2. Existence of Nonoscillatory Solutions

doi:10.1155/2010/767620

Research Article

Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

Min Liu and Zhenyu Guo

School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China

Correspondence should be addressed to Zhenyu Guo,[email protected] Received 19 March 2010; Revised 10 July 2010; Accepted 5 September 2010 Academic Editor: S. Grace

Copyrightq2010 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 neutral delay difference equationΔakn· · ·Δa2nΔa1nΔxnbnxn−dfn, xn−r1n, xn−r2n, . . . , xn−rsn 0,n≥n0, wheren0 ≥0,d >0,k >0, ands >0 are integers,{ain}_n≥n0 i1,2, . . . , kand{bn}_n≥n0are real sequences,s

j1{rjn}_n≥n0 ⊆ Z, andf : {n : n ≥ n0} ×R^s → Ris 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{bn}_n≥n0. 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 1–17and references cited therein. Some authors have paied their attention to various difference equations. For example,

ΔanΔxn pnx_gn0, n≥0 1.1

see 14,

ΔanΔxn qnx_n1, ΔanΔxn qnfxn1, n≥0 1.2

see 11,

Δ²

x_npx_n−m

p_nx_n−k−q_nx_n−l0, n≥n₀ 1.3

see 6,

Δ²

x_npx_n−k

fn, xn 0, n≥1 1.4

see 10,

Δ²

x_n−px_n−τ ^m

i1

q_if_ix_n−σi, n≥n₀ 1.5

see 9,

ΔanΔxnbx_n−τ fn, xn−d1n, x_n−d2n, . . . , x_n−dkn cn, n≥n0 1.6 see 8,

Δ^mxncx_n−k pnx_n−r 0, n≥n0 1.7

see 15,

Δ^mxncnx_n−k pnfxn−r 0, n≥n0 1.8

see 3,4,12,13,

Δ^mxncx_n−k u s1

p_n^sf_sxn−rs q_n, n≥n₀ 1.9

see 16,

Δ^mxncx_n−k pnx_n−r−qnx_n−l0, n≥n0 1.10

see 17.

Motivated and inspired by the papers mentioned above, in this paper, we investigate the following higher-order nonlinear neutral delay difference equation:

Δa_kn· · ·Δa_2nΔa_1nΔx_nb_nx_n−d fn, xn−r1n, x_n−r2n, . . . , x_n−rsn 0, n≥n₀, 1.11

wheren0≥0,d >0,k >0, ands >0 are integers,{ain}_n≥n0i1,2, . . . , kand{bn}_n≥n0are real sequences,_s

j1{rjn}_n≥n0⊆Z, andf :{n:n≥n₀} ×R^s → Ris a mapping. Clearly, difference

equations1.1–1.10are special cases of1.11. By using Schauder fixed point theorem and Krasnoselskii fixed point theorem, the existence of bounded nonoscillatory solutions of1.11 is established.

Lemma 1.1Schauder fixed point theorem. LetΩbe a nonempty closed convex subset of a Banach spaceX. LetT :Ω → Ωbe a continuous mapping such thatTΩis a relatively compact subset ofX.

ThenT has at least one fixed point inΩ.

Lemma 1.2 Krasnoselskii fixed point theorem. LetΩ be a bounded closed convex subset of a Banach spaceX, and letT₁, T₂ : Ω → X satisfy T₁xT₂y ∈ Ω for each x, y ∈ Ω. If T₁ is a contraction mapping andT₂is a completely continuous mapping, then the equationT₁xT₂xxhas at least one solution inΩ.

The forward difference Δ is defined as usual, that is,Δxn x_n1 −x_n. The higher-order difference for a positive integermis defined asΔ^mx_n ΔΔ^m−1x_n,Δ⁰x_n x_n. Throughout this paper, assume thatR −∞,∞,NandZstand for the sets of all positive integers and integers, respectively,α inf{n−rjn : 1 ≤j ≤s, n ≥n0},β min{n0−d, α}, limn→ ∞n−rjn ∞, 1≤j ≤s, andl_β^∞denotes the set of real sequences defined on the set of positive integers lager thanβ where any individual sequence is bounded with respect to the usual supremum normxsup_n≥β|xn| forx{xn}_n≥β∈l^∞_β. It is well known thatl^∞_β is a Banach space under the supremum norm. A subset Ωof a Banach spaceXis relatively compact if every sequence inΩhas a subsequence converging to an element ofX.

