On the Oscillation of Even-Order Half-Linear Functional Difference Equations with Damping Term

Abstract

We investigate the oscillatory behavior of solutions of the 𝑚th order half-linear functional difference equations with damping term of the form Δ[𝑝_𝑛𝑄(Δ^𝑚−1𝑦_𝑛)]+𝑞_𝑛𝑄(Δ^𝑚−1𝑦_𝑛)+𝑟_𝑛𝑄(𝑦_𝜏𝑛)=0, 𝑛≥𝑛₀, where 𝑚 is even and 𝑄(𝑠)=|𝑠|^𝛼−2𝑠, 𝛼>1 is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.

1. Introduction

Consider the second order half-linear difference equation: Δ𝑝_𝑛Δ𝑦_𝑛^𝛼−2Δ𝑦_𝑛+𝑟_𝑛𝑦_𝑛+1^𝛼−2𝑦_𝑛+1=0,𝑛≥𝑛₀,𝛼>1, (1) where Δ is the forward difference operator and {𝑝_𝑛}, {𝑟_𝑛} are sequences of nonnegative real numbers with {𝑝_𝑛}>0. The study of (1) has been initiated by Rehák in [1]. It is well known that there is a close similarity between (1) and the linear second order difference equation. Indeed, if {𝑦_𝑛} is a solution of (1), then so is {𝑐𝑦_𝑛} for any constant 𝑐. Thus, (1) has one half of linearity properties [2].

In the presence of damping, (1) has been extended further to the second order half-linear difference equation with damping term of the form Δ𝑝_𝑛Δ𝑦_𝑛^𝛼−2Δ𝑦_𝑛+𝑞_𝑛Δ𝑦_𝑛^𝛼−2Δ𝑦_𝑛+𝑟_𝑛𝑦_𝑛+1^𝛼−2𝑦_𝑛+1=0,𝑛≥𝑛₀∈ℕ, (2) where {𝑞_𝑛} is a sequence of nonnegative real numbers. It is to be noted that neither (1) nor (2) has involved a delaying term. There are numerous numbers of oscillation criteria established in the literature for the solutions of (1) and (2). Most of these results were obtained by using certain efficient tools among them we name the Riccati transformation, variational principle, and some inequality techniques; see, for instance, the monograph [3] in which many contributions have been cited therein and to the recent papers [4–9].

Let 𝑄∶ℝ→ℝ be defined by 𝑄(𝑠)=|𝑠|^𝛼−2𝑠; 𝛼>1 is a fixed real number and ℕ_𝑛0={𝑛₀,𝑛₀+1,…}. Consider the 𝑚th order half-linear functional difference equation with damping term of the form Δ𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛+𝑞_𝑛𝑄Δ^𝑚−1𝑦_𝑛+𝑟_𝑛𝑄𝑦_𝜏𝑛=0,𝑛∈ℕ_𝑛0, (3) where 𝑚 is even number, and (H1){𝑝_𝑛}∶ℕ_𝑛0→ℝ⁺ with Δ𝑝_𝑛≥0 for all 𝑛≥𝑛₀;(H2){𝑞_𝑛} and {𝑟_𝑛}∶ℕ_𝑛0→ℝ with 𝑞_𝑛≥0 and 𝑟_𝑛>0;(H3){𝜏_𝑛}∶ℕ_𝑛0→ℤ with 𝜏_𝑛<𝑛 and lim_𝑛→∞𝜏_𝑛=∞.For close results regarding the continuous counterparts of (1), (2), and (3), the reader is suggested to consult [10–14].

A primary purpose of this paper is to establish sufficient conditions that guarantee the oscillation of solutions of (3). Our main results are obtained via employing the generalized Riccati transformation. In view of (3), one can easily figure out that it is formulated in more general form so that it includes some particular cases which have been studied in the literature; see [15–23] for more details. To the best of authors’ observation, however, no published result has been concerned with the investigation of oscillatory behavior of solutions of (3) or its continuous counterpart. Therefore, our paper is new and presents a new approach.

