Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 328914, 24 pages
doi:10.1155/2011/328914
Research Article
Solvability of a Second Order Nonlinear Neutral
Delay Difference Equation
Zeqing Liu,1 Liangshi Zhao,1
Jeong Sheok Ume,2 and Shin Min Kang3, 4
1
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
4
Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea
2
Correspondence should be addressed to Jeong Sheok Ume, jsume@changwon.ac.kr
Received 2 August 2011; Accepted 9 October 2011
Academic Editor: Sergey V. Zelik
Copyright q 2011 Zeqing Liu et al. 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.
This paper studies the second-order nonlinear neutral delay difference equation Δan Δxn bn xn−τ fn, xf1n , . . . , xfkn gn, xg1n , . . . , xgkn cn , n ≥ n0 . By means of the Krasnoselskii
and Schauder fixed point theorem and some new techniques, we get the existence results of
uncountably many bounded nonoscillatory, positive, and negative solutions for the equation,
respectively. Ten examples are given to illustrate the results presented in this paper.
1. Introduction
We are concerned with the second-order nonlinear neutral delay difference equation of the
form
Δ an Δxn bn xn−τ f n, xf1n , . . . , xfkn g n, xg1n , . . . , xgkn cn ,
n ≥ n0 ,
1.1
0 for
where τ, k ∈ N, n0 ∈ N0 , {an }n∈Nn , {bn }n∈Nn and {cn }n∈Nn are real sequences with an /
0
0
0
each n ∈ Nn0 , f, g ∈ CNn0 × Rk , R, and fl , gl : Nn0 → Z with
lim fln lim gln ∞, l ∈ {1, 2, . . . , k}.
n→∞
n→∞
1.2
Note that a few special cases of 1.1 were studied in 1–9. In particular, González and
Jiménez-Melado 3 used a fixed-point theorem derived from the theory of measures of
2
Abstract and Applied Analysis
noncompactness to investigate the existence of solutions for the second-order difference
equation
Δ qn Δxn fn xn 0,
n ≥ 0.
1.3
By applying the Leray-Schauder nonlinear alternative theorem for condensing operators,
Agarwal et al. 1 studied the existence of a nonoscillatory solution for the second-order
neutral delay difference equation
Δ an Δ xn pxn−τ Fn 1, xn1−σ 0,
n ≥ 0,
1.4
where p ∈ R\{±1}. Using the Banach contraction principle, Cheng 5 discussed the existence
of a positive solution for the second-order neutral delay difference equation with positive and
negative coefficients
Δ2 xn pxn−m pn xn−k − qn xn−l 0,
1.5
n ≥ n0 ,
where p ∈ R \ {−1}, Liu et al. 6 and Liu et al. 7 extended the results due to cheng 5 and
got the existence of uncountably many bounded nonoscillatory solutions for 1.1 and the
second-order nonlinear neutral delay difference equation
Δan Δxn bxn−τ fn, xn−d1n , xn−d2n , . . . , xn−dkn cn ,
n ≥ n0 ,
1.6
with respect to b ∈ R, where f is Lipschitz continuous, respectively.
The purpose of this paper is to establish the existence results of uncountably many
bounded nonoscillatory, positive, and negative solutions, respectively, for 1.1 by using the
Krasnoselskii fixed point theorem, Schauder fixed point theorem, and a few new techniques.
The results obtained in this paper improve essentially the corresponding results in 5–7 by
removing the Lipschitz continuity condition. Ten nontrivial examples are given to reveal the
superiority and applications of our results.
2. Preliminaries
Throughout this paper, we assume that Δ is the forward difference operator defined by Δxn xn1 − xn , Δ2 xn ΔΔxn , R −∞, ∞, R 0, ∞, R− −∞, 0, and Z, N, and N0
stand for the sets of all integers, positive integers, and nonnegative integers, respectively,
Nn0 {n : n ∈ N0 with n ≥ n0 }, n0 ∈ N0 ,
β min n0 − τ, inf fln , gln : 1 ≤ l ≤ k, n ∈ Nn0 ,
Zβ n : n ∈ Z with n ≥ β .
2.1
Abstract and Applied Analysis
3
Let lβ∞ denote the Banach space of all bounded sequences in Zβ with norm
x sup|xn |
n∈Zβ
for x {xn }n∈Zβ ∈ lβ∞ ,
Bd, D x {xn }n∈Zβ ∈ lβ∞ : x − d ≤ D
for d {d}n∈Zβ ∈ lβ∞ , D > 0
2.2
represent the closed ball centered at d and with radius D in lβ∞ .
By a solution of 1.1, we mean a sequence {xn }n∈Zβ with a positive integer T ≥ n0 τ |β| such that 1.1 is satisfied for all n ≥ T . As is customary, a solution of 1.1 is said to be
oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to
be nonoscillatory.
Lemma 2.1 2. A bounded, uniformly Cauchy subset Y of lβ∞ is relatively compact.
Lemma 2.2 Krasnoselskii fixed point theorem 10. Let Y be a nonempty bounded closed convex
subset of a Banach space X and S, G : Y → X mappings such that Sx Gy ∈ Y for every pair
x, y ∈ Y . If S is a contraction and G is completely continuous, then
Sx Gx x
2.3
has a solution in Y .
Lemma 2.3 Schauder fixed point theorem 10. Let Y be a nonempty closed convex subset of a
Banach space X and S : Y → Y a continuous mapping such that SY is a relatively compact subset
of X. Then, S has a fixed point in Y .
Lemma 2.4. Let τ ∈ N, n0 ∈ N0 and {an }n∈Nn be a nonnegative sequence. Then,
0
∞ ∞
as < ∞ ⇐⇒
i0 sn0 iτ
Moreover, if
∞ ∞
i0
sn0 iτ as
∞
sas < ∞ .
2.4
sn0
< ∞, then
∞
∞ i0 sn0 iτ
as ≤
∞ 1
sn0
s
as < ∞.
τ
2.5
4
Abstract and Applied Analysis
Proof. For each t ∈ R, let t stand for the largest integer not exceeding t. It follows that
∞ ∞
as i0 sn0 iτ
∞
as sn0
s − n0
1
τ
lim
s→∞
∞
as sn0 τ
sn0
∞ ∞
as · · ·
sn0 2τ
as ≤
s
as ,
1
τ
∞ sn0
1 s − n0 /τ
1.
s/τ
2.6
2.7
∞
Combining 2.6 and 2.7, we infer that 2.4 holds. Assume that ∞
i0 sn0 iτ as < ∞. In
∞
∞
view of 2.4, we get that sn0 sas < ∞, which gives that sn0 as < ∞. It follows that
∞ 1
sn0
s
as < ∞.