Definition 1.3see 5. A setΩof sequences inl^∞_β is uniformly Cauchyor equi-Cauchyif, for everyε >0, there exists an integerN₀such that

xi−xj< ε, 1.12

wheneveri, j > N0for anyx{xk}_k≥βinΩ.

Lemma 1.4discrete Arzela-Ascoli’s theorem 5. A bounded, uniformly Cauchy subsetΩofl^∞_β is relatively compact.

Let

AM, N

x{xn}_n≥β∈l^∞_β :M≤x_n≤N,∀n≥β

forN > M >0. 1.13

Obviously,AM, Nis a bounded closed and convex subset ofl^∞_β. Put

blim sup

n→ ∞ b_n, blim inf

n→ ∞ b_n. 1.14

By a solution of 1.11, we mean a sequence {xn}_n≥β with a positive integerN0 ≥ n₀d|α|such that1.11is satisfied for alln≥N₀. As is customary, a solution of1.11is said to be oscillatory about zero, or simply oscillatory, if the termsx_nof the sequence{xn}_n≥β are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonoscillatory.

2. Existence of Nonoscillatory Solutions

In this section, a few sufficient conditions of the existence of bounded nonoscillatory solutions of1.11are given.

Theorem 2.1. Assume that there exist constants M and N with N > M > 0 and sequences {ain}_n≥n0 1≤i≤k,{bn}_n≥n0,{hn}_n≥n0, and{qn}_n≥n0such that, forn≥n₀,

bn ≡ −1, eventually, 2.1

fn, u1, u₂, . . . , u_s−fn, v1, v₂, . . . , v_s≤h_nmax{|ui−v_i|:u_i, v_i∈ M, N,1≤i≤s}, 2.2 fn, u1, u₂, . . . , u_s≤q_n, u_i∈ M, N, 1≤i≤s, 2.3

∞ tn0

max 1

|ait|, ht, qt: 1≤i≤k

<∞. 2.4

Then1.11has a bounded nonoscillatory solution inAM, N.

Proof. Choose L ∈ M, N. By2.1, 2.4, and the definition of convergence of series, an integerN₀> n₀d|α|can be chosen such that

b_n≡ −1, ∀n≥N₀, 2.5

∞ j1

∞ t1N0jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

≤min{L−M, N−L}. 2.6

Define a mappingT_L:AM, N → Xby

TLx_n

⎧⎪

⎨

⎪⎩

L−−1^k∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

, n≥N₀,

TLx_N0, β≤n < N0

2.7

for allx∈AM, N.

iIt is claimed thatT_Lx∈AM, N, for allx∈AM, N.

In fact, for everyx∈AM, Nandn≥N0, it follows from2.3and2.6that

TLx_n≥L−^∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

≥L−^∞

j1

∞ t1N0jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

qt

_k

i1a_iti

≥M,

TLx_n≤L^∞

j1

∞ t1N0jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

qt

_k

i1a_iti

≤N.

2.8

That is,TLxAM, N⊆AM, N.

iiIt is declared thatTLis continuous.

Letx {xn} ∈ AM, N and x^u {xn^u} ∈ AM, Nbe any sequence such that x^u_n → x_nasu → ∞. Forn≥N₀,2.2guarantees that

T_Lx^u_n −T_Lx_n

≤^∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

f t, x^u_t−r

1t, x^u_t−r

2t, . . . , x^u_t−r

st

−ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

≤^∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

htmaxx^u_t−rjt−x_t−rjt: 1≤j ≤s _k

i1a_iti

≤x^u−x^∞

j1

∞ t1N0jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

h_{t k}

i1a_iti.

2.9

This inequality and2.4imply thatTLis continuous.

iiiIt can be asserted thatT_LAM, Nis relatively compact.