2. Main Results

We start by recalling the following standard definitions.

Definition 1. A nontrivial sequence 𝑦_𝑛 is called a solution of (3) if it is defined for all 𝑛≥𝜎 where 𝑛∈ℤ, 𝜎=min_𝑖≥𝑛0{𝜏_𝑖}, and 𝑝_𝑛𝑄(Δ^𝑚−1𝑦_𝑛) is differenceable on ℕ_𝑛0 and satisfies (3) for all 𝑛∈ℕ_𝑛0.

Definition 2. A nontrivial solution 𝑦_𝑛 of (3) is said to be oscillatory if the terms of the sequence 𝑦_𝑛 are not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A difference equation is called oscillatory if all its solutions oscillate.

To obtain our main results, we need the following essential lemmas. The first of these is the discrete analogue of the well-known Kiguradze’s lemma.

Lemma 3 (see [24]). Let 𝑦_𝑛 be defined for 𝑛≥𝑛₀∈ℕ and 𝑦(𝑛)>0 with Δ^𝑚𝑦_𝑛 of constant sign for 𝑛≥𝑛₀ and not identically zero. Then, there exists an integer 𝑙, 0≤𝑙≤𝑚 with (𝑚+𝑙) odd for Δ^𝑚𝑦_𝑛≤0 and (𝑚+𝑙) even for Δ^𝑚𝑦(𝑛)≥0 such that(i)𝑙≤𝑚−1 implies (−1)^𝑙+𝑖Δ^𝑖𝑦_𝑛>0 for all 𝑛≥𝑛₀, 𝑙≤𝑖≤𝑚−1,(ii)𝑙≥1 implies Δ^𝑖𝑦_𝑛>0 for all large 𝑛≥𝑛₀, 1≤𝑖≤𝑙−1.

Lemma 4 (see [25]). Let 𝑦_𝑛 be defined for 𝑛≥𝑛₀ and 𝑦_𝑛>0 with Δ^𝑚𝑦_𝑛≤0 for 𝑛≥𝑛₀ and not identically zero. Then, there exists a large integer 𝑛₁≥𝑛₀ such that 𝑦_𝑛≥1(𝑚−1)!𝑛−𝑛₁^𝑚−1Δ^𝑚−1𝑦_{2𝑚−𝑙−1𝑛},𝑛≥𝑛₁, (4) where 𝑙 is defined as in Lemma 3. Further, if 𝑦_𝑛 is increasing, then 𝑦_𝑛≥1()!𝑛𝑚−12^{𝑚−1𝑚−1}Δ^𝑚−1𝑦_𝑛,𝑛≥2^𝑚−1𝑛₁. (5)

Lemma 5. Let 𝑦_𝑛 satisfy conditions of Lemmas 3 and 4 and Δ^𝑚−1𝑦_𝑛Δ^𝑚𝑦_𝑛≤0 for 𝑛≥𝑛₁≥𝑛₀. Further, if 𝑦_𝑛 is increasing, then Δ𝑦_𝑛−𝑘≥𝑀𝑛^𝑚−2Δ^𝑚−1𝑦_𝑛,𝑛≥𝑛₁, (6) where 𝑀=(1/((𝑚−1)!2^(𝑚−1)2))>0.

The proof of Lemma 5 is straightforward and it can be achieved by using the last inequality of Lemma 4.

Lemma 6. Let 𝑦_𝑛 be an eventually positive solution of (3). If (Λ1)lim_𝑛→∞^𝑛−1_𝑠=𝑛11𝑝_𝑠1−^𝑠−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘^1/(𝛼−1)=∞, (7) then Δ^𝑚−1𝑦_𝑛>0, Δ^𝑚𝑦_𝑛≤0, and Δ𝑦_𝑛>0 for all 𝑛≥𝑛₁≥𝑛₀.