τ
2.8
This completes the proof.
3. Existence of Uncountably Many Bounded Positive Solutions
Now, we use the Krasnoselskii fixed point theorem to prove the existence of uncountably
many bounded nonoscillatory, positive, and negative solutions of 1.1 under various
conditions relative to the sequence {bn }n∈Nβ ⊂ R.
Theorem 3.1. Assume that there exist n1 ∈ Nn0 , d ∈ R, D, b ∈ R \ {0} and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying
0
0
|d| 1 − b
3.1
<
, |bn | ≤ b, ∀n ≥ n1 ,
D
b
fn, u1 , . . . , uk ≤ Fn , gn, u1 , . . . , uk ≤ Gn , ∀n, ul ∈ Nn × d − D, d D, 1 ≤ l ≤ k,
0
3.2
⎧
⎫
∞
i−1
⎨ ⎬
1
< ∞.
3.3
max Fi ,
max Gj , cj ⎩ jn
⎭
|a |
in 1 i
1<
0
0
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
Proof. Let L ∈ d − 1 − bD b|d|, d 1 − bD − b|d|. It follows from 3.3 that there exists
T ≥ 1 n0 n1 τ |β| satisfying
⎤
⎡
∞
i−1
1 ⎣
Fi Gj cj ⎦ ≤ 1 − bD − b|d| − |L − d|.
|a
i|
jn0
iT
3.4
Abstract and Applied Analysis
5
Define two mappings SL and GL : Bd, D → lβ∞ by
SL xn ⎧
⎨L − bn xn−τ , n ≥ T,
⎩
SL xT ,
3.5
β ≤ n < T,
⎫
⎧
⎧
∞
i−1 ⎬
⎨
⎪
1
⎪
⎨
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
GL xn in ai ⎩
jn
0
⎪
⎪
⎩
GL xT ,
n ≥ T,
3.6
β ≤ n < T,
for any x {xn }n∈Zβ ∈ Bd, D.
Now, we assert that
SL x GL y ∈ Bd, D,
∀x, y ∈ Bd, D,
3.7
SL x − S L y ≤ b x − y ,
∀x, y ∈ Bd, D,
3.8
∀x ∈ Bd, D.
3.9
GL x ≤ D,
It follows from 3.1, 3.2, and 3.4–3.6 that for any x {xn }n∈Zβ , y {yn }n∈Zβ ∈ Bd, D,
and n ≥ T ,
∞
1
SL xn GL yn − d L − d − bn xn−τ a
in i
⎫
⎧
i−1 ⎨ ⎬
g j, yg1j , . . . , ygkj − cj
× f i, yf1i , . . . , yfki ⎭
⎩
jn
0
⎡
⎤
∞
i−1
1 ⎣
≤ |L − d| b|d| D Fi Gj cj ⎦
|ai |
jn0
iT
3.10
≤ |L − d| b|d| D 1 − bD − b|d| − |L − d| D,
SL xn − SL yn bn xn−τ − yn−τ ≤ b x − y ,
⎤
⎡
∞
i−1
1 ⎣
Gj cj ⎦ ≤ 1 − bD − b|d| − |L − d| ≤ D,
Fi |GL xn | ≤
|a
|
i
jn
iT
0
which imply that 3.7–3.9 hold.
Next, we prove that GL is continuous and GL Bd, D is uniformly Cauchy. It follows
from 3.3 that for each ε > 0, there exists M > T satisfying
⎤
⎡
∞
i−1
ε
1 ⎣
Fi Gj cj ⎦ < .
4
|a
|
i
jn0
iM
3.11
6
Abstract and Applied Analysis
Let xν {xnν }n∈Zβ and x {xn }n∈Zβ ∈ Bd, D satisfy that
lim xν x.
3.12
ν→∞
In view of 3.12 and the continuity of f and g, we know that there exists V ∈ N such that
M−1
iT
⎡
1 ⎣ ν
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai |
⎤
i−1 ε
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎦ < ,
2
jn0
3.13
∀ν ≥ V.
Combining 3.6, 3.11, and 3.13, we obtain that
⎧
∞
⎨ 1
GL xν − GL x ≤
f i, xfν1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai | ⎩
iT
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
⎧
M−1
1 ⎨ ≤
f i, xfν1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai | ⎩
iT
3.14
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
⎞
∞
i−1
1 ⎝
2
Gj ⎠ < ε,
Fi |a
|
jn0
iM i
⎛
∀ν ≥ V,
which means that GL is continuous in Bd, D.
In view of 3.6 and 3.11, we obtain that for any x {xn }n∈Zβ ∈ Bd, D and t, h ≥ M
⎫
⎧
i−1 ⎬
∞ 1⎨ f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
|GL xt − GL xh | ≤ ⎭
it ai ⎩
jn0
⎫
⎧
i−1 ⎬
∞ 1⎨ f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
ih ai ⎩
jn
0
⎤
⎡
∞
i−1
1 ⎣
≤2
Fi Gj cj ⎦ < ε,
|a
|
jn0
iM i
3.15
Abstract and Applied Analysis
7
which implies that GL Bd, D is uniformly Cauchy, which together with 3.9 and
Lemma 2.1 yields that GL Bd, D is relatively compact. Consequently, GL is completely
continuous in Bd, D. Thus, 3.7, 3.8, and Lemma 2.2 ensure that the mapping SL GL
has a fixed point x {xn }n∈Zβ ∈ Bd, D, which together with 3.5 and 3.6 implies that
xn L − bn xn−τ
⎫
⎧
∞
i−1 ⎬
⎨
1
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
a⎩
in i
jn
n ≥ T,
0
3.16
which yields that
Δ an Δxn bn xn−τ f n, xf1n , . . . , xfkn g n, xg1n , . . . , xgkn cn ,
n ≥ T.
3.17
That is, x {xn }n∈Zβ is a bounded nonoscillatory solution of 1.1 in Bd, D.
