Some Oscillation Results for Even Order Delay Difference Equations with a Sublinear Neutral Term

and Applied Analysis 3 Since Δzn > 0 and zn > 0 eventually, there exists a positive constant M such that zn−l ≥ M for all n ≥ n2. Using this and the positivity of anΔm−1zn in (18) and letting n 󳨀→ ∞, we obtain ∞ ∑ n=n1 Qn < ∞ (19) which is a contradiction to (12).This completes the proof. Remark 6. In the above theorem, we did not impose any condition on β and hence our result is more general than some of the existing results in the literature. In the following, we present other oscillation criteria using Lemma 3. Theorem 7. Let condition (2) hold. Assume that there is a positive decreasing real sequence {ρn} tending to zero such that P(n) is positive for all n ≥ N ∈ N(n0). If (i)

Let  = max{, ℓ}.Under a solution of (1), we mean a real sequence {  } defined for  ≥  0 −  and satisfying (1) for all  ∈ N( 0 ).As usual a solution of ( 1) is said to be oscillatory if it is neither eventually positive nor eventually negative; else it is nonoscillatory.
In the past few years, there is a great interest in studying the oscillatory and asymptotic behavior of solutions of higher order neutral type difference equations, since such type of equations naturally arises in the applications including problems in population dynamics or in cobweb models in economics and so on.The problem of finding sufficient conditions which ensure that all solutions of the neutral type difference equations are oscillatory has been investigated by many authors; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12] and the references cited therein.In all the results the neutral term is linear and few results are available when the neutral term is nonlinear; see [13][14][15][16][17][18][19][20][21].
In [20], the authors considered (1) with  ≥ 1 and   ≡ 1 and established sufficient conditions for the oscillation of all solutions.In view of these facts, in this paper our purpose is to obtain sufficient conditions for the oscillation of solution of (1) when or Thus the results presented here extend and generalize some of the results in [13,14,16,18,19,21], complement the results in [20], and correct some of the results in [8].
Lemma 2 (Discrete Kneser's Theorem).Let {  } be a sequence of real number and   > 0 with Δ    being of constant sign eventually and not identically zero eventually.en there exists an integer , 0 ≤  ≤ , with ( + ) odd for Δ    ≤ 0 and ( + ) even for Δ    ≥ 0 and  ∈ N( 0 ) such that and for all  ≥ .

Lemma 3.
Let {  } be defined for  ∈ N( 0 ) and   > 0 with Δ    ≤ 0 for all  ∈ N( 0 ) and not identically zero.en there exists a large  1 ∈ N( 0 ) such that where  is defined in Lemma .Further, if {  } is increasing, then The proof of the last two lemmas can be found in [1].
Next we define the sequence {  } by Lemma 4. Assume condition ( ) holds.Let {  } be a positive solution of ( ). en there is an integer  1 ∈ N( 0 ) such that for all  ≥  1 .
Proof.The proof is similar to that of Lemma 3 of [8], and hence the details are omitted.

Oscillation Theorems
In this section, we present some sufficient conditions for the oscillation of all solutions of (1).To simplify our notation, for any positive real sequence {  } which is decreasing to zero, we set and Theorem 5. Let condition ( ) hold.Assume that there is a positive decreasing real sequence {  } tending to zero such that () is positive for all  ≥  ∈ N( 0 ).
then every solution of ( ) is oscillatory.
Proof.Let {  } be a nonoscillatory solution of (1).Without loss of generality, we may assume that {  } is a positive solution of (1).Then there exists an integer From the definition of   , we have where we have used Lemma 1.Since   is positive and increasing and   is positive and decreasing to zero, there is an integer  2 ≥  1 such that Using ( 14) in ( 13), one obtains and substituting this in (1) yields Now summing the last inequality from  2 to  − 1, we obtain for all  ≥  2 .That is Since Δ  > 0 and   > 0 eventually, there exists a positive constant  such that  −ℓ ≥  for all  ≥  2 .Using this and the positivity of   Δ −1   in (18) and letting  → ∞, we obtain which is a contradiction to (12).This completes the proof.
Remark .In the above theorem, we did not impose any condition on  and hence our result is more general than some of the existing results in the literature.
In the following, we present other oscillation criteria using Lemma 3.
Theorem 7. Let condition ( ) hold.Assume that there is a positive decreasing real sequence {  } tending to zero such that () is positive for all  ≥  ∈ N( 0 ).
Proof.Assume that (1) has a nonoscillatory solution {  } which is eventually positive such that lim →∞   ̸ = 0. From the definition of   , we have   > 0 for all  ≥  1 ∈ N( 0 ).By virtue of (1) and Lemma 2 there are two possibilities, either Case (i).Suppose conditions (28) hold for all  ≥  1 ; then the proof for this case is similar to that of Case (i) of Theorem 7 and hence the details are omitted.

