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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 767620, 14 pages doi:10.1155/2010/767620 Research Article Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation MinLiuandZhenyuGuo School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China Correspondence should be addressed to Zhenyu Guo, guozy@163.com Received 19 March 2010; Revised 10 July 2010; Accepted 5 September 2010 Academic Editor: S. Grace Copyright q 2010 M. Liu and Z. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The existence of bounded nonoscillatory solutions of a higher-order nonlinear n eutral delay difference equation Δa kn ···Δa 2n Δa 1n Δx n b n x n−d fn, x n−r 1n ,x n−r 2n , ,x n−r sn 0, n ≥ n 0 , where n 0 ≥ 0, d>0, k>0, and s>0areintegers,{a in } n≥n 0 i  1, 2, ,k and {b n } n≥n 0 are real sequences,  s j1 {r jn } n≥n 0 ⊆ Z,andf : {n : n ≥ n 0 }×R s → R is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence {b n } n≥n 0 . Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions. 1. Introduction and Preliminaries Recently, the interest in the study of the solvability of difference equations has been increasing see 1 –17 and references cited therein. Some authors have paied their attention to various difference equations. For example, Δ  a n Δx n   p n x gn  0,n≥ 0 1.1 see 14, Δ  a n Δx n   q n x n1 , Δ  a n Δx n   q n f  x n1  ,n≥ 0 1.2 2 Advances in Difference Equations see 11, Δ 2  x n  px n−m   p n x n−k − q n x n−l  0,n≥ n 0 1.3 see 6, Δ 2  x n  px n−k   f  n, x n   0,n≥ 1 1.4 see 10, Δ 2  x n − px n−τ   m  i1 q i f i  x n−σ i  ,n≥ n 0 1.5 see 9, Δ  a n Δ  x n  bx n−τ   f  n, x n−d 1n ,x n−d 2n , ,x n−d kn   c n ,n≥ n 0 1.6 see 8, Δ m  x n  cx n−k   p n x n−r  0,n≥ n 0 1.7 see 15, Δ m  x n  c n x n−k   p n f  x n−r   0,n≥ n 0 1.8 see 3, 4, 12, 13, Δ m  x n  cx n−k   u  s1 p s n f s  x n−r s   q n ,n≥ n 0 1.9 see 16, Δ m  x n  cx n−k   p n x n−r − q n x n−l  0,n≥ n 0 1.10 see 17. Motivated and inspired by the papers mentioned above, in t his paper, we investigate the following higher-order nonlinear neutral delay difference equation: Δ  a kn ···Δ  a 2n Δ  a 1n Δ  x n  b n x n−d   f  n, x n−r 1n ,x n−r 2n , ,x n−r sn   0,n≥ n 0 , 1.11 where n 0 ≥ 0, d>0, k>0, and s>0 are integers, {a in } n≥n 0 i  1, 2, ,k and {b n } n≥n 0 are real sequences,  s j1 {r jn } n≥n 0 ⊆ Z,andf : {n : n ≥ n 0 }×R s → R is a mapping. Clearly, difference Advances in Difference Equations 3 equations 1.1–1.10 are special cases of 1.11. By using Schauder fixed point theorem and Krasnoselskii fixed point theorem, the existence of bounded nonoscillatory solutions of 1.11 is established. Lemma 1.1 Schauder fixed point theorem. Let Ω be a nonempty closed convex subset of a Banach space X.LetT : Ω → Ω be a continuous mapping such that TΩ is a relatively compact subset of X. Then T has at least one fixed point in Ω. Lemma 1.2 Krasnoselskii fixed point theorem. Let Ω be a bounded closed convex subset of a Banach space X, and let T 1 ,T 2 : Ω → X satisfy T 1 x  T 2 y ∈ Ω for each x,y ∈ Ω.IfT 1 is a contraction mapping and T 2 is a completely continuous mapping, then the equation T 1 x T 2 x  x has at least one solution in Ω. The forward difference Δ is defined as usual, that is, Δx n  x n1 − x n . The higher-order difference for a positive integer m is defined as Δ m x n ΔΔ m−1 x n , Δ 0 x n  x n . Throughout this paper, assume that R −∞, ∞, N and Z stand for the sets of all positive integers and integers, respectively, α  inf{n − r jn :1≤ j ≤ s, n ≥ n 0 }, β  min{n 0 − d, α}, lim n →∞ n − r jn ∞, 1 ≤ j ≤ s, and l ∞ β denotes the set of real sequences defined on the set of positive integers lager than β where any individual sequence is bounded with respect to the usual supremum norm x  sup n≥β |x n | for x  {x n } n≥β ∈ l ∞ β . I t is well known that l ∞ β is a Banach space under the supremum norm. A subset Ω of a Banach space X is relatively compact if every sequence in Ω has a subsequence converging to an element of X. Definition 1.3 see 5.AsetΩ of sequences in l ∞ β is uniformly Cauchy or equi-Cauchy if, for every ε>0, there exists an integer N 0 such that   x i − x j   <ε, 1.12 whenever i, j > N 0 for any x  {x k } k≥β in Ω. Lemma 1.4 discrete Arzela-Ascoli’s theorem 5. A bounded, uniformly Cauchy subset Ω of l ∞ β is relatively compact. Let A  M, N    x  { x n } n≥β ∈ l ∞ β : M ≤ x n ≤ N, ∀n ≥ β  for N>M>0. 1.13 Obviously, AM, N is a bounded closed and convex subset of l ∞ β .Put b  lim sup n →∞ b n ,b lim inf n →∞ b n . 1.14 By a solution of 1.11, we mean a sequence {x n } n≥β with a positive integer N 0 ≥ n 0  d  |α| such that 1.11 is satisfied for all n ≥ N 0 . As is customary, a solution of 1.11 is said to be oscillatory about zero, or simply oscillatory, if the terms x n of the sequence {x n } n≥β are neither eventually all positive nor eventually all negative. Otherwise, t he solution is called nonoscillatory. 4 Advances in Difference Equations 2. Existence of Nonoscillatory Solutions In this section, a few sufficient conditions of the existence of bounded nonoscillatory solutions of 1.11 are given. Theorem 2.1. Assume that there exist constants M and N with N>M>0 and sequences {a in } n≥n 0 1 ≤ i ≤ k, {b n } n≥n 0 , {h n } n≥n 0 , and {q n } n≥n 0 such that, for n ≥ n 0 , b n ≡−1, eventually, 2.1   f  n, u 1 ,u 2 , ,u s  − f  n, v 1 ,v 2 , ,v s    ≤ h n max {| u i − v i | : u i ,v i ∈  M, N  , 1 ≤ i ≤ s } , 2.2   f  n, u 1 ,u 2 , ,u s    ≤ q n ,u i ∈  M, N  , 1 ≤ i ≤ s, 2.3 ∞  tn 0 max  1 | a it | ,h t ,q t :1≤ i ≤ k  < ∞. 2.4 Then 1.11 has a bounded nonoscillatory solution in AM, N. Proof. Choose L ∈ M, N.By2.1, 2.4, and the definition of convergence of series, an integer N 0 >n 0  d  |α| can be chosen such that b n ≡−1, ∀n ≥ N 0 , 2.5 ∞  j1 ∞  t 1 N 0 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≤ min { L − M, N − L } . 2.6 Define a mapping T L : AM, N → X by  T L x  n  ⎧ ⎪ ⎨ ⎪ ⎩ L −  −1  k ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i ,n≥ N 0 ,  T L x  N 0 ,β≤ n<N 0 2.7 for all x ∈ AM, N. i It is claimed that T L x ∈ AM, N, for all x ∈ AM, N. Advances in Difference Equations 5 In fact, for every x ∈ AM, N and n ≥ N 0 , it follows from 2.3 and 2.6 that  T L x  n ≥ L − ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k   f  t, x t−r 1t ,x t−r 2t , ,x t−r st        k i1 a it i    ≥ L − ∞  j1 ∞  t 1 N 0 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≥ M,  T L x  n ≤ L  ∞  j1 ∞  t 1 N 0 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≤ N. 2.8 That