Proof. The fact that 𝑦_𝑛 is eventually positive solution of (3) implies 𝑦_𝑛>0 and 𝑦_𝜏𝑛>0 for all 𝑛≥𝑛₁≥𝑛₀. In view of (3), we get Δ𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛+𝑞_𝑛𝑄Δ^𝑚−1𝑦_𝑛<0, (8) which leads to Δ𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1𝑄Δ^𝑚−1𝑦_𝑛1<0. (9) Hence, 𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1𝑄Δ^𝑚−1𝑦_𝑛1 (10) is decreasing and Δ^𝑚−1𝑦_𝑛 is eventually positive or eventually negative.
We claim that Δ^𝑚−1𝑦_𝑛>0,𝑛≥𝑛₁. (11) Assume, on the contrary, that Δ^𝑚−1𝑦_𝑛<0, 𝑛≥𝑛₁. Then, from (10), we obtain 𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−2Δ^𝑚−1𝑦_𝑛−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1≤𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1−^𝑛1−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1, (12) where ∏^𝑛1−1_𝑘=𝑛1(1−(𝑞_𝑘/𝑝_𝑘))𝑝_𝑛1|Δ^𝑚−1𝑦_𝑛1|^𝛼−2Δ^𝑚−1𝑦_𝑛1=0. Therefore, from (12), we have 𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−2Δ^𝑚−1𝑦_𝑛−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1≤𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1−^𝑛1−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘𝑝_𝑛1Δ^𝑚−1𝑦_𝑛1^𝛼−2Δ^𝑚−1𝑦_𝑛1≡−𝑀₁^𝛼−1, (13) where 𝑀₁=𝑝_𝑛^1/(𝛼−1)₁|Δ^𝑚−1𝑦_𝑛1|>0. It follows that −Δ^𝑚−1𝑦_𝑛^𝛼−1≥𝑀₁^𝛼−1𝑝_𝑛1−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘 (14) or Δ^𝑚−1𝑦_𝑛≤−𝑀₁1𝑝_𝑛1−^𝑛−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘^1/(𝛼−1). (15) Consequently, we obtain Δ^𝑚−2𝑦_𝑛≤Δ^𝑚−2𝑦_𝑛1−𝑀₁^𝑛−1_𝑠=𝑛11𝑝_𝑠1−^𝑠−1_𝑘=𝑛1𝑞1−^𝑘𝑝_𝑘^1/(𝛼−1). (16) Letting 𝑛→∞ in the above inequality, one gets lim_𝑛→∞Δ^𝑚−2𝑦_𝑛=−∞. Hence, 𝑦_𝑛 is an eventually negative function which contradicts that 𝑦_𝑛>0. Therefore, inequality (11) holds.
From (3), we get Δ𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛=𝑝_𝑛+1ΔΔ^𝑚−1𝑦_𝑛^𝛼−1+Δ^𝑚−1𝑦_𝑛^𝛼−1Δ𝑝_𝑛≤0 (17) from which it follows that ΔΔ^𝑚−1𝑦_𝑛^𝛼−1≤0,𝑛≥𝑛₁≥𝑛₀. (18) The above inequality implies that (Δ^𝑚−1𝑦_𝑛)^𝛼−1 is nonincreasing. Therefore, we can write ΔΔ^𝑚−1𝑦_𝑛^𝛼−1=Δ^𝑚−1𝑦_𝑛+1^𝛼−1−Δ^𝑚−1𝑦_𝑛^𝛼−1=Δ^𝑚−1𝑦_𝑛+1−Δ^𝑚−1𝑦_𝑛×Δ^𝑚−1𝑦_𝑛+1^𝛼−2+Δ^𝑚−1𝑦_𝑛+1^𝛼−3Δ^𝑚−1𝑦_𝑛+Δ^𝑚−1𝑦_𝑛+1^𝛼−4Δ^𝑚−1𝑦_𝑛²+⋯+Δ^𝑚−1𝑦_𝑛^𝛼−2≤0. (19) Since (Δ^𝑚−1𝑦_𝑛)^𝛼−1 is nonincreasing and positive, then from the above inequality, we have ΔΔ^𝑚−1𝑦_𝑛^𝛼−1≤Δ^𝑚−1𝑦_𝑛+1−Δ^𝑚−1𝑦_𝑛(𝛼−1)Δ^𝑚−1𝑦_𝑛^𝛼−2=(𝛼−1)ΔΔ^𝑚−1𝑦_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−2≤(𝛼−1)Δ^𝑚𝑦_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−2≤0 (20) by which we have Δ^𝑚𝑦_𝑛≤0. (21) In virtue of (21) and Lemma 3, we deduce that since 𝑚 is even then 𝑙 is odd. Hence Δ𝑦_𝑛>0 for 𝑛≥𝑛₁≥𝑛₀. The proof is complete.

