Oscillatory and Asymptotic Behavior of Solutions of Second Order Neutral Delay Difference Equations with “ Maxima ”

In this paper, we study the oscillatory and asymptotic behavior of second order neutral delay difference equation with “maxima” of the form ( ) ( ) [ ] ( ) n n n n n s n n a x p x q x n n − − ∆ ∆ + + = ∈ α τ σ 0 , max 0, .  Examples are given to illustrate the main result.

where Δ is the forward difference operator defined by .By a solution of Equation ( 1), we mean a real sequence { } n x satisfying Equation (1) for all 0 n n θ ≥ − .Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise.
From the review of literature it is well known that there is a lot of results available on the oscillatory and asymptotic behavior of solutions of neutral difference equations, see [1]- [5], and the references cited therein.But very few results are available in the literature dealing with the oscillatory and asymptotic behavior of solutions of neutral difference equations with "maxima", see [6]- [9], and the references cited therein.Therefore, in this paper, we investigate the oscillatory and asymptotic behavior of all solutions of Equation (1).The results obtained in this paper extend that in [4] for equation without "maxima".
In Section 2, we obtain some sufficient conditions for the oscillation of all solutions of Equation (1).In Section 3, we present some sufficient conditions for the existence of nonoscillatory solutions for the Equation (1) using contraction mapping principle.In Section 4, we present some examples to illustrate the main results.

Oscillation Results
In this section, we present some new sufficient conditions for the oscillation of all solutions of Equation (1).Throughout this section we use the following notation without further mention: , x be an eventually positive solution of Equation (1).Then one of the following holds (I) 0, 0 be an eventually positive solution of Equation ( 1).Then we may assume that 0 Then inview of (C 3 ) we have 0 n z > for all 0 n n ≥ .From the Equation (1), we obtain Hence n n a z ∆ and n z are of eventually of one sign.This completes the proof. Lemma 2.2.Let { } n x be an eventually negative solution of Equation (1).Then one of the following holds (I) 0, 0 and then every solution of Equation (1) is oscillatory.
Proof.Assume to the contrary that there exists a nonoscillatory solution { } n x of Equation (1).Without loss of generality we may assume that 0 n x θ − > for all ( ) , where N is chosen so that both the cases of Lemma 2.1 hold for all n N ≥ .We shall show that in each case we are led to a contradiction.Case(I).From Lemma 2.4 and Equation (1), we have .
From Lemma 2.6, 0 From ( 8) and ( 9), we have Multiply (10) by Summation by parts formula yields Therefore, from ( 7) and (11), we have Letting n → ∞ in the last inequality, we obtain a contradiction to (3).This completes the proof. Theorem 2.2.Assume that 1 α ≥ , and there exists a positive integer k such that k σ τ ≥ − .If for all sufficiently large ( ) and for every constant 0 M > , (2) holds, and hold, then every solution of equation ( 1) is oscillatory.
Proof.Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.1 holds for ( ) . Case(I).Proceeding as in the proof of Theorem 2.1 (Case(I)) we obtain a contradiction to (12).Case(II).Proceeding as in the proof of Theorem 2.1 (Case(II)) we obtain (7) and (10).Multiplying (10) by Using the summation by parts formula in the first term of the last inequality and rearranging, we obtain Inview of (7), we have As n → ∞ in the last inequality, we obtain a contradiction to (12).This completes the proof.Theorem 2.3.Assume that 1 α ≥ , and there exists a positive integer k such that k σ τ ≥ − .If for all sufficiently large ( ) and for every constant 0 M > , (2) holds, and then every solution of equation ( 1) is oscillatory.
Proof.Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.1 holds and Case(I) is eliminated by the condition (2).
Case(II).Proceeding as in the proof of Theorem 2.1 (Case(II)) we have ( ) Letting n → ∞ in the above inequality, we obatin a contradiction to (14).This completes the proof.Next, we obtain sufficient conditions for the oscillation of all solutions of Equation (1) when 0 1 α < ≤ .Theorem 2.4.Assume that 0 1 α < ≤ , and there exists a positive integer k such that k σ τ ≥ − .If for all sufficiently large ( ) and for every constant and then every solution of equation ( 1) is oscillatory.
Proof.Proceeding as in the proof of Theorem 2.1, we see that Lemma 2.4 holds for ( ) Then 0 n w > and from Equation ( 1) and Lemma 2.2, we have Using Lemma 2.5 in (18), we obtain [ ] ( ) for some constant 1 0 M > for all large n.Using (20) in (19) and then summing the resulting inequality from  ) ( ) Letting n → ∞ in (21), we obtain a contradiction to (16).

Case(II). Define a function n v by
, .

and n n a z
∆ is negative and decreasing we have .
Since n z is a positive and decreasing, we have Now using (15) in (22), we obtain Multiplying the last inequality by Using the summation by parts formula in the first term of the above inequality and rearranging we obtain Using completing the square in the las term of the left hand side of the last inequality, we obtain Letting n → ∞ in the above inequality, we obtain a contradiction to (17).The proof is now complete.

Existence of Nonoscillatory Solutions
In this section, we provide sufficient conditions for the existence of nonoscillatory solutions of Equation (1) in case 1 α > or 0 1 α < < .Note that in this section we do not require n p p ≡ .
Clearly, T is continuous.Now for every x S ∈ and n N ≥ , (25) implies ( ) ( ) ∑ Thus, we have that TS S ⊂ .Since S is bounded, closed and convex subset of χ , we only need to show that T is contraction mapping on S in order to apply the contraction mapping principle.For , x y S ∈ and n N ≥ , we have   ( )   and we see that T is a contraction on S. Hence, T has a unique fixed point which is clearly a positive solution of Equation ( 1).This completes the proof of the theorem.

Examples
In this section we present some examples to illustrate the main results.
Consider the oscillatory and asymptotic behavior of second order neutral delay difference equation with "maxima" of the form

n
is a nonnegative integer subject to the following conditions:(C 1 ) τ and σ are positive integers; (C 2 ) α is a ratio of odd positive integers;(C 3 ) { } n p and { } n qare nonnegative real sequences with 0 for n N ≥ .Let χ be the set of all

1 α∑
is a contraction mapping, so T has a unique fixed point x S ∈ such that Tx x = .It is easy to see that solution of Equation (1).This complete the proof of the theorem. Theorem 3.2.Assume that 0 Let χ be the set of all bounded real sequences defined for all 0

Example 4 . 1 .
Consider the difference equations

4 . 2 .
Further it is easy to verify that all other conditions of Theorem 2.1 are satisfied.Therefore every solution of Equation (28) is oscillatory.Example Consider the difference equations 2 k = , we see that k σ τ ≥ − .Further it is easy to verify that all other conditions of Theorem 2.4 are satisfied.Therefore every solution of Equation (29) is oscillatory.
The proof is similar to that of Lemma 2.1. Lemma 2.4.Let { } n x be an eventually positive solution of Equation (1) and suppose Case (I) of Lemma 2.1 holds.Then there exists ( ) bounded real sequences defined for all By the Mean Value Theorem applied to the function ( ) It is easy to see that T is continuous, TS S ⊂ , and for any ,