Advances in Difference Equations

, 2010:404582 | Cite as

Further Extending Results of Some Classes of Complex Difference and Functional Equations

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Abstract

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form Open image in new window , where Open image in new window and Open image in new window are two finite index sets, Open image in new window are distinct, nonzero complex numbers, Open image in new window and Open image in new window are small functions relative to Open image in new window is a rational function in Open image in new window with coefficients which are small functions of Open image in new window . We also consider related complex functional equations in the paper.

Keywords

Difference Equation Meromorphic Function Monic Polynomial Meromorphic Solution Small Function 

1. Introduction and Main Results

Let Open image in new window be a meromorphic function in the complex plane. We assume that the reader is familiar with the standard notations and results in Nevanlinna's value distribution theory of meromorphic functions such as the characteristic function Open image in new window , proximity function Open image in new window , counting function Open image in new window , the first and second main theorems (see, e.g., [1, 2, 3, 4]). We also use Open image in new window to denote the counting function of the poles of Open image in new window whose every pole is counted only once. The notation Open image in new window denotes any quantity that satisfies the condition: Open image in new window as Open image in new window possibly outside an exceptional set of Open image in new window of finite linear measure. A meromorphic function Open image in new window is called a small function of Open image in new window if and only if Open image in new window

Recently, a number of papers (see, e.g., [5, 6, 7, 8, 9]) focusing on Malmquist type theorem of the complex difference equations emerged. In 2000, Ablowitz et al. [5] proved some results on the classical Malmquist theorem of the complex difference equations in the complex differential equation by utilizing Nevanlinna theory. They obtained the following two results.

Theorem A.

If the second-order difference equation

with polynomial coefficients Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ), admits a transcendental meromorphic solution of finite order, then Open image in new window

Theorem B.

If the second-order difference equation

with polynomial coefficients Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ), admits a transcendental meromorphic solution of finite order, then Open image in new window

One year later, Heittokangas et al. [7] extended the above two results to the case of higher-order difference equations of more general type. They got the following.

Theorem C.

Let Open image in new window . If the difference equation

with the coefficients of rational functions Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ) admits a transcendental meromorphic solution of finite order, then Open image in new window

Theorem D.

Let Open image in new window . If the difference equation

with the coefficients of rational functions Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ) admits a transcendental meromorphic solution of finite order, then Open image in new window

Laine et al. [9] and Huang and Chen [8], respectively, generalized the above results. They obtained the following theorem.

Theorem E.

Let Open image in new window be distinct, nonzero complex numbers, and suppose that Open image in new window is a transcendental meromorphic solution of the difference equation

with coefficients Open image in new window ( Open image in new window ) and Open image in new window ( Open image in new window ), which are small functions relative to Open image in new window where Open image in new window is a collection of all subsets of Open image in new window . If the order Open image in new window is finite, then Open image in new window .

In the same paper, Laine et al. also obtained Tumura-Clunie theorem about difference equation.

Theorem F.

Suppose that Open image in new window are distinct, nonzero complex numbers and that Open image in new window is a transcendental meromorphic solution of
where the coefficients Open image in new window are nonvanishing small functions relative to Open image in new window and where Open image in new window and Open image in new window are relatively prime polynomials in Open image in new window over the field of small functions relative to Open image in new window . Moreover, we assume that Open image in new window ,
and that, without restricting generality, Open image in new window is a monic polynomial. If there exists Open image in new window such that for all Open image in new window sufficiently large,

where Open image in new window is a small meromorphic function relative to Open image in new window .

Remark 1.1.

Huang and Chen [8] proved that the Theorem F remains true when the left hand side of (1.6) is replaced by the left hand side of (1.5), meanwhile, the condition (1.8) would be replaced by a corresponding form.

Moreover, Laine et al. [9] also gave the following result.

Theorem G.

Suppose that Open image in new window is a transcendental meromorphic solution of
where Open image in new window is a polynomial of degree Open image in new window is a collection of all subsets of Open image in new window . Moreover, we assume that the coefficients Open image in new window are small functions relative to Open image in new window and that Open image in new window Then

where Open image in new window

In this paper, we consider a more general class of complex difference equations. We prove the following results, which generalize the above related results.

Theorem 1.2.

Let Open image in new window be distinct, nonzero complex numbers and suppose that Open image in new window is a transcendental meromorphic solution of the difference equation
with coefficients Open image in new window , and Open image in new window are small functions relative to Open image in new window where Open image in new window and Open image in new window are two finite index sets, denote

If the order Open image in new window is finite, then Open image in new window

Corollary 1.3.

Let Open image in new window be distinct, nonzero complex numbers and suppose that Open image in new window is a transcendental meromorphic solution of the difference equation
with coefficients Open image in new window and Open image in new window , which are small functions relative to Open image in new window where Open image in new window is a finite index set, denote

If the order Open image in new window is finite, then Open image in new window

Remark 1.4.

In Corollary 1.3, if we take

then Corollary 1.3 becomes Theorem E. Therefore, Theorem 1.2 is a generalization of Theorem E.

Example 1.5.

Let Open image in new window Then it is easy to check that Open image in new window solves the following difference equation:

Example 1.6.

Let Open image in new window It is easy to check that Open image in new window satisfies the difference equation

In above two examples, we both have Open image in new window and Open image in new window Therefore, the estimations in Theorem 1.2 and Corollary 1.3 are sharp.

Theorem 1.7.

