Consider a nonlinear two dimensional difference system of the form
where and are real sequences and , and are ratio of odd positive integers.
By a solution of Equation (1.1), we mean a real sequence which is defined for all and satisfies Equation (1.1) for all .
In the last few decades there has been an increasing interest in obtaining necessary and sufficient conditions for the oscillation and nonoscillation of two dimensional difference equation. See for example  -   and the references cited therein.
Further it will be assumed that is non-negative for all , for all u, v.
The oscillation criteria for system (1.1), when
studied in  . Therefore in this paper we consider the other case that is
and investigated the oscillatory behaviour of solutions of the system (1.1). Hence the results obtained in this paper complement to that of in  .
We may introduce the function defined by
Throughout this paper condition (1.2) is tacitly assumed; always denotes the function defined by (1.3).
In Section 2, we establish necessary and sufficient conditions for the system (1.1) to have solutions which behave asymptotically like nonzero constants or linear functions and in Section 3, we present criteria for the oscillation of all solutions of the system (1.1). Examples are inserted to illustrate some of the results in Section 4.
2. Existence of Bounded/Unbounded Solutions
In this section first we obtain necessary and sufficient conditions for the system (1.1) to have solutions which behave asymptotically like nonzero constants.
Theorem 2.1. If
are satisfied, then for any constant , system (1.1) has a solution . such that
as , where
Proof. We may assume without loss of generality that . Let
choose , so that
and let be large enough such that
Let B be the space of all real sequences with the topology of pointwise convergence. We now define X to be the set of sequences . such that
where and define Y to be the set of sequences . Such that
Let and denote the mappings from defined by
Finally define by
Clearly is a bounded, closed and convex subset of .
First we show that T maps into itself. Let . From (2.11), we have
and so, using (2.6) and (2.7), we see that
Now from (2.12) it follows that
for . This implies that . Next from (2.13), we have
where conditions (2.5), (2.7) and (2.10) have been used. Thus . Hence as desired.
Now let and for each . Let be a sequence in . Such that . Then a straight forward argument and hence T is continuous.
Finally, in order to apply Schauder-Tychonoff fixed point theorem, we need to show that is relatively compact in . In view of recent result of cheng and patula  it suffices to show that is uniformly cauchy in . To prove this, it is enough to show that and are uniformly cauchy in B. To this end, let and observe that for any , we have
It is now clear that for a given , we can choose , such that , imply and . Thus and are uniformly cauchy and so is uniformly cauchy. Thus is relatively compact.
Therefore by Schauder-Tychonoff fixed point theorem, there is an element such that . From (2.12), (2.13) and (2.14)
From (2.15) and (2.16), we see that is a solution of then system (1.1) with the properties (2.3) and (2.4). This completes the proof of the theorem.
Corollary 2.2. Assume (2.1) and (2.2) are satisfied. Then for any system (1.1) has a nonoscillatory solution such that
as . The proof is left to the reader.
Before stating and proving our next results, we give a lemma which is concerned with the nonoscillatory solution of (1.1).
Lemma 2.3. Let be a solution of (1.1) for with for all . Then
for , where is a nonnegative constant.
This lemma has been proved by Graef and Thandapani  and is very useful in the following theorems. In our next theorem, we establish a necessary condition for the system (1.1) to have nonoscillatory solution satisfying condition (2.17).
Theorem 2.4. Assume that for all . Then a necessary condition for the system (1.1) to have a nonoscillatory solution satisfying (2.17) is that
Proof. Let be a nonoscillatory solution of the system (1.1) for . Since is not identically zero for . Hence is nonoscillatory, without loss of generality, we may assume that is eventually positive for . From Lemma 2.3, we have for and
Since as , from the first equation of system (1.1), we obtain for ,
Define . If , then and
If , then and (2.23) again holds. From (2.22) and (2.23), we obtain
which in view of the boundedness of implies that
From the second inequality of (2.21) and the following inequality
where “d” being the constant, we see that
Since as , from the first equation of system (1.1), we obtain for
which in view of boundedness of , implies that
The inequalities (2.24) and (2.25) clearly imply (2.20). This completes the proof.
we conclude this section with the following theorem which gives a necessary condition for the system (1.1) to have a nonoscillatory solution of the form
Theorem 2.5. Assume for . The system (1.1) has a solution of the type (2.26) for some , then
for some .
Proof. Let be a solution of (1.1) satisfying (2.26). we may assume . Then there is an integer . such that
From Lemma 2.2, it follows that
for , where is a nonnegative constant. Also from the second equation of (1.1), we have
where combining (2.28) and (2.29), we have
since by (2.29), (2.30) implies
Using the inequality in (2.31) we obtain
If either is nonincreasing or nondecreasing holds, then (2.27) follows. This completes the proof of the theorem.
3. Oscillation Results
In this section we establish criteria for all solutions of the system (1.1) to be oscillatory. First, we consider the case where the composition of functions is storngly superlinear in the sense that
Theorem 3.1. Let for and (3.1) hold. If
then the difference system (1.1) is oscillatory.
Proof. Assume the existence of nonoscillatory solution of the system (1.1) for . As in the proof of the Theorem 2.4, we may assume that for all . From Lemma 2.3, we have (2.22) Now following argument as in the proof of Theorem 2.5, we obtain
Because of condition (3.1), the last inequality implies
Next from the second inequality (2.21), we have
The last inequality implies
Again using the argument as in the proof of Theorem 2.5, we obtain
for all . So by condition on (3.1), we have
The inequalities (3.3) and (3.4) thus obtained clearly contradicts (3.2). This contradiction completes the proof of the theorem.
Our final result is for the case when the composition of function is strongly sublinear in the sense that
for all and .
Theorem 3.2. Let for and (3.5) hold. If
where , then all solutions of the system (1.1) are oscillatory.
Proof. Let be a nonoscillatory solution of the system (1.1) for . As in the proof of Theorem 2.5, we may assume that for . From the Lemma 2.3 we have (2.21). Now summing the second equation of system (1.1) from to j, we obtain
for . Note that
Since otherwise it would follow from (3.9) that as , which contradicts the first inequality of (2.21). Therefore letting in (3.9), we obtain
and in view of first inequality of (2.21) and (3.7), is convergent.
From (3.11) and (3.12), we have .
Now substituting the value in the first equation of (1.1) and then summing the resulting inequality, we obtain
Now using conditions (3.7) and (3.8)
since , the above inequality can be written as,
observe that for , we have , and therefore
Hence from (3.13) and (3.14), we obtain
which, in view of condition (3.5) and (3.8) provides a contradiction. This completes the proof of the theorem.
Example 4.1. Consider the system
Here , , , . All the necessary conditions of Theorem 3.1 are satisfied and hence the system (4.1) is oscillatory. Here, is an oscillatory solution of the system (4.1).
Example 4.2. Consider the system
Here , , and with . we see that all conditions of Theorem 3.2 are satisfied. Hence all solutions of the system (4.2) are oscillatory.
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