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Oscillatory Criteria for the Two-Dimensional Difference Systems
Journal of Inequalities and Applications volume 2010, Article number: 209309 (2010)
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.
1. Introduction
We consider the following two-dimensional nonlinear difference system as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ2_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ3_HTML.gif)
Also, if , then (1.3) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ4_HTML.gif)
Furthermore, if and
is a ratio of odd positive integers, then (1.1) reduces to the well-known quasilinear difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ6_HTML.gif)
() If
, then every solution of (1.4) oscillates if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ7_HTML.gif)
For (1.3), if one denotes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ8_HTML.gif)
and assumes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ10_HTML.gif)
() If
, then every solution of (1.3) oscillates if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ11_HTML.gif)
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 [6–8]. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ13_HTML.gif)
summing up from to
, and by (1.2) of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ15_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ16_HTML.gif)
Since is decreasing, hence there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ17_HTML.gif)
Summing up
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ18_HTML.gif)
from to
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ19_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ20_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ21_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ22_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ23_HTML.gif)
and let , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ24_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ25_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ26_HTML.gif)
which leads to a contradiction.
Necessity. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ27_HTML.gif)
for some , then there exist
and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ28_HTML.gif)
Let be the Banach space of all the real-valued sequences
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ29_HTML.gif)
let be the subset of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ30_HTML.gif)
and let be the operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ32_HTML.gif)
for any . From Knaster's fixed-point theorem, we know that there exists
such that
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ33_HTML.gif)
then and
. On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ34_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ35_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ37_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ38_HTML.gif)
Since is increasing, hence there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ39_HTML.gif)
Summing up
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ40_HTML.gif)
from to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ41_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ42_HTML.gif)
From and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ43_HTML.gif)
But
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ44_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ45_HTML.gif)
which leads to a contradiction.
Necessity. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ46_HTML.gif)
for some , then there exists
large enough, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ47_HTML.gif)
Let be the set of all bounded and real-valued sequences
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ48_HTML.gif)
and be the subset of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ49_HTML.gif)
then is a bounded, convex, and closed subset of
. Let
be the operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ50_HTML.gif)
Then maps
into
. In fact, if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ53_HTML.gif)
for any . From Schauder's fixed-point theorem (see [11]), we know that there exists
such that
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ54_HTML.gif)
then and
. On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ55_HTML.gif)
Therefore, and
which leads to a contradiction. The proof of Theorem 2.5 is completed.
Example 2.6.
Considering the difference system,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ56_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ58_HTML.gif)
where .
Corollary 2.9.
If , then every solution of (1.5) oscillation if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F209309/MediaObjects/13660_2010_Article_2086_Equ59_HTML.gif)
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.
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The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions.
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Cheng, JF., Chu, YM. Oscillatory Criteria for the Two-Dimensional Difference Systems. J Inequal Appl 2010, 209309 (2010). https://doi.org/10.1155/2010/209309
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DOI: https://doi.org/10.1155/2010/209309