Asymptotic Behavior of Higher Order Quasilinear Neutral Di ff erence Equations

We study, the asymptotic behavior of solutions to a class of higher order quasilinear neutral difference equations under the assumptions that allow applications to even and odd-order difference equations with delayed and advanced arguments, as well as to functional difference equations with more complex arguments that may for instance, alternate infinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

(c 3 ) α and β ∈ R where R stands for the set consisting of all quotients of odd positive integers.
Analysis of qualitative properties of equation ( 1) is important not only for the sake of further development of the oscillation theory, but for practical reasons too.In fact, an Emden-Fowler type differential equation ( r(t) has numerous applications in physics and engineering: that is; for instance, the papers by Ou and Wong [2004].
By a solution of equation ( 1), we mean a real sequence {x(n)} which is defined for n ≥ −µ = max{sup{τ(n), σ(n)} and which satisfies the equation (1) for n ≥ n 0 .We deal only with proper solution x(n) of (1) that satisfy the condition sup{|x(n)| : n ≥ N} > 0 for all N ≥ N µ and tacitly assume that (1) possesses such solutions.A solution of ( 1) is said to be oscillatory if it has arbitrarily large zeros on n ≥ N µ .Otherwise, it is termed nonoscillatory.
S.S. Cheng and W.T. Patula [1993] studied the difference equation where p > 1 and proved an existence theorem for the equation (3).
X. Zhou and J. Yan [1998] studied the difference equation and they obtained some comparison results and necessary and sufficient conditions for the oscillation of equation ( 4).
I. Kvbiaczyk and S.H. Sekar [2002] studied the second order sublinear delay difference equation By using Riccati transformation techniques the authors obtained oscillation criteria for the equation ( 5) under the conditions Y. Bolat and O. Alzabot [2012] considered the half-linear delay difference equation under the condition < ∞ and with using that ∆p n ≥ 0 and derived some oscillation and asymptotic criteria for the equation (6).M.K. Yildiz and O. Ocalan [2007] studied the neutral difference equation where 1 > α > 0 is a quotient of odd positive integers and {p n } satisfies −1 < p n < 1.
J. Luo in 2002Luo in [2002] ] considered the second order quasilinear neutral delay difference equation ∞ and obtained several oscillation results.
Pon. Sundar and E. Thandapani [2000], considered the second order quasilinear functional difference equation and established some necessary and sufficient conditions for the equation ( 9) to have various types of nonoscillatory solutions.
Our principal goal is to analyze the asymptotic behavior of solutions to (1) in the case where the condition holds.We provide sufficient conditions which ensure that solution to (1) are either oscillatory or approach zero at infinity.In some cases, we reveal oscillatory nature of (1).
As usual, all functional inequalities are supposed to hold for all t large enough.Without loss of generality, we deal only with positive solutions of (1) since, under our assumptions, if x(n) is a solution, then −x(n) is a solution of this equation too.
In the sequel, we denote by τ −1 the function which is inverse to τ.We also adopt the following notations for the compact presentation of our results.
where the meaning of γ, θ, η i and η 3 will be explained later.

Asymptotic Behavior of Solutions to Even-order Equations
In what follows, τ(n) can be both a delayed or an advanced argument.Throughout this section, in addition to the basic assumptions listed in the introduction, it is also supposed that (10) holds along with (c 4 ) 0 ≤ p(n) ≤ p 0 < ∞ for some constant p 0 ; We shall need the following lemmas which are useful is the sequel.
Lemma 1. [Gyori and Ladas, 1991] Assume that q(n) ≥ 0 for all n ∈ N and Then, Lemma 2. [Agarwal, 1992](Discrete Kingser's Theorem) Let z(n) be defined for n ≥ a, and z(n) > 0 with ∆ m z(n) of constant sign for n ≥ a and not identically zero.Then, there exists an integer j, 0 ≤ j ≤ m with (m + j) odd for ∆ m z(n) ≤ 0 and (m + j) even for Lemma 3. [Agarwal, 1992] Let z(n) be defined for n ≥ a and z(n) > 0 with ∆ m z(n) ≤ 0 for n ≥ a and not identically zero.Then, there exists a large n 1 ≥ a such that and j is defined us in Lemma 2. Further z(n) is increasing, then Lemma 4. Assume that A ≥ 0, B ≥ 0, α ≥ 1.Then Proof.If A = 0 or B = 0, then the inequality holds trivially.For A 0, setting x = B A .The inequality takes the form (1 + x) α ≥ 1 + x α which is for x > 0 evidently true.
