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The growth of solutions of systems of complex q-shift difference equations
Advances in Difference Equations volume 2012, Article number: 216 (2012)
Abstract
In this paper, we study the properties of systems of two types of complex q-shift difference equations with meromorphic solutions from the point of view of Nevanlinna theory. Some results obtained in this paper improve and extend the previous theorems given by Gao, and five examples show the extension of solutions of the system of complex difference equation.
MSC:39A50, 30D35.
1 Introduction and main results
In this paper, the problem of the existence and growth of solutions of complex difference equations will be studied. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used in this paper (see [1–3]). In addition, for a meromorphic function f, denotes any quantity satisfying for all r outside of a possible exceptional set E of finite logarithmic measure , denotes any quantity satisfying for all r on a set F of logarithmic density 1 (or for all r outside of a possible exceptional set of logarithmic density 0, the logarithmic density of a set F is defined by
Throughout this paper, the set E of finite logarithmic measure and F of logarithmic density will be not necessarily the same at each occurrence.
In recent years, there has been an increasing interest in studying difference equations, difference product and q-difference in the complex plane ℂ, considerable attention was paid to the growth of solutions of difference equations, value distribution and the uniqueness of differences analogues of Nevanlinna’s theory [4–13]. Chiang and Feng [6], Halburd and Korhonen [14] established a difference analogue of the logarithmic derivative lemma independently, and Barnett, Halburd, Korhonen and Morgan [4] also established an analogue of the logarithmic derivative lemma on q-difference operators. After their works, many people applied it to prove a number of results on meromorphic solutions of complex difference equations.
In 2000, Ablowitz, Halburd and Herbst [15] studied some class of complex difference equations,
where the coefficients are meromorphic functions, and obtained the following results.
Theorem 1.1 (see [15])
If the difference equation (1) (or (2)) with polynomial coefficients , admits a transcendental meromorphic solution of finite order, then .
In 2001, Heittokangas et al. [16] further investigated some complex difference equations which are similar to (1) and (2) and obtained the following results which are the improvement of Theorems 1.1 and 1.2.
Theorem 1.2 (see [[16], Proposition 8 and Proposition 9])
Let . If the equations
with rational coefficients , admit a transcendental meromorphic solution of finite order, then .
In 2002, Gundersen et al. [17] studied the growth of meromorphic solutions of q-difference equations and obtained the result as follows.
Theorem 1.3 (see [[17], Theorem 3.2])
Suppose that f is a transcendental meromorphic solution of an equation of the form
with meromorphic coefficients , and a constant (), assuming that , , , and that is irreducible in f. Then .
In 2010, Zheng and Chen [18] further considered the growth of meromorphic solutions of q-difference equations and obtained some results which extended the theorems given by Heittokangas et al. [16].
Theorem 1.4 (see [18])
Suppose that f is a transcendental meromorphic solution of the equation of the form
where , are two finite index sets satisfying
, , , and all coefficients of (3) are of growth . If , then for sufficiently large r, , where K (>0) is a constant. Thus, the lower order of f satisfies .
In 2011, Liu and Qi [11] investigated the properties of meromorphic solutions of a q-shift difference equation and obtained some results as follows.
Theorem 1.5 (see [[11], Theorem 4.1])
Suppose that f is a transcendental meromorphic solution of
where the coefficients , are rational, and . Assuming is irreducible in f, . If and , then , provided that f has infinitely many poles.
In 2012, Gao [19] investigated a complex difference equation which is similar to (3) and obtained the following theorem.
Theorem 1.6 (see [[19], Theorem 1.1])
Let . If the equation
has a transcendental meromorphic solution of finite order, then
where , , , , and are small functions of f.
Thus, the following questions arise naturally:
Question A Will the assertion of Theorem 1.6 remain valid if the transcendental meromorphic solution has infinite order?
Question B What will happen when is replaced by for in Theorem 1.6?
In this paper, we investigate the above questions and obtain the following theorem.
Theorem 1.7 Let , and , , and be of growth without an exceptional set as . If f is a transcendental meromorphic solution of the equation of the form
and satisfies one of the following conditions:
-
(i)
and ;
-
(ii)
and .
Then the conclusion of Theorem 1.6 still holds.
Remark 1.1 Theorem 1.7 is an improvement of Theorem 1.6. And Theorem 1.7 is an answer to Question A and Question B.
In this paper, we also investigated the extension of a meromorphic solution of the following system of complex q-shift difference equations of the form:
and
where , is a collection of all subsects of , , two q-shift difference polynomials , are defined by
and , are defined by
and all coefficients , , , , , and , are small functions of , , and for . Let
and
The order and hyper order of a meromorphic solution of the system (6) are defined by
where
To state our main results, we require the following definition.
Definition 1.8 (see [19])
If is a meromorphic solution of the equation system (6), and its component satisfies
then is called admissible, where E is an exceptional set with finite logarithmic measure.