Let L1 , L2 ∈ d − 1 − bD b|d|, d 1 − bD − b|d|, and L1 / L2 . Similarly, we can
prove that for each l ∈ {1, 2}, there exist a constant Tl ≥ 1 n0 n1 τ |β| and two mappings
SLl and GLl : Bd, D → lβ∞ satisfying 3.4–3.6, where T, L, SL , and GL are replaced by
Tl , Ll , SLl , and GLl , respectively, and SLl GLl has a fixed point zl ∈ Bd, D, which is a bounded
nonoscillatory solution of 1.1; that is,
zln Ll − bn zln−τ
⎫
⎧
∞
i−1 ⎬
1⎨ l
,
f i, zf1i , . . . , zlfki g j, zlg1j , . . . , zlgkj − cj
⎭
⎩
a
i
in
jn
0
∀n ≥ Tl ,
3.18
l ∈ {1, 2}.
Note that 3.3 implies that there exists T3 > max{T1 , T2 } satisfying
⎞
⎛
∞
i−1
1 ⎝
|L1 − L2 |
.
Fi Gj ⎠ <
4
|a |
jn0
iT3 i
Using 3.2, 3.18, and 3.19, we get that for any n ≥ T3 ,
1
zn − z2n bn z1n−τ − z2n−τ ⎧
∞
1 ⎨ 1
f j, zf1j , . . . , z1fkj − f j, z2f1j , . . . , z2fkj
L1 − L2 a⎩
in i
⎫
i−1 ⎬
g j, z1g1j , . . . , z1gkj − g j, z2g1j , . . . , z2gkj
⎭
jn0
3.19
8
Abstract and Applied Analysis
⎞
⎛
∞
i−1
1 ⎝
≥ |L1 − L2 | − 2
Fi Gj ⎠
|a |
jn0
iT3 i
>
|L1 − L2 |
2
> 0,
3.20
that is, z1 / z2 . Therefore, 1.1 possesses uncountably many bounded nonoscillatory solutions in Bd, D. This completes the proof.
Theorem 3.2. Assume that there exist n1 ∈ Nn0 , d ∈ R, D, b ∈ R \ {0}, and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
0
|d| 1 − b
<
,
D
b
|bn | ≤ b,
∀n ≥ n1 .
3.21
Then, 1.1 has uncountably many bounded solutions in Bd, D.
The proof of Theorem 3.2 is analogous to that of Theorem 3.1 and hence is omitted.
Theorem 3.3. Assume that there exist n1 ∈ Nn0 , d ∈ R, D, b∗ , b∗ ∈ R \ {0}, and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
0
|d| b∗ − 1
< ∗
,
D
b 1
1 < b∗ ≤ |bn | ≤ b∗ ,
∀n ≥ n1 .
3.22
Then, 1.1 has uncountably many bounded solutions in Bd, D.
Proof. Let L ∈ −b∗ − 1D b∗ 1|d|, b∗ − 1D − b∗ 1|d|. It follows from 3.3 that there
exists T ≥ 1 n0 n1 τ |β| satisfying
⎤
⎡
∞
i−1
1 ⎣
Fi Gj cj ⎦ ≤ b∗ − 1D − b∗ 1|d| − |L|.
|a
|
i
jn0
iT
3.23
Define two mappings SL and GL : Bd, D → lβ∞ by
⎧
⎨ L − xnτ , n ≥ T,
SL xn bnτ bnτ
⎩
β ≤ n < T,
SL xT ,
3.24
Abstract and Applied Analysis
⎫
⎧
⎧
i−1 ∞
⎬
⎨
⎪
1
⎪
⎨ 1
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
GL xn bnτ inτ ai ⎩
jn
0
⎪
⎪
⎩
GL xT ,
9
n ≥ T,
β ≤ n < T,
3.25
for any x {xn }n∈Zβ ∈ Bd, D.
Now, we assert that 3.7, 3.9, and the below
SL x − SL y ≤
1
x−y ,
b∗
∀x, y ∈ Bd, D
3.26
hold. It follows from 3.2 and 3.22–3.25 that for any x {xn }n∈Zβ , y {yn }n∈Zβ ∈ Bd, D,
and n ≥ T ,
SL xn GL yn − d
∞
L
1
xnτ
1 −d−
b
b
b
a
nτ
nτ
nτ
inτ i
⎫
⎧
i−1 ⎨ ⎬
g j, yg1j , . . . , ygkj − cj
× f i, yf1i , . . . , yfki ⎭
⎩
jn
0
⎤
⎡
∞
i−1
1
1 ⎣
1
|d| D
≤ |L − bnτ d| Fi Gj cj ⎦
b∗
b∗
b∗ iT |ai |
jn0
1
1
|d| D
b∗ − 1D − b∗ 1|d| − |L| ≤ D,
|L| b∗ |d| b∗
b∗
b∗
SL xn − SL yn 1 xnτ − ynτ ≤ 1 x − y ,
b
b
nτ
∗
⎤
⎡
i−1
∞
1
1 ⎣
1
Fi Gj cj ⎦ ≤ b∗ − 1D − b∗ 1|d| − |L| ≤ D,
|GL xn | ≤
b∗ iT |ai |
b
∗
jn0
≤
3.27
which imply 3.7, 3.9, and 3.26.
Next, we show that GL is continuous and GL Bd, D is uniformly Cauchy. It follows
from 3.3 that for each ε > 0, there exists M > T satisfying 3.11. Let xν {xnν }n∈Zβ and
10
Abstract and Applied Analysis
x {xn }n∈Zβ ∈ Bd, D with 3.12. It follows from 3.12 and the continuity of f and g that
there exists V ∈ N satisfying 3.13. In light of 3.11, 3.13, and 3.25, we deduce that
⎡
∞
1 ⎣ ν
1 GL x − GL x ≤
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki b∗ iT τ |ai |
ν
⎤
i−1 g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎦
jn0
⎡
1 1 M−1
⎣f i, xν , . . . , xν − f i, xf1i , . . . , xfki ≤
f1i
fki
b∗ iT |ai |
⎛
⎞
⎤
i−1 i−1
∞
2
1
⎝Fi g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎦ Gj ⎠
b∗ iM |ai |
jn0
jn0
<
ε
,
b∗
∀ν ≥ V,
3.28
which yields that GL is continuous in Bd, D.