By2.4, for anyε >0, takeN3≥N0large enough so that

∞ j1

∞ t1N3jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

< ε

2. 2.10

Then, for anyx{xn} ∈AM, Nandn1, n2≥N3,2.10ensures that

|TLxn1−TLxn2| ≤^∞

j1

∞ t1n1jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti

^∞

j1

∞ t1n2jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti

≤^∞

j1

∞ t1N3jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

qt

_k

i1a_iti

^∞

j1

∞ t1N3jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1a_iti

< ε 2 ε

2 ε,

2.11

which means thatTLAM, Nis uniformly Cauchy. Therefore, byLemma 1.4,TLAM, Nis relatively compact.

ByLemma 1.1, there existsx{xn} ∈AM, Nsuch thatTLxx, which is a bounded nonoscillatory solution of1.11. In fact, forn≥N0d,

xnL−−1^k∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti ,

x_n−dL−−1^k∞

j1

∞ t1nj−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti ,

2.12

which derives that

xn−x_n−d −1^k∞

j1 njd−1

t1nj−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti ,

Δx_n−x_n−d −1^k∞

j1 njd

t1n1j−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

−−1^k∞

j1 njd−1

t1nj−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

−−1^k∞

j1

∞ t2nj−1d

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_t−rst

a_1nj−1dk

i2aiti

−1^k∞

j1

∞ t2njd

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_t−rst a_1njdk

i2aiti

−1^k−1∞

t2n

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_t−rst a_1nk

i2a_iti .

2.13

That is,

a_1nΔx_n−x_n−d −1^k−1∞

t2n

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i2a_iti , 2.14

by which it follows that

Δa1nΔxn−x_n−d −1^{k−1 ∞}

t2n1

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i2aiti

−−1^k−1∞

t2n

∞ t3t2

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i2aiti

−1^k−2∞

t3n

∞ t4t3

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_t−rst a2n_k

i3aiti

,

...

Δa_kn· · ·Δa_2nΔa_1nΔx_nb_nx_n−d −1^k−k1fn, x_n−r1n, x_n−r2n, . . . , x_n−rsn −fn, xn−r1n, x_n−r2n, . . . , x_n−rsn.

2.15 Therefore,xis a bounded nonoscillatory solution of1.11. This completes the proof.

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 L₁, L₂ ∈ M, N with L₁/L₂. For L₁ and L₂, as the preceding proof in Theorem 2.1, there exist integers N1, N2 ≥ n0 d |α| and mappings TL1, TL2

satisfying 2.5–2.7, where L, N0 are replaced by L1, N1 and L2, N2, respectively, and _∞

j1_∞

t1N4jd_∞

t2t1· · ·_∞

tktk−1

_∞

ttkht/|_k

i1a_iti| < |L1 − L₂|/2N for some N₄ ≥ max{N1, N2}. Then the mappingsTL1andTL2have fixed pointsx, y∈AM, N, respectively, which are bounded nonoscillatory solutions of1.11inAM, N. For the sake of proving that1.11possesses uncountably many bounded nonoscillatory solutions inAM, N, it is only needed to show thatx /y. In fact, by2.7, we know that, forn≥N4,

x_nL₁−−1^k∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

,

ynL2−−1^k∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

f

t, y_t−r1t, y_t−r2t, . . . , y_{t−rst k}

i1aiti

.

2.16

Then,

xn−yn≥ |L1−L2|

−^∞

j1

∞ t1njd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_t−rst−f

t, y_t−r1t, y_t−r2t, . . . , y_{t−rst k}

i1a_iti

≥ |L1−L2| −x−y^∞

j1

∞ t1N4jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

h_{t k}

i1a_iti

≥ |L₁−L₂| −2N ∞ j1

∞ t1N4jd

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

h_{t k}

i1a_iti

>0, n≥N4,

2.17

that is,x /y.

Theorem 2.3. Assume that there exist constants M and N with N > M > 0 and sequences {ain}_n≥n0 1≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

bn≡1, eventually. 2.18

Then1.11has a bounded nonoscillatory solution inAM, N.