is, T L xAM, N ⊆ AM, N. ii It is declared that T L is continuous. Let x  {x n }∈AM, N and x u  {x u n }∈AM, N be any sequence such that x u n → x n as u →∞. For n ≥ N 0 , 2.2 guarantees that    T L x u n − T L x n    ≤ ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k    f  t, x u t−r 1t ,x u t−r 2t , ,x u t−r st  − f  t, x t−r 1t ,x t−r 2t , ,x t−r st         k i1 a it i    ≤ ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k h t max     x u t−r jt − x t−r jt    :1≤ j ≤ s      k i1 a it i    ≤    x u − x    ∞  j1 ∞  t 1 N 0 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k h t     k i1 a it i    . 2.9 This inequality and 2.4 imply that T L is continuous. iii It can be asserted that T L AM, N is relatively compact. 6 Advances in Difference Equations By 2.4, for any ε>0, take N 3 ≥ N 0 large enough so that ∞  j1 ∞  t 1 N 3 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    < ε 2 . 2.10 Then, for any x  {x n }∈AM, N and n 1 ,n 2 ≥ N 3 , 2.10 ensures that | T L x n 1 − T L x n 2 | ≤ ∞  j1 ∞  t 1 n 1 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k   f  t, x t−r 1t ,x t−r 2t , ,x t−r st        k i1 a it i     ∞  j1 ∞  t 1 n 2 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k   f  t, x t−r 1t ,x t−r 2t , ,x t−r st        k i1 a it i    ≤ ∞  j1 ∞  t 1 N 3 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i     ∞  j1 ∞  t 1 N 3 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    < ε 2  ε 2  ε, 2.11 which means that T L AM, N is uniformly Cauchy. Therefore, by Lemma 1.4, T L AM, N is relatively compact. By Lemma 1.1, there exists x  {x n }∈AM, N such that T L x  x, which is a bounded nonoscillatory solution of 1.11. In fact, for n ≥ N 0  d, x n  L −  −1  k ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , x n−d  L −  −1  k ∞  j1 ∞  t 1 nj−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , 2.12 Advances in Difference Equations 7 which derives that x n − x n−d   −1  k ∞  j1 njd−1  t 1 nj−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , Δ  x n − x n−d    −1  k ∞  j1 njd  t 1 n1j−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i −  −1  k ∞  j1 njd−1  t 1 nj−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i  −  −1  k ∞  j1 ∞  t 2 nj−1d ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st  a 1nj−1d  k i2 a it i   −1  k ∞  j1 ∞  t 2 njd ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st  a 1njd  k i2 a it i   −1  k−1 ∞  t 2 n ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st  a 1n  k i2 a it i . 2.13 That is, a 1n Δ  x n − x n−d    −1  k−1 ∞  t 2 n ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i2 a it i , 2.14 by which it follows that Δ  a 1n Δ  x n − x n−d    −1  k−1 ∞  t 2 n1 ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i2 a it i −  −1  k−1 ∞  t 2 n ∞  t 3 t 2 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i2 a it i   −1  k−2 ∞  t 3 n ∞  t 4 t 3 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st  a 2n  k i3 a it i , . . . Δ  a kn ···Δ  a 2n Δ  a 1n Δ  x n  b n x n−d    −1  k−k1 f  n, x n−r 1n ,x n−r 2n , ,x n−r sn   −f  n, x n−r 1n ,x n−r 2n , ,x n−r sn  . 2.15 Therefore, x is a bounded nonoscillatory solution of 1.11. This completes the proof. 