BOUNDED NONOSCILLATORY SOLUTIONS OF NEUTRAL TYPE DIFFERENCE SYSTEMS

This paper deals with the existence of a bounded nonoscillatory solution of nonlinear neutral type difference systems. Examples are provided to illustrate the main results.

In this paper, we obtain some sufficient conditions for the existence of a nonoscillatory bounded solutions of the system (1) via fixed point theorems and some new techniques.Here we allow the sequences {p i (n)} and {e i (n)} , i = 1, 2, to be oscillatory.The existence of nonoscillatory solutions of neutral type difference equations has been treated in [1,2,6,12] and in papers cited therein.

Main Results
In this section, we obtain sufficient conditions for the existence of bounded nonoscillatory solutions of the system (1).We consider the following cases: , and their combination.
In this sequel we use the following fixed point theorem.
Lemma 2.1 [Krasnoselskii's fixed point theorem] Let B be a Banach space.Let S be a bounded, closed, convex subset of B and let F , T be maps of S into B such that F x + T y ∈ S for all x, y ∈ S. If F is contractive and T is completely continuous, then the equation The details of Lemma 2.1 can be found in [1] and [4] . and Then the system (1) has a bounded nonoscillatory solution.
Proof.In view of conditions (2) and (3), one can choose N ∈ N (n 0 ) sufficiently large that where ) is a Banach space.We define a closed, bounded and convex subset S of B (n 0 ) as follows: Let maps F = (F 1 , F 2 ) and T = (T 1 , T 2 ) : S → B (n 0 ) be defined by and First we show that if x, y ∈ S, then F x + T y ∈ S. Hence, for x = {x (n)} and y = {y (n)} ∈ S and n ≥ N, we have Further we obtain Hence F x + T y ∈ S for any x, y ∈ S, that is, F S ∪ T S ⊂ S. Next we show that F is a contraction on S. In fact for x, y ∈ S and n ≥ N, we have This implies that , we conclude that F is a contraction mapping on S. Next we show that T is completely continuous.For this, first we show that T is continuous.
Next we show that T S is relatively compact.Using the result [[3], Theorem 3.3], we need only to show that T S is uniformly Cauchy.Let x = ({x 1 (n)} , {x 2 (n)}) be in S. From ( 2) and (3) it follows that for ǫ > 0, there exists Then for n 2 > n 2 ≥ N * , we have Thus T S is uniformly Cauchy.Hence it is relatively compact.Thus by Lemma 2.1, there is a We see that {x 0 (n)} is a bounded nonoscillatory solution of the system (1).The proof is now complete.Theorem 2.2 Suppose that 1 < a i ≡ a i (n) < ∞ and conditions (2) and (3) hold.Then the system (1) has a bounded nonoscillatory solution.
Proof.In view of conditions (2) and (3), we can choose N > n 0 sufficiently large that where D i = max Let B (n 0 ) be the Banach space defined in the proof of Theorem 2.1.We define a closed,bounded and convex subset S of B(n 0 ) as follows: Now we show that F is a contractive mapping on S. For any x, y ∈ S and n ≥ N, we obtain This implies that Since 0 < max 1 a i , i = 1, 2 < 1 we conclude that F is a contraction mapping on S.
Next we show that for any x, y ∈ S, F x + T y ∈ S. For every x, y ∈ S and n ≥ N, we have Further, we obtain Hence Thus we proved that F x + T y ∈ S for any x, y ∈ S.
Proceeding similarly as in the proof of Theorem 2.1, we obtain that the mapping T is completely continuous.By Lemma 2.1, there is a x 0 ∈ S such that F x 0 + T x 0 = x 0 .We see that {x 0 (n)} is a non oscillatory bounded solution of the system (1).This completes the proof of Theorem 2.2.2) and (3) hold.Then the difference system (1) has a bounded nonoscillatory solution.
Proof.In view of conditions ( 2) and (3), we can choose a N > n 0 sufficiently large that Let B (n 0 ) be the Banach space defined as in Theorem 2.1.We define a closed, bounded and convex subset S of B (n 0 ) as follows: Let maps F = (F 1 , F 2 ) and T = (T 1 , T 2 ) : S → B (n 0 ) be defined by As in the proof of Theorems 2.1 and 2.2 one can show that F 1 , F 2 are contractive mappings on S. It is easy to show that for any x, y ∈ S. It is easy to show that for any x, y ∈ S, F 1 x + T 1 y ∈ S and also F 2 x + T 2 y ∈ S. Proceeding as in the proof of Theorem 2.1, we obtain that the mappings T 1 , T 2 are completely continuous.By Lemma 2.1, there are x 01 , x 02 ∈ S such that F 1 x 01 + T 1 x 01 = x 01 , F 2 x 02 + T 2 x 02 = x 02 .We see that x 0 (n) = ({x 01 (n)} , {x 02 (n)}) is a nonoscillatory bounded solution of the difference system (1).The proof is now complete.
In the following we provide some examples to illustrate the results.
Example 2.1 Consider the difference system It is easy to see that all conditions of Theorem 2.1 are satisfied, and hence the system (4) has a bounded nonoscillatory solution.In fact 3 n is one such solution of the system (4).
Example 2.2 Consider the difference system (5) By Theorem 2.2, the system (5) has a bounded nonoscillatory solution.In fact is one such solution of the system (5).
Example 2.3 Consider the difference system It is easy to see that all conditions of Theorem 2.3 are satisfied, and hence the system (6) has a bounded nonoscillatory solution.In fact x (n) = 1 2 n , is one such solution of the system.
Remark 2.1 It is easy to see that the difference system (1) includes different types of (ordinary, delay, neutral) fourth order difference equations, and hence the results obtained in this paper generalize many of the existing results for the fourth order difference equations, see for example [5,7,8,9,10,11] and the references cited therein.