Suppose that Open image in new window are distinct, nonzero complex numbers and that Open image in new window is a transcendental meromorphic solution of
where the coefficients Open image in new window are nonvanishing small functions relative to Open image in new window and Open image in new window and Open image in new window are relatively prime polynomials in Open image in new window over the field of small functions relative to Open image in new window , Open image in new window and Open image in new window are two finite index sets, denote
Moreover, we assume that Open image in new window ,
and that, without restricting generality, Open image in new window is a monic polynomial. If there exists Open image in new window such that for all Open image in new window sufficiently large,

where Open image in new window is a small meromorphic function relative to Open image in new window .

If the left hand side of (1.19) in Theorem 1.7 is replaced by the left hand side of (1.14) in Corollary 1.3, then (1.23) implies (1.22). Since we have

by the fundamental property of counting function. Therefore, we get the following result easily.

Corollary 1.8.

Suppose that Open image in new window are distinct, nonzero complex numbers and that Open image in new window is a transcendental meromorphic solution of
where the coefficients Open image in new window are nonvanishing small functions relative to Open image in new window and Open image in new window and Open image in new window are relatively prime polynomials in Open image in new window over the field of small functions relative to Open image in new window , Open image in new window is a finite index set, denote
Moreover, we assume that Open image in new window ,
and that, without restricting generality, Open image in new window is a monic polynomial. If there exists Open image in new window such that for all Open image in new window sufficiently large,

where Open image in new window is a small meromorphic function relative to Open image in new window .

Finally, we give a result corresponding to Theorem G.

Theorem 1.9.

Let Open image in new window be distinct, nonzero complex numbers and suppose that Open image in new window is a transcendental meromorphic solution of
where Open image in new window is a polynomial of degree Open image in new window , Open image in new window and Open image in new window are two finite index sets. Denote
Moreover, we assume that the coefficients Open image in new window and Open image in new window are small functions relative to Open image in new window and that Open image in new window Then

where Open image in new window

2. Main Lemmas

In order to prove our results, we need the following lemmas.

Lemma 2.1 (see [10]).

Let Open image in new window be a meromorphic function. Then for all irreducible rational functions in Open image in new window ,
such that the meromorphic coefficients Open image in new window satisfy

Lemma 2.2 (see [11]).

Let Open image in new window be distinct meromorphic functions and

where Open image in new window and Open image in new window are two finite index sets, and Open image in new window ( Open image in new window ).

Remark 2.3.

If we suppose that Open image in new window and Open image in new window hold for all Open image in new window , and denote Open image in new window and Open image in new window then we have the following estimation by the proof of Lemma 2.2

Lemma 2.4 (see [6]).

Let Open image in new window be a meromorphic function with order Open image in new window and let Open image in new window be a fixed nonzero complex number, then for each Open image in new window one has

Lemma 2.5 (see [12]).

Let Open image in new window be a meromorphic function and let Open image in new window be given by
where Open image in new window are small meromorphic functions relative to Open image in new window . Then either

Lemma 2.6 (see [9, 13]).

Let Open image in new window be a nonconstant meromorphic function and let Open image in new window , Open image in new window be two polynomials in Open image in new window with meromorphic coefficients small relative to Open image in new window . If Open image in new window and Open image in new window have no common factors of positive degree in Open image in new window over the field of small functions relative to Open image in new window , then

Lemma 2.7 (see [14]).

Let Open image in new window be a transcendental meromorphic function, and Open image in new window be a nonconstant polynomial of degree Open image in new window . Given Open image in new window denote Open image in new window and Open image in new window . Then given Open image in new window and Open image in new window one has

for all Open image in new window large enough.

Lemma 2.8 (see [15]).

Let Open image in new window be positive and bounded in every finite interval, and suppose that Open image in new window holds for all Open image in new window large enough, where Open image in new window and Open image in new window are real constants. Then

where Open image in new window .

3. Proof of Theorems

Proof of Theorem 1.2.

We assume that Open image in new window is a meromorphic solution of finite order of (1.12). It follows from Lemmas 2.1, 2.2, and 2.4 that for each Open image in new window

This yields the asserted result.

Proof of Theorem 1.7.

Suppose Open image in new window is a transcendental meromorphic solution of (1.19) and the second alternative of the conclusion is not true. Then according to Lemmas 2.5 and 2.6, we get
Thus, we have
Now assuming the order Open image in new window , then we have Open image in new window and
for all Open image in new window . By using Lemmas 2.1 and 2.2, we conclude that
It follows from this that
We prove the following inequality by induction:
The case Open image in new window has been proved. We assume that above inequality holds when Open image in new window . Next, we prove that inequality (3.7) holds for Open image in new window We have
Noting that Open image in new window thus we have
This implies that
It follows from (3.7) that
Let Open image in new window be large enough such that

Finally, let Open image in new window and we conclude that the order Open image in new window Therefore, we get a contradiction and the assertion follows.

Proof of Theorem 1.9.

We assume Open image in new window is a transcendental meromorphic solution of (1.31). Denoting again Open image in new window According to the last assertion of Lemmas 2.7 and 2.2, we get that
Since Open image in new window holds for Open image in new window large enough for Open image in new window we may assume Open image in new window to be large enough to satisfy
outside a possible exceptional set of finite linear measure. By the standard idea of removing the exceptional set (see [4, page 5]), we know that whenever Open image in new window
holds for all Open image in new window large enough. Denote Open image in new window , thus inequality (3.20) may be written in the form
By Lemma 2.8, we have

Denoting now Open image in new window , thus we obtain the required form. Theorem 1.9 is proved.

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The research was supported by NSF of China (Grant no. 10871089).

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Copyright information

© Jian-jun Zhang and Liang-wen Liao. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina

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