Theorem 1.9 For the system (6), are two collections of all non-empty subsets of for , () are distinct complex constants, and is a polynomial in u of (>0), its coefficients of are all small functions of , . Let be a meromorphic solution of the system (6) such that , are non-rational meromorphic, and all the coefficients of (6) are small functions relative to , . Thus,
-
(i)
if and , we have
(8) -
(ii)
if and , then
(9) -
(iii)
if and , then ,
where is the lower order of f.
Some examples will show that the conclusions (8) and (9) in Theorem 1.9 are sharp.
Example 1.1 The function satisfies the system
where c, η are any nonzero complex constants, , and
Thus, we have
where , and . This example shows that the equality in (8) can be arrived at.
Example 1.2 The function satisfies the system
where c is any nonzero complex constant, , , and
We note that , are small functions relative to , . Thus, we have
where , and . This example shows that the inequality (8) is true.
Example 1.3 The function satisfies the system
where c, η are any nonzero complex constants, , and
Thus, we have
where , and . This example shows that the equality in (9) can be arrived at.
Example 1.4 The function satisfies the system
where c is a nonzero constant, , ,
and
We note that , , , , are small functions relative to , . Thus, we have
where , and . This example shows that the inequality in (9) is true.
Example 1.5 The function satisfies the following system:
We have . Thus, it shows that (iii) in Theorem 1.9 is true when , , , and .
Theorem 1.10 Let be an admissible meromorphic solution of (7), , and satisfy one of the following conditions:
-
(i)
and ;
-
(ii)
and .
Then
Theorem 1.11 Let be an admissible meromorphic solution of (7), the coefficients , , , , and of (7) be meromorphic functions and not necessarily small functions of , and
If , and q satisfy one of the following conditions:
-
(i)
and ;
-
(ii)
and .
Then the solutions in the pair are either both admissible or both non-admissible.
Remark 1.2 It is easy to see that Theorem 1.7, Theorem 1.10 and Theorem 1.11 are the improvement of some results in [19].
2 Some lemmas
Lemma 2.1 (Valiron-Mohon’ko [20])
Let be a meromorphic function. Then, for all irreducible rational functions in f,
with meromorphic coefficients , , the characteristic function of satisfies
where and .
Lemma 2.2 ([[21], Theorem 5.1])
Let f be a transcendental meromorphic function of , , ε be a number small enough. Then
for all r outside of a set of finite logarithmic measure.
Lemma 2.3 ([[21], Lemma 8.3])
Let be a non-decreasing continuous function and let . If the hyper order of T is strictly less that one, that is,
and , then
for all r runs to infinity outside of a set of finite logarithmic measure.
From Lemma 2.3, we can get the following lemma easily.
Lemma 2.4 Let be a transcendental meromorphic function of hyper order and c be a non-zero complex constant. Then we have
Lemma 2.5 ([[22], p.36] or [[13], Theorem 1.1 and Theorem 1.3])
Let be a transcendental meromorphic function of zero order and q be a nonzero complex constant. Then
on a set of logarithmic density 1 or for all r outside of a possible exceptional set of logarithmic density 0.
Lemma 2.6 Let be a transcendental meromorphic function of zero order and q, c be two nonzero complex constants. Then
Proof By using the idea of [[22], p.47], we can get the conclusions of Lemma 2.6 easily. Now, we will give another method to prove this lemma as follows.
From Lemma 2.4 and Lemma 2.5, we get the first equality of this lemma.
Next, the idea of the proof of the other inequalities of this lemma is from [8, 23]. Since f is of zero order, by Lemma 2.5, we have
By Lemma 2.5 and since f is a meromorphic function of zero order, we have . And by Lemma 2.3, we have
outside of a possible exceptional set with finite logarithmic measure.
From (10), (11) and , we have and . □
Lemma 2.7 ([4, 14] or [[11], Theorem 2.1])
Let be a nonconstant zero-order meromorphic function and . Then
on a set of logarithmic density 1 for all r outside of a possible exceptional set of logarithmic density 0.
Lemma 2.8 ([[22], p.44] or [[19], Lemma 2.5])
If
then
where , are sets with finite logarithmic measure.
Lemma 2.9 Let , , be two transcendental meromorphic functions and , ,
where (i) is an index set consisting of finite elements.
If and for , then
for all r outside of a possible exceptional set E of finite logarithmic measure.
If and for , then
for all r on a set F of logarithmic density 1 or for all r outside of a possible exceptional set of logarithmic density 0, where , are as stated in Section 1, M a real constant and not necessarily the same at each occurrence.
Proof Set .
Let be a pole of of order τ that is not the zero and pole of , then there exists at least one index such that has a pole at of order . Thus, there exists a subset of such that each one of () has a pole at . If (≥1) () are the order of at respectively, then we have
Hence, we have , that is,
We can rewrite in the following form:
It follows
Thus,
We will consider two cases as follows.
Case 1. If and . By Lemma 2.2, since the coefficients are small functions of , , and from (13) and the definitions of , , we have
where E is a set of finite logarithmic measure and , are as stated in Section 1.
Since and , by Lemma 2.3 and Lemma 2.4, from (12), we have
for all r outside of a possible exceptional set E of finite logarithmic measure. From (14) and (15), we get
for all r outside of a possible exceptional set E of finite logarithmic measure.