Theorem 6.Let m ≥ 2 be even and 0 < β ≤ 1. Assume that conditions (c 4 ) and (c 5 ) are satisfied, and there exist two real numbers γ, λ ∈ R such that γ ≤ β ≤ λ and γ < α < λ.Suppose further that there exist two sequence η Then every solution x(n) of (1) is either oscillatory or satisfies Proof.Assume that equation (1) has a nonoscillatory solutions x(n) which is eventually positive and such that In view of (1), we have Using ( 22), we obtain It follows from equation ( 1), ( 22) and ( 23) that Then there exist two possible cases: for n ≥ n 1 where n 1 ≥ n 0 is large enough.
Case 1: Suppose first that conditions (25) hold.Using inequality ( 24) and assumption η 1 (n) ≤ σ(n), we conclude that Furthermore, by the monotonicity of z(n), there exists a constants M > 0 such that Combining ( 27) and ( 28), we have where M 1 = M β−γ .An application of conditions (25) allows us to deduce that the sequence is positive and nonincreasing.By Lemma 3, we have for every λ ∈ (0, 1) and for sufficiently large n.Using (31) in ( 29), we conclude that w(n) is a positive solution of a delay difference inequality Define now a function y(n) by Then by the monotonicity of w(n), Substituting ( 34) into (32), we observe that y(n) is a positive solution of a delay difference inequality Then, by virtue of [Gyori and Ladas, 1991], the associated delay difference equation also has a positive solution.However, by Lemma 1 implies that, under assumption (17), equation ( 36) is oscillatory.Therefore, (1) cannot have positive solutions.
Case 2: Assume now that conditions (26) hold.By virtue of (20), we have that An application of Lemma 3 yields for any λ ∈ (0, 1) and for sufficiently large n.Hence, by ( 24) and ( 38), we obtain Using conditions ∆ m−1 z(n) < 0, σ(n) ≤ η 2 (n), and inequality (39), we have Furthermore, by the monotonicity of ∆ m−2 z(n), there exists a constant N > 0 such that Combining ( 40) and ( 41), we arrive at where N 1 = N β−λ .Using the monotonicity of w(n), for s ≥ n ≥ n 1 we conclude that Dividing ( 43) by r 1/α (s) and summing the resulting inequality from n to l − 1, we obtain Passing to the limit as l → ∞, we deduce that Combining ( 42) and ( 46), we have Using the monotonicity of w(n), we conclude that Using ( 48) into (47), we observe that y(n) is a negative solution of an advanced difference inequality consequently, by virtue of [Gyori and Ladas, 1991] the associated advanced difference equation also has a positive solution.However, by [Sundar and Murugesan, 2010] implies that, under assumption (18) equation ( 51) is oscillatory.Therefore, (1) cannot have a positive solution this contradiction with initial assumption completes the proof.
Theorem 7. Let m ≥ 2 be even, and let 0 < α = β ≤ 1. Assume that conditions (c 4 ) and (c 5 ) hold, and there exist two sequences η 1 (n), η 2 (n) satisfying (16).Suppose also that conditions where k is the delay argument and where l is the advance argument are satisfied.Then conclusion of Theorem 6 remains in fact.
Proof.Assume that x(n) is an eventually positive solution of equation ( 1) that satisfies (20).Proceeding as in the proof of Theorem 6, one comes to the conclusion that, for every λ ∈ (0, 1), a delay difference equation and an advanced difference equation both have positive solutions.On the other hand, condition (52) and [Gyori and Ladas, 1991] imply that equation ( 54) is oscillatory, a contradiction.Likewise, by virtue of [Sundar and Murugesan, 2010, Lemma 2.3.2]condition (53) yields that equation ( 55) is oscillatory.This contradiction completes the proof.
Proof.As above let x(n) be an eventually positive solution of equation ( 1) that satisfies (20).As in the proof of Theorem 6, we split the arguments into two parts.