Theorem 7. Let condition (Λ1) hold. Further, assume that there exists a constant 𝜆>𝛼−1 such that (Λ2)limsup_𝑛→∞1𝑛−𝑛₁^𝜆𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘−(𝛼−1)^𝛼−1𝛼^𝛼𝑋^𝛼_𝑛𝑘𝑌^1−𝛼_𝑛𝑘=∞, (22) where 𝑋_𝑛𝑘=(𝑛+1−𝑘)^𝜆−(𝑛−𝑘)^𝜆𝑞1+^𝑘𝑝_𝑘+1,𝑌_𝑛𝑘=(𝛼−1)𝑀𝜏_𝑘^𝑚−2(𝑛−𝑘)^𝜆𝑝_𝑘𝑝_𝑘+1>0, (23) and 𝑀 is as in Lemma 5. Then, (3) is oscillatory.

Proof. For the sake of contradiction, assume that (1) has a nonoscillatory solution 𝑦_𝑛. Without loss of generality, we assume that 𝑦_𝑛 is eventually positive (the proof is similar when 𝑦_𝑛 is eventually negative). That is, 𝑦_𝑛>0,𝑦_𝜏𝑛>0 and 𝑦_{𝜏𝑛−𝑘}>0 for all 𝑛≥𝑛₁≥𝑛₀. By Lemma 6, we have Δ^𝑚−1𝑦_𝑛>0, Δ^𝑚𝑦_𝑛≤0, and Δ𝑦_𝑛>0 for 𝑛≥𝑛₁. Consider the function 𝑤_𝑛=𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛𝑄𝑦_{𝜏𝑛−𝑘}=𝑝_𝑛Δ^𝑚−1𝑦_𝑛𝑦_{𝜏𝑛−𝑘}^𝛼−1>0,𝑛≥𝑛₁. (24) Taking into account that Δ𝑦_𝑛>0 and 𝑦_𝑛 is increasing and 𝜏_𝑛−𝑘<𝜏_𝑛, we deduce that Δ^𝑚𝑦_𝑛≤0 and Δ^𝑚−1𝑦_𝑛 is nonincreasing. Lemmas 3 and 4, (1), and (24) yield Δ𝑤_𝑛=−𝑞_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−1−𝑟_𝑛𝑦_𝜏𝑛^𝛼−1𝑦_{𝜏𝑛+1−𝑘}^𝛼−1−𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−1Δ𝑦_{𝜏𝑛−𝑘}^𝛼−1𝑦_{𝜏𝑛−𝑘}^𝛼−1𝑦_{𝜏𝑛+1−𝑘}^𝛼−1≤−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−(𝛼−1)𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−1𝑦_{𝜏𝑛−𝑘}^𝛼−2Δ𝑦_{𝜏𝑛−𝑘}𝑦_{𝜏𝑛−𝑘}^𝛼−1𝑦_{𝜏𝑛+1−𝑘}^𝛼−1≤−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−(𝛼−1)𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼−1𝑦_{𝜏𝑛−𝑘}^𝛼−2𝑀𝜏_𝑛^𝑚−2Δ^𝑚−1𝑦_𝜏𝑛𝑦_{𝜏𝑛−𝑘}^𝛼−1𝑦_{𝜏𝑛+1−𝑘}^𝛼−1≤−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