Oscillatory Criteria for the Two-Dimensional Difference Systems
© Jin-Fa Cheng and Yu-Ming Chu. 2010
Received: 5 April 2010
Accepted: 17 June 2010
Published: 8 July 2010
(H3): it is
For (1.4), the following statements are true.
then the following theorem is proved in .
If (1.9) holds, then the following statements are true.
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 , 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.
This contradiction completes the proof of the lemma.
which leads to a contradiction.
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.
which leads to a contradiction.
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 .
for any . From Schauder's fixed-point theorem (see ), we know that there exists such that .
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.
It is easy to see that Theorems A and B are the special cases of our Corollaries 2.8 and 2.9, respectively.
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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