Case 1: Assume first that (25) is satisfied.If has been established in the proof of Theorem 6 that the sequence w(n) defined by ( 30) is positive, nonincreasing and satisfies inequality (32).Introducing again y(n) by ( 33) and using the monotonicity of w(n), we conclude that substituting of ( 57) into (32) implies that, for sufficiently large n, y(n) is a positive solution of a delay difference inequality Then, the associated difference equation also has a positive solution.However, by [Sundar and Murugesan, 2010, Lemma 2.3.2]implies that , under assumption ( 17) equation ( 59) is oscillatory.Therefore equation ( 1) cannot have positive solutions.
Case 2: Assume that ( 26) is satisfied.It has been established in the proof of Theorem 6 that the sequence w(n) defined by (30) is negative, nonincreasing and satisfies the inequality (47).Introducing again y(n) by ( 33) and using the monotonicity of w(n), we conclude that substituting ( 60) into (47), we observe that y(n) is a negative solution of an advanced difference inequality Then, by virtue of [Gyori and Ladas, 1991] the associated advanced difference equation also has a positive solution.However, by [Sundar and Murugesan, 2010, Lemma 2.3.2]implies that; under assumption (18), equation ( 63) is oscillatory.Therefore, equation (1) cannot have positive solutions.This contradiction with our initial assumption complete the proof.
Theorem 9. Let m ≥ 2 be even and 0 < α = β ≤ 1. Assume that conditions (c 4 ) and (c 5 ) are satisfied, and there exist two real sequences η 1 (n), η 2 (n) satisfying (56).Suppose also that 1 where k denotes the delay arguments and where l denotes the advance arguments, are satisfied.Then conclusion of Theorem 6 remains infact.
Proof.Assume that x(n) is an eventually positive solution of equation ( 1) that satisfies (20) and proceeding as in the proof of Theorem 8, one concludes that; for every λ ∈ (0, 1) a delay difference equation and an advanced difference equation have positive solutions.On the other hand, application of condition (64) along with [Gyori and Ladas, 1991] imply that equation ( 66) is oscillatory, a contradiction.Likewise, by virtue of [Sundar and Murugesan, 2010, Lemma 2.3.2]condition (65) yields that equation ( 67) is oscillatory.This contradiction completes the proof.
Note that Theorems 6 to 9 ensure that every solution x(n) of equation ( 1) is either oscillatory or tends to zero as n → ∞ and unfortunately cannot distinguish solutions with different behaviors.In the remaining part of this section, we establish several results which guarantee that all solutions of equation ( 1) are oscillatory.
Proof.Without loss of generality, suppose that x(n) is a nonoscillatory solution of equation ( 1) which is eventually positive.As in the proof of Theorem 6, we obtain (24).In view of equation ( 1) and Lemma 2, in addition to the case (25), there are two more possible types of behavior of solutions for n ≥ n 1 where n 1 ≥ n 0 is large enough in the proof of Theorem 6. Namely, one can also have for all odd integers j ∈ {1, 2, • • • , m − 3}.However conditions ( 17) and ( 18) yields that neither ( 24) nor ( 70) is possible.
Therefore, we have to analyze the only remaining case, and we assume now that all the conditions in (71) are satisfied.Then, inequality (46) holds.Summing (46) from n to ∞ (n − 2) times, we obtain where w(n) is defined by ( 30).Taking into account that ∆z(n) < 0, σ(n) ≤ η 3 (n) and using ( 24), we have By virtue of monotonicity of z(n), there exists a constant M 2 > 0 such that combining ( 73) and ( 74), we obtain Using ( 72) in ( 75), we conclude that in this case, the sequence w(n) defined by ( 30) is negative, nonincreasing, and satisfies the inequality Introduction again y(n) by ( 33) and using the monotonicity of w(n), we arrive at (48).Substitution of ( 48) into (76) leads to the conclusion that y(n) is a negative solution of an advanced difference equation In which case the function u(n) = −y(n) is a positive solution of an advanced difference inequality Then, by [Gyori and Ladas, 1991], the associated advanced difference equation also has a positive solution.However, by [Sundar and Murugesan, 2010, Lemma 2.3.2]implies that ( 79) is oscillatory under assumption (69).Therefore, equation (1) cannot have positive solutions.This contradiction with our initial assumption completes the proof.
Proof.Let x(n) be a nonoscillatory solution of (1) which is eventually positive.As in the proof of Theorem 10, one can have either ( 25) or ( 70) or (71).However, conditions ( 51) and ( 53) exclude cases ( 25) and ( 70).Then all the inequalities in (71) should be satisfied.Along the same lines as in the proof of Theorem 10, one comes to the conclusion that an advanced difference equation has positive solutions.On the other hand, if condition (80) holds, then by virtue of [Sundar and Murugesan, 2010] implies that equation ( 80) is oscillatory.This contradiction completes the proof.
