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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,
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 Δa
kn
···Δa
2n
Δa
1n
Δx
n
b
n
x
n−d
fn, 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
j1
{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
gn
0,n≥ 0 1.1
see 14,
Δ
a
n
Δx
n
q
n
x
n1
, Δ
a
n
Δx
n
q
n
f
x
n1
,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
i1
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
s1
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
j1
{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
n1
− x
n
. The higher-order
difference for a positive integer m is defined as Δ
m
x
n
ΔΔ
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, AM, 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
∞
tn
0
max
1
|
a
it
|
,h
t
,q
t
:1≤ i ≤ k
< ∞.
2.4
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Choose L ∈ M, N.By2.1, 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
∞
j1
∞
t
1
N
0
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≤ min
{
L − M, N − L
}
.
2.6
Define a mapping T
L
: AM, N → X by
T
L
x
n
⎧
⎪
⎨
⎪
⎩
L −
−1
k
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,n≥ N
0
,
T
L
x
N
0
,β≤ n<N
0
2.7
for all x ∈ AM, N.
i It is claimed that T
L
x ∈ AM, N, for all x ∈ AM, N.
Advances in Difference Equations 5
In fact, for every x ∈ AM, N and n ≥ N
0
, it follows from 2.3 and 2.6 that
T
L
x
n
≥ L −
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
≥ L −
∞
j1
∞
t
1
N
0
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≥ M,
T
L
x
n
≤ L
∞
j1
∞
t
1
N
0
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≤ N.
2.8
That is, T
L
xAM, N ⊆ AM, N.
ii It is declared that T
L
is continuous.
Let x {x
n
}∈AM, N and x
u
{x
u
n
}∈AM, 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
≤
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
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
i1
a
it
i
≤
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
h
t
max
x
u
t−r
jt
− x
t−r
jt
:1≤ j ≤ s
k
i1
a
it
i
≤
x
u
− x
∞
j1
∞
t
1
N
0
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
h
t
k
i1
a
it
i
.
2.9
This inequality and 2.4 imply that T
L
is continuous.
iii It can be asserted that T
L
AM, 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
∞
j1
∞
t
1
N
3
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
<
ε
2
.
2.10
Then, for any x {x
n
}∈AM, N and n
1
,n
2
≥ N
3
, 2.10 ensures that
|
T
L
x
n
1
− T
L
x
n
2
|
≤
∞
j1
∞
t
1
n
1
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
∞
j1
∞
t
1
n
2
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
≤
∞
j1
∞
t
1
N
3
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
∞
j1
∞
t
1
N
3
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
<
ε
2
ε
2
ε,
2.11
which means that T
L
AM, N is uniformly Cauchy. Therefore, by Lemma 1.4, T
L
AM, N is
relatively compact.
By Lemma 1.1, there exists x {x
n
}∈AM, 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
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
x
n−d
L −
−1
k
∞
j1
∞
t
1
nj−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
2.12
Advances in Difference Equations 7
which derives that
x
n
− x
n−d
−1
k
∞
j1
njd−1
t
1
nj−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
Δ
x
n
− x
n−d
−1
k
∞
j1
njd
t
1
n1j−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
−
−1
k
∞
j1
njd−1
t
1
nj−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
−
−1
k
∞
j1
∞
t
2
nj−1d
∞
t
3
t
2
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
a
1nj−1d
k
i2
a
it
i
−1
k
∞
j1
∞
t
2
njd
∞
t
3
t
2
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
a
1njd
k
i2
a
it
i
−1
k−1
∞
t
2
n
∞
t
3
t
2
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
a
1n
k
i2
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
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i2
a
it
i
,
2.14
by which it follows that
Δ
a
1n
Δ
x
n
− x
n−d
−1
k−1
∞
t
2
n1
∞
t
3
t
2
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i2
a
it
i
−
−1
k−1
∞
t
2
n
∞
t
3
t
2
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i2
a
it
i
−1
k−2
∞
t
3
n
∞
t
4
t
3
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
a
2n
k
i3
a
it
i
,
.
.
.
Δ
a
kn
···Δ
a
2n
Δ
a
1n
Δ
x
n
b
n
x
n−d
−1
k−k1
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
∞
j1
∞
t
1
N
4
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
h
t
/|
k
i1
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 ∈ AM, N, respectively,
which are bounded nonoscillatory solutions of 1.11 in AM, N. For the sake of proving
that 1.11 possesses uncountably many bounded nonoscillatory solutions in AM, N,itis
only needed to show that x
/
y.Infact,by2.7, we know that, for n ≥ N
4
,
x
n
L
1
−
−1
k
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
y
n
L
2
−
−1
k
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, y
t−r
1t
,y
t−r
2t
, ,y
t−r
st
k
i1
a
it
i
.
2.16
Then,
x
n
− y
n
≥
|
L
1
− L
2
|
−
∞
j1
∞
t
1
njd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
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
i1
a
it
i
≥
|
L
1
− L
2
|
−
x − y
∞
j1
∞
t
1
N
4
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
h
t
k
i1
a
it
i
≥
|
L
1
− L
2
|
− 2N
∞
j1
∞
t
1
N
4
jd
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
h
t
k
i1
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 AM, N.
Advances in Difference Equations 9
Proof. Choose L ∈ M, N.By2.18 and 2.4, an integer N
0
>n
0
d |α| can be chosen such
that
b
n
≡ 1, ∀n ≥ N
0
,
∞
j1
N
0
2jd−1
t
1
N
0
2j−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≤ min
{
L − M, N − L
}
.