−(𝛼−1)𝑝_𝑛Δ^𝑚−1𝑦_𝑛^𝛼𝑀𝜏_𝑛^𝑚−2𝑦_{𝜏𝑛+1−𝑘}^𝛼≤−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−(𝛼−1)𝑀𝜏_𝑛^𝑚−2𝑝_𝑛Δ^𝑚−1𝑦_𝑛+1^𝛼𝑦_{𝜏𝑛+1−𝑘}^𝛼=−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−𝑝_𝑛(𝛼−1)𝑀𝜏_𝑛^𝑚−2𝑝_𝑛+1Δ^𝑚−1𝑦_𝑛+1^𝛼𝑝_𝑛+1𝑦_{𝜏𝑛+1−𝑘}^𝛼=−𝑟_𝑛−𝑞_𝑛𝑝_𝑛+1𝑤_𝑛+1−(𝛼−1)𝑀𝜏_𝑛^𝑚−2𝑝_𝑛𝑝_𝑛+1𝑤^{𝛼/(𝛼−1)}_𝑛+1. (25) Multiplying by (𝑛−𝑘)^𝜆 and summing up from 𝑛₁ to 𝑛−1, we obtain ^𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘≤−^𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆Δ𝑤_𝑘−^𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑞_𝑘𝑝_𝑘+1𝑤_𝑘+1−^𝑛−1_𝑘=𝑛1(𝛼−1)𝑀𝜏_𝑘^𝑚−2(𝑛−𝑘)^𝜆𝑝_𝑘𝑝_𝑘+1𝑤^{𝛼/(𝛼−1)}_𝑘+1≤𝑛−𝑛₁^𝜆𝑤_𝑛1−𝑤_𝑛+^𝑛−1_𝑘=𝑛1(𝑛+1−𝑘)^𝜆−(𝑛−𝑘)^𝜆𝑞1+^𝑘𝑝_𝑘+1𝑤_𝑘+1−^𝑛−1_𝑘=𝑛1(𝛼−1)𝑀𝜏_𝑘^𝑚−2(𝑛−𝑘)^𝜆𝑝_𝑘𝑝_𝑘+1𝑤^{𝛼/(𝛼−1)}_𝑘+1≤𝑛−𝑛₁^𝜆𝑤_𝑛1+^𝑛−1_𝑘=𝑛1(𝑛+1−𝑘)^𝜆−(𝑛−𝑘)^𝜆𝑞1+^𝑘𝑝_𝑘+1𝑤_𝑘+1−^𝑛−1_𝑘=𝑛1(𝛼−1)𝑀𝜏_𝑘^𝑚−2(𝑛−𝑘)^𝜆𝑝_𝑘𝑝_𝑘+1𝑤^{𝛼/(𝛼−1)}_𝑘+1 (26) or 1𝑛−𝑛₁^𝜆𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘−^𝑛−1_𝑘=𝑛1𝑋_𝑛𝑘𝑤_𝑘+1−𝑌_𝑛𝑘𝑤^{𝛼/(𝛼−1)}_𝑘+1≤𝑤_𝑛1, (27) where 𝑋_𝑛𝑘=(𝑛+1−𝑘)^𝜆−(𝑛−𝑘)^𝜆𝑞1+^𝑘𝑝_𝑘+1,𝑌_𝑛𝑘=(𝛼−1)𝑀𝜏_𝑘^𝑚−2(𝑛−𝑘)^𝜆𝑝_𝑘𝑝_𝑘+1>0. (28) Let 𝐹𝑤_𝑘+1=𝑋_𝑛𝑘𝑤_𝑘+1−𝑌_𝑛𝑘𝑤^{𝛼/(𝛼−1)}_𝑘+1. (29) Then, 𝐹 has maximum value at 𝑤_𝑘+1=((𝛼−1)/𝛼)^𝛼−1𝑋^𝛼−1_𝑛𝑘𝑌^1−𝛼_𝑛𝑘. That is, 𝐹_max=(𝛼−1)^𝛼−1𝛼^𝛼𝑋^𝛼_𝑛𝑘𝑌^1−𝛼_𝑛𝑘. (30) Therefore, (27) can be rewritten as 1𝑛−𝑛₁^𝜆𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘−(𝛼−1)^𝛼−1𝛼^𝛼𝑋^𝛼_𝑛𝑘𝑌^1−𝛼_𝑛𝑘≤𝑤_𝑛1. (31) Hence, we have limsup_𝑛→∞1𝑛−𝑛₁^𝜆𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘−(𝛼−1)^𝛼−1𝛼^𝛼𝑋^𝛼_𝑛𝑘𝑌^1−𝛼_𝑛𝑘≤𝑤_𝑛1 (32) which contradicts condition (Λ2). The proof is complete.