Using 3.1 and 3.25, we get that for any x {xn }n∈Zβ ∈ Bd, D and t, h ≥ M
⎫
⎧
i−1 ∞
⎨ ⎬
1 1
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
|GL xt − GL xh | ≤ ⎭
btτ itτ ai ⎩
jn0
⎫
⎧
∞
i−1 ⎨ ⎬
1 1
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
bhτ ihτ ai ⎩
jn0
⎡
⎤
≤
i−1
∞
2 1 ⎣
Gj cj ⎦
Fi b∗ iM |ai |
jn0
<
ε
,
b∗
3.29
which means that GL Bd, D is uniformly Cauchy, which together with 3.9 and Lemma 2.1
yields that GL Bd, D is relatively compact. Consequently, GL is completely continuous in
Bd, D. Thus, 3.22, 3.26, and Lemma 2.2 ensure that the mapping SL GL has a fixed point
x {xn }n∈Zβ ∈ Bd, D; that is,
xn L
bnτ
⎫
⎧
i−1 ∞
⎬
1⎨ xnτ
1 g j, xg1j , . . . , xgkj − cj
,
−
f i, xf1i , . . . , xfki ⎭
bnτ bnτ inτ ai ⎩
jn0
n ≥ T,
3.30
Abstract and Applied Analysis
11
which gives that
Δ an Δxn bn xn−τ f n, xf1n , . . . , xfkn g n, xg1n , . . . , xgkn cn ,
n ≥ T τ.
3.31
That is, x {xn }n∈Zβ is a bounded solution of 1.1 in Bd, D.
L2 . Similarly,
Let L1 , L2 ∈ −b∗ − 1D b∗ 1|d|, b∗ − 1D − b∗ 1|d| and L1 /
we conclude that for each l ∈ {1, 2}, there exist a constant Tl ≥ 1 n0 n1 τ |β| and two
mappings SLl and GLl : Bd, D → lβ∞ satisfying 3.23–3.25, where T, L, SL , and GL are
replaced by Tl , Ll , SLl , and GLl , respectively, and SLl GLl has a fixed point zl ∈ Bd, D, which
is a bounded solution of 1.1; that is,
⎫
⎧
i−1 ∞
⎨ l
⎬
z
1
L
1
l
,
f i, zlf1i , . . . , zlfki g j, zlg1j , . . . , zlgkj − cj
zln − nτ ⎭
bnτ bnτ bnτ inτ ai ⎩
jn0
3.32
for all n ≥ Tl and l ∈ {1, 2}. Note that 3.3 implies that there exists T3 > max{T1 , T2 } satisfying
3.19. By means of 3.2, 3.19, and 3.32, we infer that for any n ≥ T3 ,
zlnτ − z2nτ 1
2
zn − zn bnτ
⎧
∞
⎨ L1 − L 2
1
1
f j, z1f1j , . . . , z1fkj − f j, z2f1j , . . . , z2fkj
bnτ inτ ai ⎩
bnτ
⎫
i−1 ⎬
g j, z1g1j , . . . , z1gkj − g j, z2g1j , . . . , z2gkj
⎭
jn
0
3.33
⎛
⎞
i−1
∞
1 ⎝
|L1 − L2 | 2 ≥
Fi −
Gj ⎠
b∗
b∗ iT3 |ai |
jn0
>
|L1 − L2 |
2b∗
> 0,
that is, z1 / z2 . Therefore, 1.1 possesses uncountably many bounded solutions in Bd, D.
This completes the proof.
Similar to the proofs of Theorems 3.1 and 3.3, we have the following results.
Theorem 3.4. Assume that there exist n1 ∈ Nn0 , d ∈ R, D, b∗ , b∗ ∈ R \ {0} and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
0
12
Abstract and Applied Analysis
b∗2 b∗ b∗ b∗2 − b∗2 − b∗2 D > b∗2 − b∗2 − b∗2 b∗ b∗ b∗2 |d|,
3.34
1 < b∗ ≤ bn ≤ b∗ , ∀n ≥ n1 .
|d| > D,
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
Theorem 3.5. Assume that there exist n1 ∈ Nn0 , d, D ∈ R \ {0}, b∗ , b∗ ∈ R− and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
d > D,
0
D2 b∗ b∗ < db∗ − b∗ ,
b∗ ≤ bn ≤ b∗ < −1,
∀n ≥ n1 .
3.35
Then, 1.1 has uncountably many bounded positive solutions in Bd, D.
Theorem 3.6. Assume that there exist n1 ∈ Nn0 , D ∈ R \ {0}, d, b∗ ∈ R− \ {0}, b∗ ∈ R− and two
nonnegative sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
−d > D,
0
db∗ − b∗ > D2 b∗ b∗ ,
b∗ ≤ bn ≤ b∗ ,
∀n ≥ n1 .
3.36
Then, 1.1 has uncountably many bounded negative solutions in Bd, D.
Theorem 3.7. Assume that there exist n1 ∈ Nn0 , d ∈ R \ {0}, b∗ , D ∈ R \ {0} and two negative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
0
1<
|d| 2 − b∗
<
,
D
b∗
0 ≤ bn ≤ b ∗ ,
∀n ≥ n1 .
3.37
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
Theorem 3.8. Assume that there exist n1 ∈ Nn0 , d ∈ R \ {0}, b∗ ∈ R− \ {0}, D ∈ R \ {0} and two
negative sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
1<
0
|d| b∗ 2
<
,
D
−b∗
b∗ ≤ bn ≤ 0,
∀n ≥ n1 .
3.38
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
Next, we investigate the existence of uncountably bounded nonoscillatory solutions
for 1.1 with the help of the Schauder fixed point theorem under the conditions of bn ±1.
Theorem 3.9. Assume that there exist n1 ∈ Nn0 , d ∈ R, D ∈ R \{0} and two nonnegative sequences
{Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2, 3.3, and
0
0
|d| > D,
bn 1,
∀n ≥ n1 .
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
3.39
Abstract and Applied Analysis
13
Proof. Let L ∈ d − D, d D. It follows from 3.3 that there exists T ≥ 1 n0 n1 τ |β|
satisfying
⎤
⎡
∞
i−1
1 ⎣
Gj cj ⎦ ≤ D − |L − d|.
