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  • Research Article
  • Open Access

Oscillatory Criteria for the Two-Dimensional Difference Systems

Journal of Inequalities and Applications20102010:209309

https://doi.org/10.1155/2010/209309

  • Received: 5 April 2010
  • Accepted: 17 June 2010
  • Published:

Abstract

We establish some necessary and sufficient conditions for oscillation of the solutions of the following two-dimensional difference system: , , where and are strongly superlinear or sublinear functions.

Keywords

  • Banach Space
  • Positive Integer
  • Physical Application
  • Difference Equation
  • Difference System

1. Introduction

We consider the following two-dimensional nonlinear difference system as follows:
(1.1)

where , , and are strongly superlinear or sublinear functions.

Now we pose some conditions on functions and :

(H1): and for ;

(H2): and are continuous real-valued functions, and nondecreasing with respect to ;

(H3): it is

(1.2)

for each .

Definition 1.1.

Suppose that are real-valued functions. and are the quotients of positive odd numbers.

( ) and are said to be strongly superlinear if there exist constants and with , such that and are nondecreasing with respect to for each fixed .

( ) and are said to be strongly sublinear if there exist constants and with , such that and are nonincreasing with respect to for each fixed .

The solutions of (1.1) are said to be nonoscillatory if or is eventually positive or negative. Otherwise the solutions are called oscillatory.

Some oscillation results for the difference system (1.1) in the case of with have been established by many authors. In particular, if and , then the difference system (1.1) is reduced to the well-known second-order nonlinear difference equation:
(1.3)
Also, if , then (1.3) becomes
(1.4)
Furthermore, if and is a ratio of odd positive integers, then (1.1) reduces to the well-known quasilinear difference equation:
(1.5)

For (1.4), the following well-known Theorem A was established by Hooker and Patula [1, 2].

Theorem A.

For (1.4), the following statements are true.

( ) If , then every solution of (1.4) oscillates if and only if
(1.6)
( ) If , then every solution of (1.4) oscillates if and only if
(1.7)
For (1.3), if one denotes
(1.8)
and assumes that
(1.9)

then the following theorem is proved in [3].

Theorem B.

If (1.9) holds, then the following statements are true.

( ) If , then every solution of (1.3) oscillates if and only if
(1.10)
( ) If , then every solution of (1.3) oscillates if and only if
(1.11)

The problem of oscillation of second-order nonlinear difference equations has attracted the attention of many mathematicians because of its physical applications [2, 4]. For some results regarding the growth of solutions of some equations related to the above mentioned see book [5], as well as the following papers [68]. It is an interesting problem to extend oscillation criteria for second-order nonlinear difference equations to the case of nonlinear two-dimensional difference systems since such systems include, in particular, the second-order nonlinear, half-linear, and quasilinear difference equations that are the special cases of the nonlinear two-dimensional difference systems [5, 9, 10].

The main purpose of this paper is to establish some necessary and sufficient conditions for oscillation of the nonlinear two-dimensional difference systems.

2. Main Results

In order to establish our main results, we need the following lemma.

Lemma 2.1.

Suppose that conditions are satisfied. If and are nonoscillatory solutions of (1.1) for , then
(2.1)

Proof.

Without loss of generality, we assume that is eventually positive; that is, for . From (1.1), we clearly see that , then we know that either or eventually holds.

If for , then we have
(2.2)
summing up from to , and by (1.2) of , we get
(2.3)

This contradiction completes the proof of the lemma.

Theorem 2.2.

If and are strongly sublinear (i.e., ), then a necessary and sufficient condition for (1.1) to oscillate is that
(2.4)

for every , where .

Proof.

Sufficiency. If (1.1) has a nonoscillatory solution , then without loss of generality, we assume that is eventually positive. Then, by Lemma 2.1, for sufficiently large,
(2.5)
Since is decreasing, hence there exists such that
(2.6)
Summing up
(2.7)
from to , we obtain
(2.8)
and so
(2.9)
Therefore
(2.10)
Since
(2.11)
we have
(2.12)
and let , we have
(2.13)
From
(2.14)
we get
(2.15)

which leads to a contradiction.