Proof.Let x(n) be an eventually positive nonoscillatory solution of equation ( 1).The same argument as in the proof of Theorem 10 yields that (71) holds.Define the sequence w(n) by ( 30).From the proof of Theorem 10, we already know that w(n) is negative, nonincreasing and satisfies the inequality (76).Introducing the sequence y(n) by ( 32) and using the monotonicity of w(n), we arrive at (60).Substituting ( 60) into (76) we observe that y(n) is a negative solution of an advanced difference inequality while u(n) = −y(n) is a positive solution of an advanced difference inequality In this case, the result due to Gyori [1991] allows one to deduce that the associated advanced difference equation also has a positive solution.However, it has been established [Sundar and Murugesan, 2010, Lemma 2.3.2] that if condition ( 69) is satisfied, then equation ( 83) is oscillatory.Therefore equation ( 1) cannot have positive solutions, and this contradiction with the assumptions of the theorem completes the Theorem 13.Let m ≥ 4 be even and 0 < α = β ≤ 1. Assume that conditions (c 4 ) and (c 5 ) are satisfied, and there exist three real sequences η 1 (n), η 2 (n) and η 3 (n) as in Theorem 12. Suppose further that (64), ( 65) hold, and where l denotes the advanced argument.Then equation (1) is oscillatory.
Proof.Assume that x(n) is an eventually positive nonoscillatory solution of equation ( 1) and increasing as in the proof of Theorem 10 one concludes that (71) holds.As in the proof of Theorem 12 we observed that an advance difference equation has positive solution.On the other hand, if condition ( 84) is satisfied, a result reported by [Sundar and Murugesan, 2010, Lemma 2.3.2]yields that equation ( 85) is oscillatory.This contradiction completes the proof.

Asymptotic Behavior of Solutions to Odd-Order Equations
In this section, in addition to conditions (c 4 ), (c 5 ) and ( 10), we also assume that The validity of the following four propositions can be established in the same manner as it has been done for Theorems 6, 9. Therefore, to avoid unnecessary repetition, we only formulate counterparts of Theorems 9 and following for the case of odd-order equations.
Theorem 14.Let m ≥ 3 be odd and 0 < β ≤ 1. Assume that conditions (c 4 ) − (c 6 ) are satisfied, and there exist two real numbers γ, λ ∈ R as in Theorem 6 and a real sequence Then the conclusion of Theorem 6 remains intact.
Theorem 15.Let m ≥ 3 be odd, and let 0 < α = β ≤ 1. Assume that conditions (c 4 ) − (c 6 ) are satisfied, and there exists a real sequence η 4 (n) as in Theorem 14. Suppose also that Then the conclusion of Theorem 6 remains intact.
Note that Theorems 14-17 apply only if σ is a delayed argument, σ(n) < n.Hence it is important to complement such results with the following theorems that can be applied in the case where σ is an advanced argument, σ(n) ≥ n.
Proof.Assume that equation ( 1) has an eventually positive solution x(n) satisfying (20).Proceeding as in the proof of Theorem 6, we arrive at (23) and observe that equation (1) yields that either (24) or (25) holds.Indeed, it follows from the condition ∆ this immediately leads us to conditions (25).

Examples and Discussions
The following examples illustrate applications of some of theoretical results presented in the previous sections.In all the examples, p 0 is a constant such that 0 ≤ p 0 < ∞.Remark 1.By using inequality which holds for any β ≥ 1 and for all x 1 , x 2 ∈ [0, ∞), results reported in this paper can be extended to equation (1) for all β ∈ R which satisfy β > 1.In this case one has to replace Q(n) = min{q(n), q(τ(n))} with a function Q(n) = 2 1−β min{q(n), q(τ(n))} and proceed in above.
Remark 2. Our main assumptions on functional arguments do not specify whether τ(n) is delayed or an advanced argument.Remarkably, σ(n) can even switch its nature between an advanced and delayed argument.However, such flexibility is achieved at the cost of requiring that the function τ is monotonic and satisfies τ • σ = σ • τ.The question regarding the analysis of the asymptotic behavior of solutions to (1) with other methods that do not require these assumptions remains open at the moment.