Fi |ai |
jn0
iT
3.40
Define a mapping SL : Bd, D → lβ∞ by
SL xn ⎫
⎧
i−1 ⎬
⎨
1
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
a⎩
jn
s1 in2s−1τ i
⎧
∞
⎪
⎪
⎨L ⎪
⎪
⎩
n2sτ−1
n ≥ T,
0
β ≤ n < T,
3.41
SL xT ,
for any x {xn }n∈Zβ ∈ Bd, D.
Now, we prove that
SL x ∈ Bd, D,
SL x ≤ |L| D,
∀x ∈ Bd, D.
3.42
It follows from 3.2 and 3.39–3.41 that for any x {xn }n∈Zβ ∈ Bd, D and n ≥ T ,
⎫
⎧
∞ n2sτ−1
i−1 ⎨ ⎬
1
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
|SL xn − d| L − d ⎭
⎩
a
i
jn0
s1 in2s−1τ
⎤
⎡
∞
i−1
1 ⎣
≤ |L − d| Fi Gj cj ⎦
|ai |
jn0
iT
≤ |L − d| D − |L − d|
D,
⎤
⎡
∞
i−1
1 ⎣
Fi Gj cj ⎦ ≤ |L| D − |L − d| ≤ |L| D,
|SL xn | ≤ |L| |a
|
i
jn0
iT
3.43
which imply 3.42.
Next, we prove that SL is continuous and SL Bd, D is uniformly Cauchy. It follows
from 3.3 that for each ε > 0, there exists M > T satisfying 3.11. Let xν {xnν }n∈Zβ and
14
Abstract and Applied Analysis
x {xn }n∈Zβ ∈ Bd, D satisfying 3.12. It follows from 3.12 and the continuity of f and g
that there exists V ∈ N satisfying 3.13. Combining 3.11, 3.13, and 3.41, we infer that
SL xν − SL x
⎧
⎧
∞ n2sτ−1
⎨
1 ⎨ ν
≤ sup
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki n≥T ⎩ s1 in2s−1τ |ai | ⎩
⎫⎫
i−1 ⎬⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭⎭
jn
0
⎧
∞
1 ⎨ ν
≤
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai | ⎩
iT
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
⎧
M−1
1 ⎨ ≤
f i, xfν1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai | ⎩
iT
⎫
⎞
⎛
i−1 ∞
i−1
⎬
1 ⎝
ν
ν
g j, xg1j , . . . , xgkj − g j, xg1j , . . . , xgkj 2
Fi Gj ⎠
⎭
|a |
jn
jn
iM i
0
< ε,
0
∀ν ≥ V,
3.44
which implies that SL is continuous in Bd, D.
By means of 3.11 and 3.41, we get that for any x {xn }n∈Zβ ∈ Bd, D and t, h ≥ M
⎫
⎧
i−1 ⎨ ⎬
∞ t2sτ−1
1
g j, xg1j , . . . , xgkj − cj
f i, xf1i , . . . , xfki |SL xt − SL xh | ≤ ⎭
s1 it2s−1τ ai ⎩
jn0
⎫
⎧
i−1 ⎨ ⎬
∞ h2sτ−1
1
g j, xg1j , . . . , xgkj − cj
f i, xf1i , . . . , xfki ⎭
s1 ih2s−1τ ai ⎩
jn
0
⎡
⎤
∞
i−1
1 ⎣
Fi Gj cj ⎦
|a |
jn0
iM i
≤2
< ε,
3.45
Abstract and Applied Analysis
15
which means that SL Bd, D is uniformly Cauchy, which together with 3.42 and
Lemma 2.1 yields that SL Bd, D is relatively compact. It follows from Lemma 2.3 that the
mapping SL has a fixed point x {xn }n∈Zβ ∈ Bd, D; that is,
⎫
⎧
i−1 ⎬
⎨
1
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
xn L ⎭
a⎩
jn
s1 in2s−1τ i
∞
n2sτ−1
n ≥ T,
0
3.46
which give that
Δ an Δxn xn−τ f n, xf1n , . . . , xfkn g n, xg1n , . . . , xgkn cn ,
n ≥ T τ.
3.47
That is, x {xn }n∈Zβ ∈ Bd, D is a bounded nonoscillatory solution of 1.1.
L2 . Similarly, we infer that for each l ∈ {1, 2},
Let L1 , L2 ∈ d − D, d D and L1 /
there exist a constant Tl ≥ 1 n0 n1 τ |β| and a mapping SLl : Bd, D → lβ∞ satisfying
3.41, where L, T, and SL are replaced by Tl , Ll , and SLl , respectively, and SLl has a fixed
point zl ∈ Bd, D, which is a bounded nonoscillatory solution of 1.1; that is,
⎫
⎧
i−1 ⎬
⎨
1
,
f i, zlf1i , . . . , zlfki g j, zlg1j , . . . , zlgkj − cj
zln Ll ⎭
a⎩
jn
s1 in2s−1τ i
∞
n2sτ−1
n ≥ Tl ,
0
3.48
for l ∈ {1, 2}. Note that 3.3 implies that there exists T3 > max{T1 , T2 } satisfying 3.19. Using
3.2, 3.19, and 3.48, we conclude that for any n ≥ T3
1
zn − z2n ⎧
∞
n2sτ−1
⎨ 1
f i, z1f1i , . . . , z1fki − f i, z2f1i , . . . , z2fki
L1 − L2 a⎩
s1 in2s−1τ i
⎫
i−1 ⎬
g j, z1g1j , . . . , z1gkj − g j, z2g1j , . . . , z2gkj
⎭
jn0
⎞
⎛
i−1
1 ⎝
Gj ⎠
Fi |a |
jn0
s1 in2s−1τ i
∞
≥ |L1 − L2 | − 2
n2sτ−1
⎞
⎛
∞
i−1
1 ⎝
≥ |L1 − L2 | − 2
Gj ⎠
Fi |a |
jn0
iT3 i
>
|L1 − L2 |
2
> 0,
3.49
16
Abstract and Applied Analysis
z2 . Therefore, 1.1 possesses uncountably many bounded nonoscillawhich gives that z1 /
tory solutions in Bd, D. This completes the proof.