Necessity. If
(2.16)
for some , then there exist and , such that
(2.17)
Let be the Banach space of all the real-valued sequences with the norm
(2.18)
let be the subset of defined by
(2.19)
and let be the operator defined by
(2.20)
Then the mapping satisfies the assumptions of Knaster's fixed-point theorem (see [11, page 8]): maps into itself and is increasing. The latter statement is easy to see, and the former statement follows from
(2.21)
for any . From Knaster's fixed-point theorem, we know that there exists such that . Let
(2.22)
then and . On the other hand, we have
(2.23)

Then by (1.2) and the continuity of function , we have that and which leads to a contradiction and the proof of Theorem 2.2 is completed.

Example 2.3.

Considering the difference system,
(2.24)

Here , , and and are strongly sublinear. It is easy to verify that the conditions of Theorem 2.2 are satisfied and hence all solutions are oscillatory. In fact, we clearly see that the sequence is such a solution for the difference system.

Example 2.4.

Considering the difference system,
(2.25)

Here , , and and are strong sublinear. We clearly see that the conditions of Theorem 2.2 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence is such a solution.

Theorem 2.5.

If and are strongly superlinear (i.e., ), then a necessary and sufficient condition for (1.1) to oscillate is that
(2.26)

for every .

Proof.

Sufficiency. If (1.1) has a nonoscillatory solution , then without loss of generality, we may assume that is eventually positive. Then by Lemma 2.1, we have for sufficiently large,
(2.27)
Since is increasing, hence there exists such that
(2.28)
Summing up
(2.29)
from to , we have
(2.30)
Therefore
(2.31)
From and , we get
(2.32)
But
(2.33)
Therefore
(2.34)

which leads to a contradiction.

Necessity. If
(2.35)
for some , then there exists large enough, such that
(2.36)
Let be the set of all bounded and real-valued sequences with the norm
(2.37)
and be the subset of defined by
(2.38)
then is a bounded, convex, and closed subset of . Let be the operator defined by
(2.39)
Then maps into . In fact, if , then
(2.40)
Next, we show that is continuous. Let be a convergent sequence in such that , then from that is closed ( ) and the definition of , we have
(2.41)

Since , now from the continuity of and together with the well-known Lebesgue's dominated convergence theorem (see [11, page 263]), we know that for .

Finally, we show that is precompact. Let , , then for large enough we have
(2.42)

for any . From Schauder's fixed-point theorem (see [11]), we know that there exists such that .

Let
(2.43)
then and . On the other hand, we have
(2.44)

Therefore, and which leads to a contradiction. The proof of Theorem 2.5 is completed.

Example 2.6.

Considering the difference system,
(2.45)

Here , , and and are strongly suplinear. We clearly see that the conditions of Theorem 2.5 are satisfied and hence all solutions are oscillatory. In fact, the sequence is such a solution.

Example 2.7.

Considering the difference system,
(2.46)

Here , , and and are strong sublinear. However, it is easy to see that the conditions of Theorem 2.5 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence is such a solution.

If we set , then the difference system (1.1) is reduced to (1.5). From Theorems 2.2 and 2.5, we get the following results for (1.5).

Corollary 2.8.

If , then every solution of (1.5) oscillation if and only if
(2.47)

where .

Corollary 2.9.

If , then every solution of (1.5) oscillation if and only if
(2.48)

Remark 2.10.

It is easy to see that Theorems A and B are the special cases of our Corollaries 2.8 and 2.9, respectively.

Declarations

Acknowledgment

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Xiamen University, Xiamen, 361005, China
(2)
Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China

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Copyright

© Jin-Fa Cheng and Yu-Ming Chu. 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.

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