Proof. ChooseL∈M, N. By2.18and2.4, an integerN0> n0d|α|can be chosen such that

b_n≡1, ∀n≥N₀, ∞

j1

N02jd−1

t1N02j−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

qt

_k

i1a_iti ≤min{L−M, N−L}. 2.19 Define a mappingT_L:AM, N → Xby

TLx_n

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

L −1^k∞

j1

n2jd−1

t1n2j−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

, n≥N₀,

TLx_N0, β≤n < N₀

2.20

for allx∈AM, N.

The proof that TL has a fixed point x {xn} ∈ AM, N is analogous to that in Theorem 2.1. It is claimed that the fixed pointxis a bounded nonoscillatory solution of1.11.

In fact, forn≥N₀d,

x_nL −1^k∞

j1

n2jd−1

t1n2j−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti ,

x_n−dL −1^k∞

j1

n2j−1d−1

t1n2j−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

,

2.21

by which it follows that

xnx_n−d2L −1^k∞

j1 njd−1

t1nj−1d

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, xt−r1t, x_t−r2t, . . . , x_{t−rst k}

i1a_iti . 2.22

The rest of the proof is similar to that inTheorem 2.1. This completes the proof.

Theorem 2.4. Assume that there exist constantsb,M, andN with N > M > 0 and sequences {ain}_n≥n01≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

|bn| ≤b < N−M

2N , eventually. 2.23

Then1.11has a bounded nonoscillatory solution inAM, N.

Proof. ChooseL∈MbN, N−bN. By2.23and2.4, an integerN0> n0d|α|can be chosen such that

|bn| ≤b < N−M

2N , ∀n≥N₀, ∞

t1N0

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

≤min{L−bN−M, N−bN−L}. 2.24

Define two mappingsT_1L, T_2L:AM, N → Xby

T1Lx_n

⎧⎨

⎩

L−b_nx_n−d, n≥N₀, T1Lx_N0, β≤n < N₀, T2Lx_n

⎧⎪

⎨

⎪⎩

−1^k∞

t1n

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

, n≥N₀,

T2Lx_N0, β≤n < N₀

2.25

for allx∈AM, N.

iIt is claimed thatT_1LxT_2Ly∈AM, N, for allx, y∈AM, N.

In fact, for everyx, y∈AM, Nandn≥N₀, it follows from2.3,2.24that T_1LxT_2Ly

n≥L−bN− ^∞

t1N0

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

≥M, T_1LxT_2Ly

n≤LbN ^∞

t1N0

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

≤N.

2.26

That is,T1LxT_2LyAM, N⊆AM, N.

iiIt is declared thatT_1Lis a contraction mapping onAM, N.

In reality, for anyx, y∈AM,Nandn≥N0, it is easy to derive that T1Lx_n−

T1Ly

n≤ |bn|x_n−d−y_n−d≤bx−y, 2.27 which implies that

T_1Lx−T_1Ly≤bx−y. 2.28 Then,b <N−M/2N <1 ensures thatT1Lis a contraction mapping onAM, N.

iiiSimilar toiiandiiiin the proof ofTheorem 2.1, it can be showed thatT_2Lis completely continuous.

ByLemma 1.2, there existsx{xn} ∈AM, Nsuch thatT1LxT2Lxx, which is a bounded nonoscillatory solution of1.11. This completes the proof.

Theorem 2.5. Assume that there exist constantsMandNwithN >2−b/1−bM >0 and sequences{ain}_n≥n01≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

b_n≥0, eventually, and 0≤b≤b <1. 2.29

Then1.11has a bounded nonoscillatory solution inAM, N.

Proof. ChooseL ∈ M 1b/2N, N b/2M. By 2.29and 2.4, an integerN0 >

n₀d|α|can be chosen such that

b

2 ≤bn≤ 1b

2 , ∀n≥N0, ∞

t1N0

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

qt

_k

i1a_iti≤min

L−M−1b

2 N, N−Lb 2M

.

2.30

Define two mappingsT1L, T2L:AM, N → Xas2.25. The rest of the proof is analogous to that inTheorem 2.4. This completes the proof.

Similar to the proof ofTheorem 2.5, we have the following theorem.