Theorem 3.10. Assume that there exist n1 ∈ Nn0 , d ∈ R, D ∈ R \ {0} and two nonnegative
sequences {Fn }n∈Nn and {Gn }n∈Nn satisfying 3.2,
0
0
bn −1, ∀n ≥ n1 ,
&
%
∞
∞
i−1
1
< ∞.
max Gj , cj
max Fi ,
|a |
jn0
s1 in0 sτ i
3.50
|d| > D,
3.51
Then, 1.1 has uncountably many bounded nonoscillatory solutions in Bd, D.
Proof. Let L ∈ d − D, d D. It follows from 3.41 that there exists T ≥ 1 n0 n1 τ |β|
satisfying
⎤
⎡
i−1
1 ⎣
Fi Gj cj ⎦ ≤ D − |L − d|.
|a
|
i
jn0
s1 iT sτ
∞
∞ 3.52
Define a mapping SL : Bd, D → lβ∞ by
⎫
⎧
⎧
∞
∞ i−1 ⎨ ⎬
⎪
1
⎪
⎨L −
,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
a⎩
SL xn jn0
s1 insτ i
⎪
⎪
⎩
SL xT ,
n≥T
β ≤ n < T,
3.53
for any x {xn }n∈Zβ ∈ Bd, D. It follows from 3.2, 3.52, and 3.53 that for any x {xn }n∈Zβ ∈ Bd, D and n ≥ T
⎫
⎧
∞
∞ i−1 ⎨ ⎬
1
g j, xg1j , . . . , xgkj − cj
f i, xf1i , . . . , xfki |SL xn − d| L − d −
⎭
a⎩
jn0
s1 insτ i
⎤
⎡
i−1
1 ⎣
≤ |L − d| Fi Gj cj ⎦
|a |
jn0
s1 iT sτ i
∞
∞ ≤ |L − d| D − |L − d|
D
⎡
⎤
i−1
1 ⎣
Gj cj Fi ⎦ ≤ |L| D,
|SL xn | ≤ |L| |a | jn0
s1 iT sτ i
∞
∞ which imply 3.42.
3.54
Abstract and Applied Analysis
17
Next, we show that SL is continuous and SL Bd, D is uniformly Cauchy. It follows
from 3.51 and Lemma 2.4 that for each ε > 0, there exists M > 1 T τ satisfying
∞ 1
iM
i
τ
⎡
⎤
ε
Gj cj ⎦ < .
4
i−1
1 ⎣
Fi |ai |
jn0
3.55
Let xν {xnν }n∈Zβ and x {xn }n∈Zβ ∈ Bd, D satisfying 3.12. By means of 3.12 and the
continuity of f and g, we deduce that there exists V ∈ N satisfying
M−1
iT τ
i
1
τ
⎧
1 ⎨ ν
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki |ai | ⎩
⎫
i−1 ⎬ ε
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj < ,
⎭ 2
jn0
3.56
∀ν ≥ V.
In light of 3.2, 3.53–3.56 and Lemma 2.4, we conclude that
⎧
∞
∞ 1 ⎨ ν
SL xν − SL x ≤
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki |a | ⎩
s1 iT sτ i
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
⎧
∞ 1 ⎨ ν
i
≤
1
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki τ |ai | ⎩
iT τ
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
⎧
M−1
1 ⎨ ν
i
≤
1
f i, xf1i , . . . , xfνki − f i, xf1i , . . . , xfki τ |ai | ⎩
iT τ
⎫
i−1 ⎬
g j, xgν1j , . . . , xgνkj − g j, xg1j , . . . , xgkj ⎭
jn
0
2
∞ iM
< ε,
i
1
τ
⎞
i−1
1 ⎝
Gj ⎠
Fi |ai |
jn0
⎛
∀ν ≥ V,
3.57
which implies that SL is continuous in Bd, D.
18
Abstract and Applied Analysis
By virtue of 3.53, 3.55, and Lemma 2.4, we get that for any x {xn }n∈Zβ ∈ Bd, D
and t, h ≥ M,
⎫
⎧
∞
i−1 ⎨ ⎬
∞ 1
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
|SL xt − SL xh | ≤ ⎭
s1 itsτ ai ⎩
jn0
⎫
⎧
∞
i−1 ⎬
⎨
∞ 1
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
⎭
s1 ihsτ ai ⎩
jn
0
∞
∞
≤2
⎤
⎡
3.58
i−1
1 ⎣
Fi Gj cj ⎦
|a |
jn0
s1 iMsτ i
⎤
⎡
∞ i−1
1 ⎣
i
≤2
Fi 1
Gj cj ⎦
τ
|a
|
i
jn0
iM
< ε,
which means that SL Bd, D is uniformly Cauchy, which together with 3.42 and
Lemma 2.1 yields that SL Bd, R is relatively compact. It follows from Lemma 2.3 that the
mapping SL has a fixed point x {xn }n∈Zβ ∈ Bd, R; that is,
⎫
⎧
∞
∞ i−1 ⎬
1⎨ ,
f i, xf1i , . . . , xfki g j, xg1j , . . . , xgkj − cj
xn L −
⎭
a⎩
jn0
s1 insτ i
n ≥ T,
3.59
which means that
Δ an Δxn − xn−τ f n, xf1n , . . . , xfkn g n, xg1n , . . . , xgkn cn ,
n ≥ T τ.
3.60
That is, x {xn }n∈Zβ is a bounded nonoscillatory solution of 1.1 in Bd, D.
L2 . Similarly, we conclude that for each l ∈ {1, 2},
Let L1 , L2 ∈ d − D, d D and L1 /
there exist a positive integer Tl ≥ 1 n0 n1 τ |β| and a mapping SLl : Bd, D → lβ∞
satisfying 3.53, where T, L, and SL are replaced by Tl , Ll , and SLl , respectively, and SLl has a
fixed point zl ∈ Bd, D, which is a bounded nonoscillatory solution of 1.1; that is,
⎫
⎧
∞
∞ i−1 ⎬
⎨
1
zln Ll −
g j, zlg1j , . . . , zlgkj − cj
,
f i, zlf1i , . . . , zlfki ⎭
a⎩
jn0
s1 insτ i
n ≥ T,
3.61
for l ∈ {1, 2}. Note that 3.41 implies that there exists T3 > max{T1 , T2 } satisfying
∞
∞
⎛
i−1
⎞
1 ⎝
|L1 − L2 |
,
Fi Gj ⎠ <
4
|a
|
i
jn0
s1 iT3 sτ
3.62
Abstract and Applied Analysis
19
which together with 3.2, 3.53, and 3.61 gives that
1
zn − z2n ⎧
∞ ∞
⎨ 1
f j, z1f1j , . . . , z1fkj − f j, z2f1j , . . . , z2fkj
L1 − L2 −
a⎩
s1 insτ i
⎫
i−1 ⎬
g j, z1g1j , . . . , z1gkj − g j, z2g1j , . . . , z2gkj
⎭
jn0
⎞
⎛
∞
∞
i−1
1 ⎝
≥ |L1 − L2 | − 2
Gj ⎠
Fi |a |
jn0
s1 iT3 sτ i
3.63
|L1 − L2 |
2
> 0, ∀n ≥ T3 ,
>
z2 . Therefore, 1.1 possesses uncountably many bounded nonoscillatory soluthat is, z1 /
tions in Bd, D. This completes the proof.