Theorem 8. Let condition (Λ1) hold. Further, assume that there exists a function 𝛿_𝑛∶ℕ→ℝ⁺ such that (Λ3)limsup_𝑛→∞^𝑛−1_𝑘=𝑛1𝛿_𝑘+1𝑟_𝑘−1𝛼^𝛼Δ𝛿_𝑘𝛿_𝑘^𝛼𝑀𝜏_𝑘^{𝑚−21−𝛼}=∞,𝑛₁≥𝑛₀, (33) where 𝑀 is as in Lemma 5. Then, (3) is oscillatory.

Proof. For the sake of contradiction, assume that (3) has a nonoscillatory solution 𝑦_𝑛. Without loss of generality, we assume that 𝑦_𝑛 is eventually positive (the proof is similar when 𝑦_𝑛 is eventually negative). That is, 𝑦_𝑛>0, 𝑦_𝜏𝑛>0 and 𝑦_{𝜏𝑛−𝑘}>0 for all 𝑛≥𝑛₁≥𝑛₀. By Lemma 6, we have Δ^𝑚−1𝑦_𝑛>0, Δ^𝑚𝑦_𝑛≤0, and Δ𝑦_𝑛>0 for 𝑛≥𝑛₁. Consider the function 𝑤_𝑛=𝛿_𝑛𝑝_𝑛Δ^𝑚−1𝑦_𝑛𝑦_{𝜏𝑛−𝑘}^𝛼−1>0,𝑛≥𝑛₁. (34) By utilizing the same approach as in the proof of Theorem 7, we arrive at Δ𝑤_𝑛≤−𝛿_𝑛+1𝑟_𝑛+1𝛼^𝛼Δ𝛿_𝑛𝛿_𝑛^𝛼𝑀𝜏_𝑛^{𝑚−21−𝛼}. (35) Summing up (35) from 𝑛₁ to 𝑛−1, we have ^𝑛−1_𝑘=𝑛1𝛿_𝑘+1𝑟_𝑘−1𝛼^𝛼Δ𝛿_𝑘𝛿_𝑘^𝛼𝑀𝜏_𝑘^{𝑚−21−𝛼}≤𝑤_𝑛1. (36) Letting 𝑛→∞ in the above inequality and taking the upper limit, we get a contradiction to (Λ3). The proof is complete.

Remark 9. In view of the statements of Theorems 7 and 8, one can easily deduce that condition (Λ3) is a generalization of (Λ2).