Case (ii)
. Assume now that conditions (29) hold for all  ≥  1 .Since   Δ −1   is decreasing, then we have Dividing the last inequality by   and summing the resulting inequality from  to  − 1, we obtain Letting  → ∞, we obtain Define and then   > 0, and using ( 16), we have where we have used Δ −2   as positive and decreasing.Now using (  − 1)/  =   Δ −1   /Δ −2   in the above inequality, it follows that Now from Lemma 3, we obtain Since Δ −1   < 0 and  − ℓ <  + 1, we have Combining the inequalities (35) and (37), we have where  = (1/( − 2)!)(1/2 −2 ) −2 .Completing the square in the above inequality, we have By summing the last inequality from  1 to , we obtain Taking lim sup as  → ∞, in the above inequality we obtain a contradiction with (27).This completes the proof.
Theorem 9. Assume that ( ) and 0 <  < 1 hold.Further assume that there is a positive decreasing real sequence {  } tending to zero such that () is positive for all  ≥  ∈ N( 0 ).

If ( ) holds and
for some constant  1 > 0, then every solution of ( ) either is oscillatory or tends to zero as  → ∞.
Proof.Assume that {  } is an eventually positive solution of (1) such that lim →∞   ̸ = 0. Proceeding as in the proof of Theorem 8, we see that {  } satisfies two possible cases (28) and (29) for all  ≥  1 .
Case (i).Suppose conditions (28) hold for all  ≥  1 ; then the proof for this case is similar to that of Case (ii) of Theorem 7 and hence the details are omitted.

Case (ii)
. Assume now that conditions (29) hold for all  ≥  1 ; proceeding as in Case (ii) of Theorem 8 we have Now using (36) and (37) in (43), we obtain Since {Δ −2   } is positive and decreasing and  < 1, there is a constant for all  ≥  2 ≥  1 .Using this in (44) and then summing the resulting inequality from  2 to , we obtain Taking lim sup as  → ∞, in the above inequality, we obtain a contradiction with (42).This completes the proof.
Theorem 10.Assume that ( ) and  > 1 hold.Further assume that there is a positive decreasing and sequence {  } tending to zero such that () is positive for all  ≥  ∈ N( 0 ).If ( ) holds and for some constant  2 > 0, then every solution of ( ) either is oscillatory or tends to zero as  → ∞.
Proof.Let us assume that {  } is an eventually positive solution of (1) such that lim →∞   ̸ = 0. Proceeding as in the proof of Theorem 8, we see that {  } satisfies two possible cases (28) and ( 29) for all  ≥  1 .

Case (i).
If conditions (28) hold for all  ≥  1 , then the proof is similar to that of Case (iii) of Theorem 7 and hence the details are omitted.

Case (ii)
. Assume now that conditions (29) hold for all  ≥  1 .Proceeding as in Case (ii) of Theorem 9, we have Now from (32), one can see that Δ −2   /  is nondecreasing and hence there is a constant  2 > 0 such that Δ −2   /  ≥  2 for all  ≥  1 .Using this in (47) and since  > 1, we have Summing the last inequality from  1 to , we obtain Taking lim sup as  → ∞ in the above inequality, we get a contradiction with (46).This completes the proof.

Examples
In this section, we present two examples to illustrate the importance of the main results.

Conclusion
The results obtained in this paper extend and complement some of the results reported in the literature.Further, Theorem 8, where  = 1, corrects the conclusion of Theorem 4 established in [8].The results reported in the papers [3, 4, 6-12, 17, 20] cannot be applicable to (50) and (51) to yield this conclusion since these equations have sublinear neutral terms.It would be interesting to improve Theorems 8, 9, and 10 so that all solutions are oscillatory instead of either being oscillatory or tending to zero.