Complex oscillation of meromorphic solutions for difference Riccati equation
Advances in Difference Equations volume 2014, Article number: 247 (2014)
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].
In , Ishizaki gave some surveys of the basic properties of the difference Riccatiequation
where is a rational function, which have analogs in thedifferential case . In the proof of the celebrated classification theorem, Halburd and Korhonen  were concerned with the difference Riccati equation of the form
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 is a transcendental finite order meromorphic solution of the difference Riccatiequation
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.
Theorem 1.1 Letbe a constant andbe a nonconstant rational function.Set. If there exists a nonconstant rationalfunctionsuch that, then every finite order transcendental meromorphicsolutionof the difference Riccati equation (1), itsdifference, and divided differenceare oscillatory and satisfy
Theorem 1.2 Letbe a nonconstant rational function.If α is a non-zero complex constant, thenevery finite order transcendental meromorphic solutionof the difference Riccati equation
if, then, ;
if there is a rational function satisfying
if there is a rational function satisfying
Example 1.1 The function satisfies the difference Riccati equation
where , is a periodic function with period 1. Note that for any, there exists a prime periodic entire function of order by Ozawa . Thus .
Also, this solution satisfies
Using the same discussion as Lemma 2.1, we easily see that and (or ) have at most finitely many common zeros. Thus,
2 Lemmas for proofs of theorems
Firstly we need the following lemmas for the proof of Theorem 1.1.
Lemma 2.1 Letbe a nonconstant rational function,andbe a nonconstant meromorphic function.Then
have at most finitely many common zeros.
Proof Suppose that is a common zero of and . Then . Thus, . Substituting into , we obtain
Since is a nonconstant rational function, has only finitely many zeros. Thus, and have at most finitely many common zeros. □
Lemma 2.2 Let be a nonconstant finite order transcendental meromorphic solution of the differenceequation of
whereis a difference polynomialin. Iffor a meromorphic functionsatisfying, then
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 .
By (1) and the fact that , we obtain
Since and are rational functions, we know that (or ) and have the same poles, except possibly finitely many. ByLemma 2.1, we see that and have at most finitely many common zeros. Hence, by (3), weonly need to prove that
Suppose that . By and Hadamard factorization theorem, can be rewritten in the form
where is a polynomial with , and are canonical products ( may be a polynomial) formed by non-zero zeros and poles of, respectively, t is an integer, if, then , ; if , then , . Combining Theorem A with the property of thecanonical product, we have
By (5), we obtain
where . Thus, by (6), we have
Substituting (7) into (1), we obtain
By (8) and the fact that , we have
Since is a nonconstant rational faction, we see that and , so that
Thus, by (6), (9), and Lemma 2.2, we obtain for any given ε(),
holds for all r outside of a possible exceptional set with finite logarithmicmeasure.
On the other hand, by and the fact that is an entire function, we see that
Thus (10) is a contradiction. Hence, (4) holds, that is, .
Secondly, we prove that . By (1), we obtain
Thus, by this and (4), we see that .
4 Proof of Theorem 1.2
Suppose that is a finite order transcendental meromorphic solution of(2).
First, we prove that the conclusion holds when . Set . Thus, is transcendental, , and . Substituting into (2), we obtain
By the condition that is a nonconstant rational function, we obtain. By Lemma 2.2,
holds for all r outside of a possible exceptional set with finite logarithmicmeasure. That is,
holds for all r outside of a possible exceptional set with finite logarithmicmeasure. Thus, we obtain .
Now suppose that . By (2) and , we see that
Using the same discussion as Lemma 2.1, we easily see that and have at most finitely many common zeros. Thus, we onlyneed to prove that
Using the same method as in the proof of (4)-(11), we can prove that (13) holds. Hence.
Now in (12), we replace z by (), and we obtain
Set . Then (14) is transformed as
Since is a nonconstant rational function too, applying theconclusion for to (15), we obtain
Suppose that there is a rational function satisfying(16)
Now we prove
By (2), we have
If , then
Since is a rational function, and have the same poles, except possibly finitely many. By(19) and Theorem A, we obtain
If , by (16) and (18), we have
Using the same discussion as Lemma 2.1, we easily see that and have at most finitely many common zeros. Thus, by (20), inorder to prove (17), we only need to prove that
Without loss of generality, we prove (21). Suppose that . Using the same method as in the proof of (4)-(11), we seethat can be rewritten as
where , , are non-zero entire functions, such that
Substituting (22) into (2), we obtain
By the above equation and (16), we have
where . Since is a constant, to prove , we need to prove that is nonconstant.
Now we prove that
is nonconstant. Since is a nonconstant rational function and due to (16), is a nonconstant rational function too. First, if is a polynomial with , then
is the maximal degree in (since ). Thus is a polynomial with . Secondly, if , where and are polynomials with , then , where
Thus is nonconstant. Lastly, if , where and are polynomials with , then
where , , and are polynomials with and . By the above discussion, we know that is nonconstant. Hence , and, by Lemma 2.2, we see that (21) holds.
Suppose that there is a rational function satisfying(23)
In what follows, we prove that
By (2) and (23), we obtain
Using the same discussion as Lemma 2.1, we easily see that and have at most finitely many common zeros. Thus, by (25), weknow that to prove (24), we only need to prove that
Using the same method as in the proof of (21), we can prove that the above equationholds.
Thus, Theorem 1.2 is proved.
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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).
The authors declare that they have no competing interests.
Y-YJ completed the main part of this article, Y-YJ, Z-QM, and MW corrected the maintheorems. All authors read and approved the final manuscript.
An erratum to this article is available at http://dx.doi.org/10.1186/s13662-014-0339-9.
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Jiang, YY., Mao, ZQ. & Wen, M. Complex oscillation of meromorphic solutions for difference Riccati equation. Adv Differ Equ 2014, 247 (2014). https://doi.org/10.1186/1687-1847-2014-247