Case 2. If and for . By using the same argument as in Case 1, from Lemma 2.6, Lemma 2.7 and Lemma 2.8, we can get
for all r on a set of logarithmic density 1. □
Lemma 2.10 ([[24], Lemma 4])
Suppose that a meromorphic function f has finite lower order λ. Then, for every constant and a given ε, there exists a sequence such that
Lemma 2.11 ([25])
Let be a transcendental meromorphic function and be a complex polynomial of degree . For given , let , , then for given and for r large enough,
Let , be monotone increasing functions such that outside of an exceptional set E with finite linear measure, or , , where is a set of finite logarithmic measure. Then, for any , there exists such that for all .
3 The proof of Theorem 1.9
From the assumptions of Theorem 1.9, we get that , are transcendental meromorphic functions.
Denote , , and . Since (Ref. [15]), by applying Lemma 2.1 to (6) and using the same argument as in Lemma 2.9, we have
for sufficiently large r and any given , , . By Lemma 2.11 and (16), (17), for (, ), and sufficiently large r, we get
outside of a possible exceptional set , of finite linear measure, respectively. From Lemma 2.12, for any given () and sufficiently large r, we can obtain
that is,
Case 3.1 Since , , , then we have , . From (18), and by Lemma 2.10, for any given , there exists a sequence such that
for . From the above inequalities, we have
Thus, let , , and for and , since and , from (19), we can get
Thus, (8) holds.
Case 3.2 Suppose that . By using the same argument as above, we can get
where ( is chosen to be such that ), and r is sufficiently large. We can choose sufficiently small such that . Thus, it follows
outside of a possible exceptional set () of finite logarithmic measure, respectively.
Since , , , and , are transcendental, by applying Lemma 3.1 in [17] and Lemma 2.12, for and , we have
which implies that (9) is true.
Case 3.3 and . By using the same argument as in Case 3.1, we can get that .
From Case 3.1, Case 3.2 and Case 3.3, the proof of Theorem 1.9 is completed.
4 Proofs of Theorems 1.7, 1.10 and 1.11
4.1 The proof of Theorem 1.7
Two cases will be considered as follows.
Case 4.1 Suppose that and . By applying Lemma 2.1 and Lemma 2.9 for equation (5), we can get
for all r outside of a possible exceptional set E of finite logarithmic measure. From the above inequality, we have
for all r outside of a possible exceptional set E of finite logarithmic measure. Since f is transcendental, we can get
Case 4.2 Suppose that and . By using the same argument as in Case 4.1, we can get
for all r on a set of logarithmic density 1. Since f is transcendental, from the definition of logarithmic density, we can get
Thus, this completes the proof of Theorem 1.7.
4.2 The proof of Theorem 1.10
Two cases will be considered to prove Theorem 1.10.
Case 4.3 If and . By applying Lemma 2.1 and Lemma 2.9 for the system of complex equations (7), and from the assumptions of Theorem 1.10, we have
where , are the sets of finite logarithmic measure. Since is a transcendental meromorphic solution of (5), from (20) and (21), we have
for all r outside of a possible exceptional set of finite logarithmic measure. Thus, from (22), we can get
Case 4.4 Suppose that and . By using a similar method as that in Case 4.3, we have
where , are the sets of logarithmic density 0. Thus, we get
for all r outside of a possible exceptional set of logarithmic density 0. Thus, from the definition of logarithmic density, we can get
Thus, the proof of Theorem 1.10 is completed.
4.3 The proof of Theorem 1.11
Two cases will be considered as follows.
Case 4.5 If and . Since all coefficients , , , , and are not necessarily small functions, by using the same argument as in Lemma 2.9, from Lemma 2.1, we have
where , are two sets of finite logarithmic measure, is as stated in Section 1.
Suppose that the component is admissible and is non-admissible, from (24) we have
for all r outside of a possible exceptional set of finite logarithmic measure. Thus, from Lemma 2.8, we have
outside of a possible exceptional set with finite logarithmic measure, which contradicts the assumptions of Theorem 1.11.
Suppose that the component is admissible and is non-admissible, from (25) we have
for all r outside of a possible exceptional set of finite logarithmic measure. Thus, from Lemma 2.8, we have
outside of a possible exceptional set with finite logarithmic measure, which contradicts the assumptions of Theorem 1.11.
Case 4.6 If and . By using the same argument as in Case 4.5, noting the definitions of logarithmic measure and logarithmic density, we can prove that the conclusion of Theorem 1.11 is true.
Thus, the proof of Theorem 1.11 is completed.
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Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11101201 and 61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20122BAB201016 and No. 20122BAB201044) and the National Science and Technology Support Plan (2012BAH25F02).
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HYX completed the main part of this article, HYX, TBC and BXL corrected the main theorems. All authors read and approved the final manuscript.
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Xu, HY., Cao, TB. & Liu, BX. The growth of solutions of systems of complex q-shift difference equations. Adv Differ Equ 2012, 216 (2012). https://doi.org/10.1186/1687-1847-2012-216
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DOI: https://doi.org/10.1186/1687-1847-2012-216