8 Advances in Difference Equations Remark 2.2. The conditions of Theorem 2.1 ensure the 1.11 has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions. In fact, let L 1 ,L 2 ∈ M, N with L 1 /  L 2 . For L 1 and L 2 , as the preceding proof in Theorem 2.1, there exist integers N 1 ,N 2 ≥ n 0  d  |α| and mappings T L 1 ,T L 2 satisfying 2.5–2.7, where L, N 0 are replaced by L 1 , N 1 and L 2 , N 2 , respectively, and  ∞ j1  ∞ t 1 N 4 jd  ∞ t 2 t 1 ···  ∞ t k t k−1  ∞ tt k h t /|  k i1 a it i | < |L 1 − L 2 |/2N for some N 4 ≥ max{N 1 ,N 2 }. Then the mappings T L 1 and T L 2 have fixed points x, y ∈ AM, N, respectively, which are bounded nonoscillatory solutions of 1.11 in AM, N. For the sake of proving that 1.11 possesses uncountably many bounded nonoscillatory solutions in AM, N,itis only needed to show that x /  y.Infact,by2.7, we know that, for n ≥ N 4 , x n  L 1 −  −1  k ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , y n  L 2 −  −1  k ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, y t−r 1t ,y t−r 2t , ,y t−r st   k i1 a it i . 2.16 Then,   x n − y n   ≥ | L 1 − L 2 | − ∞  j1 ∞  t 1 njd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k   f  t, x t−r 1t ,x t−r 2t , ,x t−r st  − f  t, y t−r 1t ,y t−r 2t , ,y t−r st        k i1 a it i    ≥ | L 1 − L 2 | −   x − y   ∞  j1 ∞  t 1 N 4 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k h t     k i1 a it i    ≥ | L 1 − L 2 | − 2N ∞  j1 ∞  t 1 N 4 jd ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k h t     k i1 a it i    > 0,n≥ N 4 , 2.17 that is, x /  y. Theorem 2.3. Assume that there exist constants M and N with N>M>0 and sequences {a in } n≥n 0 1 ≤ i ≤ k, {b n } n≥n 0 , {h n } n≥n 0 , {q n } n≥n 0 , satisfying 2.2–2.4 and b n ≡ 1, eventually. 2.18 Then 1.11 has a bounded nonoscillatory solution in AM, N. Advances in Difference Equations 9 Proof. Choose L ∈ M, N.By2.18 and 2.4, an integer N 0 >n 0  d  |α| can be chosen such that b n ≡ 1, ∀n ≥ N 0 , ∞  j1 N 0 2jd−1  t 1 N 0 2j−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≤ min { L − M, N − L } . 2.19 Define a mapping T L : AM, N → X by  T L x  n  ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ L   −1  k ∞  j1 n2jd−1  t 1 n2j−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i ,n≥ N 0 ,  T L x  N 0 ,β≤ n<N 0 2.20 for all x ∈ AM, N. The proof that T L has a fixed point x  {x n }∈AM, N is analogous to that in Theorem 2.1. It is claimed that the fixed point x is a bounded nonoscillatory solution of 1.11. In fact, for n ≥ N 0  d, x n  L   −1  k ∞  j1 n2jd−1  t 1 n2j−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , x n−d  L   −1  k ∞  j1 n2j−1d−1  t 1 n2j−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i , 2.21 by which it follows that x n  x n−d  2L   −1  k ∞  j1 njd−1  t 1 nj−1d ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i . 2.22 The rest of the proof is similar to that in Theorem 2.1. This completes the proof. Theorem 2.4. Assume that there exist constants b, M, and N with N>M>0 and sequences {a in } n≥n 0 1 ≤ i ≤ k, {b n } n≥n 0 , {h n } n≥n 0 , {q n } n≥n 0 , satisfying 2.2–2.4 and | b n | ≤ b< N − M 2N , eventually. 2.23 Then 1.11 has a bounded nonoscillatory solution in AM, N. 10 Advances in Difference Equations Proof. Choose L ∈ M  bN, N − bN.By2.23 and 2.4, an integer N 0 >n 0  d  |α| can be chosen such that | b n | ≤ b< N − M 2N , ∀n ≥ N 0 , ∞  t 1 N 0 ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≤ min { L − bN − M, N − bN − L } . 