Remark 3.11. Theorems 3.1 and 3.4–3.10 generalize Theorem 1 in 5 and Theorems 2.1–2.7 in
6, 7, respectively. The examples in Section 4 reveal that Theorems 3.1 and 3.4–3.10 extend
authentically the corresponding results in 5–7.
4. Examples and Applications
Now, we construct ten examples to explain the advantage and applications of the results
presented in Section 3. Note that Theorem 1 in 5 and Theorem 2.1–2.7 in 6, 7 are invalid
for Examples 4.1–4.10, respectively.
Example 4.1. Consider the second-order nonlinear neutral delay difference equation
(
'
5n3 x2n1
−1n n − 1
6
n2 xn3 1 x2n2 3 n − 12 ,
xn−τ Δ n ln n Δ xn 2
3n 1
n x3n−15
n ≥ 2,
4.1
where n0 2 and τ ∈ N are fixed. Let n1 6, k 2, d ±6, D 5, b 1/3, β min{2 − τ, −9},
and
an n6 ln n,
f1n 2n 1,
bn −1n n − 1
,
3n 1
f2n 3n − 15,
Fn 55n2 ,
cn n − 12 ,
gn, u, v uvn2 ,
Gn 121n2 ,
fn, u, v g1n n3 1,
5un3
,
n v2
g2n 2n2 3,
n, u, v ∈ Nn0 × d − D, d D2 .
4.2
20
Abstract and Applied Analysis
It is easy to show that 3.1–3.3 hold. It follows from Theorem 3.1 that 4.1 possesses
uncountably many bounded nonoscillatory solutions in Bd, D.
Example 4.2. Consider the second-order nonlinear neutral delay difference equation
)
'
−1n−1 3 sin n
xn−τ
Δ n 1 − 2n Δ xn 4 sin n
*
3
2
(
2nx2n−9
n2 ,
1 n3 |x3n3 |
3n xn2 x5n 2
n ≥ 2,
4.3
where n0 2 and τ ∈ N are fixed. Let n1 5, k 2, d ±2, D 11, b 5/6, β min{2 − τ, −5},
and
an n2 1 − 2n3 ,
f1n n2 ,
bn f2n 5n,
Fn 507n2 ,
−1n−1 3 sin n
,
4 sin n
2nu
,
1 n3 |v|
gn, u, v Gn 26n,
cn n2 ,
fn, u, v 3n2 uv,
g1n 2n − 9,
g2n 3n3 ,
4.4
n, u, v ∈ Nn0 × d − D, d D2 .
It is clear that 3.2, 3.3, and 3.21 hold. It follows from Theorem 3.2 that 4.3 possesses
uncountably many bounded solutions in Bd, D.
Example 4.3. Consider the second-order nonlinear neutral delay difference equation
*
(
)
3
4n2 xn1
−1n−1 4n3 1
−1n
2 3
n4 x2n
Δ xn xn−τ xn−3 n3 ,
n sin
n
n3 2n 2
1 nxn2 2 −1
'
Δ
8
n ≥ 1,
4.5
where n0 1 and τ ∈ N are fixed. Let n1 10, k 2, d ±1, D 5, b∗ 3, b∗ 4,
β min{1 − τ, −2}, and
−1n−1 4n3 1
bn ,
n3 2n 2
−1n
,
an n sin
n
8
f1n n 1,
f2n n2 − 1,
Fn 864n2 ,
cn n3 ,
gn, u, v n4 u2 v3 ,
Gn 7776n4 ,
fn, u, v g1n 2n,
4nu3
,
1 nv2
g2n n − 3,
4.6
n, u, v ∈ Nn0 × d − D, d D2 .
It is easy to see that 3.2, 3.3, and 3.22 hold. It follows from Theorem 3.3 that 4.5
possesses uncountably many bounded solutions in Bd, D.
Example 4.4. Consider the second-order nonlinear neutral delay difference equation
' 1 −1/ln
5
Δ n sin
n
n
)
3n2 1
xn−τ
Δ xn 2
n n2
*
nxn2 2 3
1 n cos2 x2n
(
n4 − x3n
−1n
,
2
n−1
n2 xn1
n ≥ 2,
4.7
Abstract and Applied Analysis
21
where n0 2 and τ ∈ N are fixed. Let n1 5, k 2, d ±4, D 3, b∗ 2, b∗ 3, β 2 − τ, and
1 −1/ln n
,
an n5 sin
n
f1n n2 3,
bn f2n 2n,
Fn 49n,
3n2 1
,
n2 n 2
cn gn, u, v Gn n2 7,
−1n
,
n−1
n4 − u
,
n2 v2
fn, u, v g1n 3n,
nu2
,
1 n cos2 v
g2n n 1,
n, u, v ∈ Nn0 × d − D, d D2 .
4.8
It is easy to show that 3.2, 3.3, and 3.34 hold. It follows from Theorem 3.4 that 4.7 has
uncountably many bounded nonoscillatory solutions in Bd, D.
Example 4.5. Consider the second-order nonlinear neutral delay difference equation
' Δ n6 1 1
1n
*
(
3n )
6 − 2 ln 1 n2
4
3
xn−τ nx2n1 xn−9 n2 x3n x4n
Δ xn −
2n2 ,
5 ln1 n2 n ≥ 0,
4.9
where n0 0 and τ ∈ N are fixed. Let n1 10, k 2, d 9, D 7, b∗ −2, b∗ −17/15,
β min{−τ, −9}, and
an n 1 6
1
1n
3n
,
f1n 2n 1,
6 − 2 ln 1 n2
,
bn −
5 ln1 n2 f2n n 2,
Fn 1048576n,
gn, u, v n2 uv3 ,
Gn 65536n2 ,
cn 2n2 ,
fn, u, v nuv4 ,
g1n 3n,
g2n 4n,
n, u, v ∈ Nn0 × d − D, d D2 .
