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
Min Liu and Zhenyu Guo
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 neutral delay
difference equation Δakn · · · Δa2n Δa1n Δxn bn xn−d fn, xn−r1n , xn−r2n , . . . , xn−rsn 0, n ≥ n0 ,
where n0 ≥ 0, d > 0, k > 0, and s > 0 are integers, {ain }n≥n0 i 1, 2, . . . , k and {bn }n≥n0 are real
sequences, sj1 {rjn }n≥n0 ⊆ Z, and f : {n : n ≥ n0 } × Rs → 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 {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 1–17
and references cited therein. Some authors have paied their attention to various
difference equations. For example,
Δan Δxn pn xgn 0,
n≥0
1.1
see 14
,
Δan Δxn qn xn1 ,
Δan Δxn qn fxn1 ,
n≥0
1.2
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see 11
,
Δ2 xn pxn−m pn xn−k − qn xn−l 0,
1.3
n ≥ n0
see 6
,
Δ2 xn pxn−k fn, xn 0,
n≥1
1.4
n ≥ n0
1.5
see 10
,
m
Δ2 xn − pxn−τ qi fi xn−σi ,
i1
see 9
,
Δan Δxn bxn−τ fn, xn−d1n , xn−d2n , . . . , xn−dkn cn ,
n ≥ n0
1.6
see 8
,
Δm xn cxn−k pn xn−r 0,
n ≥ n0
1.7
see 15
,
Δm xn cn xn−k pn fxn−r 0,
1.8
n ≥ n0
see 3, 4, 12, 13
,
Δm xn cxn−k u
pns fs xn−rs qn ,
n ≥ n0
1.9
Δm xn cxn−k pn xn−r − qn xn−l 0,
n ≥ n0
1.10
s1
see 16
,
see 17
.
Motivated and inspired by the papers mentioned above, in this paper, we investigate
the following higher-order nonlinear neutral delay difference equation:
Δakn · · · Δa2n Δa1n Δxn bn xn−d fn, xn−r1n , xn−r2n , . . . , xn−rsn 0,
n ≥ n0 ,
1.11
where n0 ≥ 0, d > 0, k > 0, and s > 0 are integers, {ain }n≥n0 i 1, 2, . . . , k and {bn }n≥n0 are real
sequences, sj1 {rjn }n≥n0 ⊆ Z, and f : {n : n ≥ n0 } × Rs → R is a mapping. Clearly, difference
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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. Let T : Ω → Ω be a continuous mapping such that T Ω 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 T1 , T2 : Ω → X satisfy T1 x T2 y ∈ Ω for each x, y ∈ Ω. If T1 is a
contraction mapping and T2 is a completely continuous mapping, then the equation T1 x T2 x x has
at least one solution in Ω.
The forward difference Δ is defined as usual, that is, Δxn xn1 − xn . The higher-order
difference for a positive integer m is defined as Δm xn ΔΔm−1 xn , Δ0 xn xn . Throughout this
paper, assume that R −∞, ∞, N and Z stand 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, and lβ∞ 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 supn≥β |xn |
for x {xn }n≥β ∈ lβ∞ . It is well known that lβ∞ 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
. A set Ω of sequences in lβ∞ is uniformly Cauchy or equi-Cauchy if,
for every ε > 0, there exists an integer N0 such that
xi − xj < ε,
1.12
whenever i, j > N0 for any x {xk }k≥β in Ω.
Lemma 1.4 discrete Arzela-Ascoli’s theorem 5
. A bounded, uniformly Cauchy subset Ω of lβ∞
is relatively compact.
Let
for N > M > 0.
AM, N x {xn }n≥β ∈ lβ∞ : M ≤ xn ≤ N, ∀n ≥ β
1.13
Obviously, AM, N is a bounded closed and convex subset of lβ∞ . Put
b lim sup bn ,
n→∞
b lim inf bn .
n→∞
1.14
By a solution of 1.11, we mean a sequence {xn }n≥β with a positive integer N0 ≥
n0 d |α| such that 1.11 is satisfied for all n ≥ N0 . As is customary, a solution of 1.11 is
said to be oscillatory about zero, or simply oscillatory, if the terms xn of the sequence {xn }n≥β
are neither eventually all positive nor eventually all negative. Otherwise, the solution is called
nonoscillatory.