Example 10. Consider the fourth order half-linear functional difference equation with damping Δ𝑛Δ³𝑦_𝑛²+𝑛Δ³𝑦_𝑛²+1𝑛𝑦²_𝑛−1=0,𝑛≥2, (37) where 𝑝_𝑛=𝑛, 𝑞_𝑛=𝑛, 𝑟_𝑛=1/𝑛, 𝜏_𝑛=𝑛−1, 𝑚=4, and 𝛼=3. It is easy to see that conditions (H1)–(H3) are satisfied. It remains to check the validity of conditions Λ1 and Λ2.
For 𝑛≥2, we have Γ1∶=^𝑛−1_𝑠=𝑛11𝑝_𝑠1−^𝑠−1_𝜈=𝑛1𝑞1−^𝜈𝑝_𝜈^1/(𝛼−1)=^𝑛−1_𝑠=21𝑠^1/2. (38) It is clear that Γ1→∞ as 𝑛→∞. Therfore, condition (Λ1) holds. For 𝑛≥2 and 𝜆=3>𝛼−1=2, we have 1Γ2∶=𝑛−𝑛₁^𝜆𝑛−1_𝑘=𝑛1(𝑛−𝑘)^𝜆𝑟_𝑘−(𝛼−1)^𝛼−1𝛼^𝛼𝑋^𝛼_𝑛𝑘𝑌^1−𝛼_𝑛𝑘=1(𝑛−2)^3𝑛−1_𝑘=2(𝑛−𝑘)³𝑘−4𝑋27^3𝑛𝑘𝑌²_𝑛𝑘, (39) where 𝑋³_𝑛𝑘=(𝑘+1)(𝑛+1−𝑘)³−(2𝑘+1)(𝑛−𝑘)³𝑘+1³,𝑌²_𝑛𝑘=(𝑘+1)²4𝑀²𝑘²(𝑘−2)⁸(𝑛−𝑘)⁶. (40) It is clear that Γ2→∞ as 𝑛→∞. Then, condition (Λ2) holds. Thus, by the conclusion of Theorem 7, (37) is oscillatory.

Example 11. Consider the sixth order half-linear functional difference equation with damping Δ𝑛Δ⁵𝑦_𝑛²+𝑛Δ⁵𝑦_𝑛²+𝑛²𝑦²_𝑛−1=0,𝑛≥2, (41) where 𝑝_𝑛=𝑛, 𝑞_𝑛=𝑛, 𝑟_𝑛=𝑛², 𝜏_𝑛=𝑛−1, 𝑚=6, and 𝛼=3. It is easy to see that conditions (H1)–(H3) are satisfied. In Example 10, we have seen that (Λ1) is satisfied. It remains to check the validity of condition (Λ3).
For 𝑛≥2 and 𝛿_𝑛=𝑛, we have Γ3∶=^𝑛−1_𝑘=𝑛1𝛿_𝑘+1𝑟_𝑘−1𝛼^𝛼Δ𝛿_𝑘𝛿_𝑘^𝛼𝑀𝜏_𝑘^{𝑚−21−𝛼}=^𝑛−1_𝑘=2𝑘²()−1𝑘+127𝑀²𝑘³(𝑘−1)⁸=^𝑛−1_𝑘=227𝑀²𝑘⁵(𝑘+1)(𝑘−1)⁸−127𝑀²𝑘³(𝑘−1)⁸. (42) It is clear that Γ3→∞ as 𝑛→∞. Then, condition (Λ3) holds. Thus, by the conclusion of Theorem 8, (41) is oscillatory.

Remark 12. It is not possible to decide the oscillatory behavior of solutions of (37) and (41) by using any of the results reported in [12, 13]. This implies that the results of our paper extend and generalize some known theorems.

Remark 13. The main results of this paper remain valid for nondelay difference equations of the form Δ𝑝_𝑛𝑄Δ^𝑚−1𝑦_𝑛+𝑞_𝑛𝑄Δ^𝑚−1𝑦_𝑛+𝑟_𝑛𝑄𝑦_𝑛=0,𝑛∈ℕ_𝑛0. (43)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to express thier sincere thanks to the referee for pointing out several suggestions and corrections that helped making the contents of this paper more accurate.