2.19
Define a mapping T
L
: AM, N → X by
T
L
x
n
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
L
−1
k
∞
j1
n2jd−1
t
1
n2j−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,n≥ N
0
,
T
L
x
N
0
,β≤ n<N
0
2.20
for all x ∈ AM, N.
The proof that T
L
has a fixed point x {x
n
}∈AM, 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
∞
j1
n2jd−1
t
1
n2j−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
x
n−d
L
−1
k
∞
j1
n2j−1d−1
t
1
n2j−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,
2.21
by which it follows that
x
n
x
n−d
2L
−1
k
∞
j1
njd−1
t
1
nj−1d
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
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 AM, N.
10 Advances in Difference Equations
Proof. Choose L ∈ M bN, N − bN.By2.23 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
∞
tt
k
q
t
k
i1
a
it
i
≤ min
{
L − bN − M, N − bN − L
}
.
2.24
Define two mappings T
1L
,T
2L
: AM, 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
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,n≥ N
0
,
T
2L
x
N
0
,β≤ n<N
0
2.25
for all x ∈ AM, N.
i It is claimed that T
1L
x T
2L
y ∈ AM, N, for all x, y ∈ AM, N.
In fact, for every x, y ∈ AM, 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
∞
tt
k
q
t
k
i1
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
∞
tt
k
q
t
k
i1
a
it
i
≤ N.
2.26
That is, T
1L
x T
2L
yAM, N ⊆ AM, N.
ii It is declared that T
1L
is a contraction mapping on AM, N.
In reality, for any x, y ∈ AM, 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 AM, 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
}∈AM, N such that T
1L
x T
2L
x x, which is a
bounded nonoscillatory solution of 1.11. This completes the proof.
Advances in Difference Equations 11
Theorem 2.5. Assume that there exist constants M and N with N>2 − b
/1 − bM>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
≥ 0, eventually, and 0 ≤ b ≤ b<1.
2.29
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Choose L ∈ M 1
b/2N, N b/2M.By2.29 and 2.4, an integer N
0
>
n
0
d |α| can be chosen such that
b
2
≤ b
n
≤
1
b
2
, ∀n ≥ N
0
,
∞
t
1
N
0
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≤ min
L − M −
1
b
2
N, N − L
b
2
M
.
2.30
Define two mappings T
1L
,T
2L
: AM, N → X as 2.25. The rest of the proof is analogous to
that in Theorem 2.4. This completes the proof.
Similar to the proof of Theorem 2.5, we have the following theorem.
Theorem 2.6. Assume that there exist constants M and N with N>2
b/1 bM>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
≤ 0, eventually, and − 1 <b≤ b ≤ 0.
2.31
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Theorem 2.7. Assume that there exist constants M and N with N>b
b
2
− b/bb
2
− bM>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, 1 <band b<b
2
< ∞.
2.32
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Take ε ∈ 0,b
− 1 sufficiently small satisfying
1 <b
− ε<
b
ε<
b
− ε
2
,
b ε
b
− ε
2
−
b ε
2
N>
b ε
2
b
− ε
−
b − ε
2
M.
2.33
12 Advances in Difference Equations
Choose L ∈
b εM b ε/b − εN, b − εN b − ε/b εM.By2.33, an integer
N
0
>n
0
d |α| can be chosen such that
b
− ε<b
n
< b ε, ∀b ≥ N
0
,
∞
t
1
N
0
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
q
t
k
i1
a
it
i
≤ min
b
− ε
b ε
L −
b
− ε
M − N,
b
− ε
b ε
M
b
− ε
N − L
.
2.34
Define two mappings T
1L
,T
2L
: AM, N → X by
T
1L
x
n
⎧
⎪
⎪
⎨
⎪
⎪
⎩
L
b
nd
−
x
nd
b
nd
,n≥ N
0
,
T
1L
x
N
0
,β≤ n<N
0
,
T
2L
x
n
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−1
k
b
nd
∞
t
1
n
∞
t
2
t
1
···
∞
t
k
t
k−1
∞
tt
k
f
t, x
t−r
1t
,x
t−r
2t
, ,x
t−r
st
k
i1
a
it
i
,n≥ N
0
,
T
2L
x
N
0
,β≤ n<N
0
2.35
for all x ∈ AM, N. The rest of the proof is analogous to that in Theorem 2.4. This completes
the 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 bM>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, −∞ <band b<−1.
2.36
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Remark 2.9. Similar to Remark 2.2, we can also prove that the conditions of Theorems 2.3–2.8
ensure that 1.11 has 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
2.1–2.7ofLiuetal.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:
Δ
4
n
Δ
3
n
Δ
2
n
Δ
x
n
− x
n−1
0,n≥ 1. 3.1
Advances in Difference Equations 13
Choose M 1andN 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 A1, 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 neutral delay difference equation:
Δ
2
n
− n
Δ
n
2
− n 1
Δ
x
n
2
n
− 1
3
n
x
n−4
sin
2x
n−2
n
2
−
cos
3x
n−3
n
3
0,n≥ 5,
3.2
where
a
1n
n
2
− n 1,a
2n
2
n
− n, b
n
2
n
− 1
3
n
,
f
n, u
1
,u
2
sin
2u
1
n
2
−
cos
3u
2
n
3
,h
n
q
n
2
n
2
.
3.3
Choose M 1andN 5. It can be verified that the assumptions of Theorem 2.5 are fulfilled.
It follows from Theorem 2.5 that 3.2 has a nonoscillatory solution in A1, 5. However, the
results in 3, 4, 6, 8–17 are unapplicable for 3.2.
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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