4.10
It is easy to show that 3.2, 3.3, and 3.35 hold. It follows from Theorem 3.5 that 4.9 has
uncountably many bounded positive solutions in Bd, D.
Example 4.6. Consider the second-order nonlinear neutral delay difference equation
)
'
3 − 2n4
n n − 4 Δ xn xn−τ
5 n n4
n−1 17
Δ −1
5
*
2
n18 − n7 x3n−19
ln3 n5 |x2n |
(
2
n2 xn5
2
1 x2n3
n15 ,
n ≥ 5,
4.11
22
Abstract and Applied Analysis
where n0 5 and τ ∈ N are fixed. Let n1 5, k 2, d −9, D 7, b∗ −2, b∗ −
β min{5 − τ, −4}, and
an −1n−1 n17 n − 45 ,
f1n 3n − 18,
bn f2n 2n,
3 − 2n4
,
5 n n4
gn, u, v Fn n18 256n7 ,
cn n15 ,
n 2 u2
,
1 v2
fn, u, v g1n n 5,
17
,
15
n18 − n7 u2
,
ln3 n5 |v|
g2n 2n 3,
n, u, v ∈ Nn0 × d − D, d D2 .
Gn 256n2 ,
4.12
It is easy to show that 3.2, 3.3, and 3.36 hold. It follows from Theorem 3.6 that 4.11 has
uncountably many bounded negative solutions in Bd, D.
Example 4.7. Consider the second-order nonlinear neutral delay difference equation
*
(
)
3
n2 − x2n−3
n2 x2n
π
5n2 − 2n 170
Δ n ln cos
−
Δ xn x
n2 2 − n,
n−τ
2
n
1 n|xn−6 |
6n2 n 1
n x4n
'
8
n ≥ 3,
4.13
where n0 3 and τ ∈ N are fixed. Let n1 60, k 2, d ±7, D 6, b∗ 5/6, β min{3−τ, −3},
and
π
,
an n8 ln cos
n
f1n 2n,
bn 5n2 − 2n 170
,
6n2 n 1
f2n 4n,
Fn 13n,
gn, u, v Gn 2197 n2 ,
cn n2 2 − n,
n 2 − u3
,
1 n|v|
fn, u, v −
g1n 2n − 3,
n2 u
,
n v2
g2n n − 6,
n, u, v ∈ Nn0 × d − D, d D2 .
4.14
It is clear 3.2, 3.3, and 3.37 hold. It follows from Theorem 3.7 that 4.13 has uncountably
many bounded nonoscillatory solutions in Bd, D.
Example 4.8. Consider the second-order nonlinear neutral delay difference equation
)
'
1 − 8n4
Δ n Δ xn xn−τ
3 9n4
6
*
2 (
n3 xn1 − n2 1 xn−3
2
1 nxn−3
3
n2 x2n5
−1n n3 ,
2 n2 |x3n−1 |
n ≥ 4,
4.15
Abstract and Applied Analysis
23
where n0 4 and τ ∈ N are fixed. Let n1 4, k 2, d ±7, D 6, b∗ −8/9, β min{4 − τ, 1},
and
1 − 8n4
bn ,
3 9n4
an n ,
6
f1n n 1,
f2n n − 3,
n 3
cn −1 n ,
gn, u, v Fn 13n3 169 n2 1 ,
n3 u − n2 1 v2
fn, u, v ,
1 nv2
n 2 u3
,
2 n2 |v|
Gn n2 2197,
g1n 2n 5,
g2n 3n − 1,
n, u, v ∈ Nn0 × d − D, d D2 .
4.16
It is clear 3.2, 3.3, and 3.38 hold. It follows from Theorem 3.8 that 4.15 has uncountably
many bounded nonoscillatory solutions in Bd, D.
Example 4.9. Consider the second-order nonlinear neutral delay difference equation
(
' 2
1 − n3 nx7n
1 n
−1n n2
n − 12 − nx3n1
6
,
Δxn xn−τ Δ n 1
2
5 3 n
n3 1
n2 ln 3 nx6n
1 n n5 x3n
x7n
n ≥ 1,
4.17
where n0 1 and τ ∈ N are fixed. Let n1 1, k 2, d ±6, D 2, β 1 − τ, and
an n
6
1
1
n
f2n 6n,
n
,
cn gn, u, v −1n n2
,
n3 1
fn, u, v 1 − n3 nu2
,
1 n n5 |v5 u3 |
Gn 1 64n n3 ,
n − 12 − nu
,
n2 ln3 nv2 g1n 7n,
f1n 3n 1,
g2n 3n,
Fn 1 8
,
n
n, u, v ∈ Nn0 × d − D, d D2 .
4.18
It is clear 3.2, 3.3, and 3.39 hold. It follows from Theorem 3.9 that 4.17 has uncountably
many bounded nonoscillatory solutions in Bd, D.
Example 4.10. Consider the second-order nonlinear neutral delay difference equation
5nx3n3 −n2
Δ n4 1 − 2n2n − 32 Δxn − xn−τ n2 x3n3 4 x4n3 −5 n2 ,
2 n3 |x5n5 3 |
n ≥ 2,
4.19
where n0 2 and τ ∈ N are fixed. Let n1 2, k 2, d ±10, D 6, β 2 − τ, and
an n4 1 − 2n2n − 32 ,
f2n 5n5 3,
cn n2 ,
gn, u, v uvn2 ,
Gn 256n2 ,
fn, u, v g1n 3n3 4,
5nu
,
2 n3 |v|
f1n 3n3 − n 2,
g2n 4n3 − 5,
Fn 40n,
n, u, v ∈ Nn0 × d − D, d D2 .
4.20
24
Abstract and Applied Analysis
It is clear 3.2, 3.50, and 3.51 hold. It follows from Theorem 3.10 that 4.19 possesses
uncountably bounded nonoscillatory solutions in Bd, D.
Acknowledgment
This research is financially supported by Changwon National University in 2011.
References
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