynomial, . In , Chen and Shon investigated the existence and forms of rational solutions,and the Borel exceptional value, zeros, poles, and fixed points of transcendentalsolutions, and they proved the following theorem.
Theorem A Letbe a constant andbe an irreducible nonconstant rational function,whereandare polynomials withand.
if, thenhas at most one Borel exceptional value;
if, then the exponent of convergence of fixed points ofsatisfies.
In , the first author investigated fixed points of meromorphic functions for difference Riccati equation (1), and obtain someestimates of exponents of convergence of fixed points of and shifts , differences , and divided differences .
In this paper, we investigate zeros and α-points of meromorphic solutions for difference Riccati equations (1), and we obtain someestimates of the exponents of convergence of zeros and α-points of and shifts , differences , and divided differences of meromorphic solutions of (1). We prove the followingtheorem.
if, then, ;
- (ii)if there is a rational function satisfying
- (iii)if there is a rational function satisfying
where , is a periodic function with period 1. Note that for any, there exists a prime periodic entire function of order by Ozawa . Thus .
2 Lemmas for proofs of theorems
Firstly we need the following lemmas for the proof of Theorem 1.1.
have at most finitely many common zeros.
Since is a nonconstant rational function, has only finitely many zeros. Thus, and have at most finitely many common zeros. □
holds for all r outside of a possible exceptional set with finitelogarithmic measure.
3 Proof of Theorem 1.1
Suppose that . We only prove the case . We can use the same method to prove the case.
First, we prove that .
holds for all r outside of a possible exceptional set with finite logarithmicmeasure.
Thus (10) is a contradiction. Hence, (4) holds, that is, .
Thus, by this and (4), we see that .
4 Proof of Theorem 1.2
- (i)First, we prove that the conclusion holds when . Set . Thus, is transcendental, , and . Substituting into (2), we obtain
holds for all r outside of a possible exceptional set with finite logarithmicmeasure. Thus, we obtain .
Using the same method as in the proof of (4)-(11), we can prove that (13) holds. Hence.
- (ii)Suppose that there is a rational function satisfying(16)
where . Since is a constant, to prove , we need to prove that is nonconstant.
- (iii)Suppose that there is a rational function satisfying(23)
Using the same method as in the proof of (21), we can prove that the above equationholds.
Thus, Theorem 1.2 is proved.
The authors thank the referee for his/her valuable suggestions. This work issupported by PhD research startup foundation of Jiangxi Science and Technology NormalUniversity, and it is partly supported by Natural Science Foundation of GuangdongProvince, China (Nos. S2012040006865, S2013040014347) and the Natural ScienceFoundation of Jiangxi, China (No. 20132BAB201008).
- Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.MATHGoogle Scholar
- Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleMATHGoogle Scholar
- Agarwal RP, Bohner M, Grace SR, O’Regan D: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York; 2005.View 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
- Halburd RG, Korhonen R: Difference analogue of the lemma on the logarithmic derivative with applicationsto difference equations. J. Math. Anal. Appl. 2006, 314: 477-487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen R: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463-478.MathSciNetMATHGoogle Scholar
- Chen ZX: On growth, zeros and poles of meromorphic functions of linear and nonlineardifference equations. Sci. China Ser. A 2011, 54: 2123-2133.View ArticleMATHGoogle Scholar
- Ishizaki K: On difference Riccati equations and second order linear difference equations. Aequ. Math. 2011, 81: 185-198. 10.1007/s00010-010-0060-zMathSciNetView ArticleMATHGoogle Scholar
- Laine I, Yang CC: Clunie theorems for difference and q -difference polynomials. J. Lond. Math. Soc. 2007, 76: 556-566. 10.1112/jlms/jdm073MathSciNetView ArticleMATHGoogle Scholar
- Chen ZX, Shon KH: Some results on Riccati equations. Acta Math. Sin. 2011, 27: 1091-1100. 10.1007/s10114-011-9175-yMathSciNetView ArticleMATHGoogle Scholar
- Chen ZX, Huang Z, Zhang R: On difference equations relating to gamma function. Acta Math. Sin. 2011, 31: 1281-1294.MathSciNetMATHGoogle Scholar
- Halburd RG, Korhonen R: Existence of finite-order meromorphic solutions as a detector of integrability indifference equations. Physica D 2006, 218: 191-203. 10.1016/j.physd.2006.05.005MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen R: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 2007, 94: 443-474.MathSciNetView ArticleMATHGoogle Scholar
- Jiang YY, Chen ZX: Value distribution of meromorphic solutions to some difference equations. J. South China Norm. Univ., Nat. Sci. Ed. 2013, 45(1):19-23.MathSciNetMATHGoogle Scholar
- Jiang YY, Chen ZX: Fixed points of meromorphic solutions for difference Riccati equation. Taiwan. J. Math. 2013, 17(4):1413-1423.MathSciNetView ArticleMATHGoogle Scholar
- Bank SB, Gundersen G, Laine I: Meromorphic solutions of the Riccati differential equation. Ann. Acad. Sci. Fenn., Ser. A I Math. 1982, 6(2):369-398.MathSciNetMATHGoogle Scholar
- Ozawa M: On the existence of prime periodic entire functions. Kodai Math. Semin. Rep. 1978, 29: 308-321. 10.2996/kmj/1138833654View ArticleMathSciNetMATHGoogle Scholar
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