Theorem 2.6. Assume that there exist constantsMandNwithN >2b/1bM >0 and sequences{ain}_n≥n01≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

b_n≤0, eventually, and −1< b≤b≤0. 2.31

Then1.11has a bounded nonoscillatory solution inAM, N.

Theorem 2.7. Assume that there exist constantsMandNwithN >bb²−b/bb²−bM >0 and sequences{ain}_n≥n0 1≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

b_n>1, eventually, 1< bandb < b²<∞. 2.32

Then1.11has a bounded nonoscillatory solution inAM, N. Proof. Takeε∈0, b−1sufficiently small satisfying

1< b−ε < bε <

b−ε2 , bε

b−ε2−

bε₂

N >

bε₂ b−ε

−

b−ε2 M.

2.33

ChooseL∈bεM bε/b−εN,b−εN b−ε/bεM. By2.33, an integer N0> n0d|α|can be chosen such that

b−ε < b_n< bε, ∀b≥N₀, ∞

t1N0

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

q_{t k}

i1aiti

≤min b−ε bεL−

b−ε

M−N,b−ε bεM

b−ε N−L

.

2.34 Define two mappingsT1L, T2L:AM, N → Xby

T1Lx_n

⎧⎪

⎪⎨

⎪⎪

⎩ L

b_nd −x_nd

b_nd, n≥N0, T1Lx_N0, β≤n < N0, T2Lx_n

⎧⎪

⎪⎨

⎪⎪

⎩

−1^k b_nd

∞ t1n

∞ t2t1

· · · ^∞

tktk−1

∞ ttk

ft, x_t−r1t, x_t−r2t, . . . , x_{t−rst k}

i1aiti

, n≥N₀,

T2Lx_N0, β≤n < N₀

2.35

for allx∈AM, N. The rest of the proof is analogous to that inTheorem 2.4. This completes the proof.

Similar to the proof ofTheorem 2.7, we have

Theorem 2.8. Assume that there exist constantsMandNwithN >1b/1bM >0 and sequences{ain}_n≥n0 1≤i≤k,{bn}_n≥n0,{hn}_n≥n0,{qn}_n≥n0, satisfying2.2–2.4and

b_n <−1, eventually, −∞< bandb <−1. 2.36

Then1.11has a bounded nonoscillatory solution inAM, N.

Remark 2.9. Similar toRemark 2.2, we can also prove that the conditions of Theorems2.3–2.8 ensure that 1.11has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Remark 2.10. Theorems 2.1–2.8 extend and improve Theorem 1 of Cheng 6, Theorems 2.1–2.7 of Liu et al. 8, and corresponding theorems in 3,4,9–17.

3. Examples

In this section, two examples are presented to illustrate the advantage of the above results.

Example 3.1. Consider the following fourth-order nonlinear neutral delay difference equation:

Δ4ⁿΔ3ⁿΔ2ⁿΔxn−x_n−1 0, n≥1. 3.1

ChooseM1 andN2. It is easy to verify that the conditions ofTheorem 2.1are satisfied.

ThereforeTheorem 2.1ensures that3.1has a nonoscillatory solution inA1,2. However, the results in 3,4,6,8–17are not applicable for3.1.

Example 3.2. Consider the following third-order nonlinear neutral delay difference equation:

Δ

2ⁿ−nΔ

n²−n1 Δ

xn2ⁿ−1 3ⁿ x_n−4

sin2x_n−2

n² −cos3x_n−3

n³ 0, n≥5, 3.2

where

a1nn²−n1, a2n2ⁿ−n, bn 2ⁿ−1 3ⁿ , fn, u1, u2 sin2u1

n² −cos3u2

n³ , hnqn 2 n².

3.3

ChooseM1 andN5. It can be verified that the assumptions ofTheorem 2.5are fulfilled.

It follows fromTheorem 2.5that3.2has a nonoscillatory solution inA1,5. However, the results in 3,4,6,8–17are unapplicable for3.2.

Acknowledgment

The authors are grateful to the editor and the referee for their kind help, careful reading and editing, valuable comments and suggestions.

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