2.24 Define two mappings T 1L ,T 2L : AM, N → X by  T 1L x  n  ⎧ ⎨ ⎩ L − b n x n−d ,n≥ N 0 ,  T 1L x  N 0 ,β≤ n<N 0 ,  T 2L x  n  ⎧ ⎪ ⎨ ⎪ ⎩  −1  k ∞  t 1 n ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k f  t, x t−r 1t ,x t−r 2t , ,x t−r st   k i1 a it i ,n≥ N 0 ,  T 2L x  N 0 ,β≤ n<N 0 2.25 for all x ∈ AM, N. i It is claimed that T 1L x  T 2L y ∈ AM, N, for all x, y ∈ AM, N. In fact, for every x, y ∈ AM, N and n ≥ N 0 , it follows from 2.3, 2.24 that  T 1L x  T 2L y  n ≥ L − bN − ∞  t 1 N 0 ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≥ M,  T 1L x  T 2L y  n ≤ L  bN  ∞  t 1 N 0 ∞  t 2 t 1 ··· ∞  t k t k−1 ∞  tt k q t     k i1 a it i    ≤ N. 2.26 That is, T 1L x  T 2L yAM, N ⊆ AM, N. ii It is declared that T 1L is a contraction mapping on AM, N. In reality, for any x, y ∈ AM, N and n ≥ N 0 , it is easy to derive that    T 1L x  n −  T 1L y  n   ≤ | b n |   x n−d − y n−d   ≤ b   x − y   , 2.27 which implies that   T 1L x − T 1L y   ≤ b   x − y   . 2.28 Then, b<N − M/2N<1 ensures that T 1L is a contraction mapping on AM, N. iii Similar to ii and iii in the proof of Theorem 2.1, it can be showed that T 2L is completely continuous. By Lemma 1.2, there exists x  {x n }∈AM, N such that T 1L x  T 2L x  x, which is a bounded nonoscillatory solution of 1.11. This completes the proof. [...]... 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Thandapani, M M S Manuel, J R Graef, and P W Spikes, “Monotone properties of certain classes of solutions of second-order difference equations,” Computers & Mathematics with Applications, vol 36, no 10–12, pp 291–297, 1998 12 F J Yang and J C Liu, “Positive solution of even-order nonlinear neutral difference equations with variable delay, ” Journal of Systems Science and Mathematical Sciences, vol 22,... proof Similar to the proof of Theorem 2.7, we have Theorem 2.8 Assume that there exist constants M and N with N > 1 b / 1 b M > 0 and sequences {ain }n≥n0 1 ≤ i ≤ k , {bn }n≥n0 , {hn }n≥n0 , {qn }n≥n0 , satisfying 2.2 – 2.4 and bn < −1, eventually, −∞ < b and b < −1 2.36 Then 1.11 has a bounded nonoscillatory solution in A M, N Remark 2.9 Similar to Remark 2.2, we can also prove that the conditions of. .. nonlinear neutral delay difference equation: Δ 4n Δ 3n Δ 2n Δ xn − xn−1 0, n ≥ 1 3.1 Advances in Difference Equations 13 Choose M 1 and N 2 It is easy to verify that the conditions of Theorem 2.1 are satisfied Therefore Theorem 2.1 ensures that 3.1 has a nonoscillatory solution in A 1, 2 However, the results in 3, 4, 6, 8–17 are not applicable for 3.1 Example 3.2 Consider the following third-order nonlinear . Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 767620, 14 pages doi:10.1155/2010/767620 Research Article Solvability of a Higher-Order Nonlinear Neutral Delay. 2009. 9 Q. Meng and J. Yan, “Bounded oscillation for second-order nonlinear neutral difference equations in critical and non-critical states,” Journal of Computational and Applied Mathematics, vol 2002. 16 Y. Zhou and Y. Q. Huang, “Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations,” Journal of Mathematical Analysis and Applications, vol. 280,