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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
{ain }n≥n0 1 ≤ i ≤ k, {bn }n≥n0 , {hn }n≥n0 , and {qn }n≥n0 such that, for n ≥ n0 ,
bn ≡ −1, eventually,
2.1
fn, u1 , u2 , . . . , us − fn, v1 , v2 , . . . , vs ≤ hn max{|ui − vi | : ui , vi ∈ M, N
, 1 ≤ i ≤ s},
2.2
fn, u1 , u2 , . . . , us ≤ qn , ui ∈ M, N
, 1 ≤ i ≤ s,
2.3
∞
1
, ht , qt : 1 ≤ i ≤ k < ∞.
max
2.4
|ait |
tn0
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Choose L ∈ M, N. By 2.1, 2.4, and the definition of convergence of series, an
integer N0 > n0 d |α| can be chosen such that
bn ≡ −1,
∞
∞
∞
j1 t1 N0 jd t2 t1
···
∀n ≥ N0 ,
2.5
∞ ∞
qt
≤ min{L − M, N − L}.
k
tk tk−1 ttk i1 aiti 2.6
Define a mapping TL : AM, N → X by
⎧
∞
∞
∞
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ⎪
⎨L − −1k
···
,
k
TL xn tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti
⎪
⎩
TL xN0 ,
for all x ∈ AM, N.
i It is claimed that TL x ∈ AM, N, for all x ∈ AM, N.
n ≥ N0 ,
β ≤ n < N0
2.7
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In fact, for every x ∈ AM, N and n ≥ N0 , it follows from 2.3 and 2.6 that
∞ ∞ ft, xt−r1t , xt−r2t , . . . , xt−rst ···
TL xn ≥ L −
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti ∞
∞
∞
≥L−
∞
∞
∞
···
j1 t1 N0 jd t2 t1
∞ ∞
qt
k
tk tk−1 ttk a
i1 iti 2.8
≥ M,
TL xn ≤ L ∞
∞
∞
j1 t1 N0 jd t2 t1
···
∞ ∞
qt
k
tk tk−1 ttk i1 aiti ≤ N.
That is, TL xAM, N ⊆ AM, N.
ii It is declared that TL is continuous.
u
Let x {xn } ∈ AM, N and xu {xn } ∈ AM, N be any sequence such that
u
xn → xn as u → ∞. For n ≥ N0 , 2.2 guarantees that
u
TL xn − TL xn u
u
u
∞ ∞ f t, x
−
ft,
x
,
x
,
.
.
.
,
x
,
x
,
.
.
.
,
x
t−r
t−r
t−r
1t
2t
st t−r1t
t−r2t
t−rst
≤
···
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti ∞
∞
∞
u
∞ ∞ ht max x
:1≤j≤s
−
x
t−r
jt t−rjt
≤
···
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti ∞
∞
∞
∞
≤ xu − x
∞
∞
j1 t1 N0 jd t2 t1
···
∞ ∞
h
t .
k
tk tk−1 ttk i1 aiti 2.9
This inequality and 2.4 imply that TL is continuous.
iii It can be asserted that TL AM, N is relatively compact.
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Advances in Difference Equations
By 2.4, for any ε > 0, take N3 ≥ N0 large enough so that
∞
∞
∞
···
j1 t1 N3 jd t2 t1
∞ ∞
qt
ε
< .
k
2
tk tk−1 ttk i1 aiti 2.10
Then, for any x {xn } ∈ AM, N and n1 , n2 ≥ N3 , 2.10 ensures that
|TL xn1 − TL xn2 | ≤
∞
∞
∞
···
j1 t1 n1 jd t2 t1
∞ ∞ ft, x
t−r1t , xt−r2t , . . . , xt−rst k
tk tk−1 ttk
i1 aiti ∞ ∞ ft, xt−r1t , xt−r2t , . . . , xt−rst ···
k
tk tk−1 ttk
j1 t1 n2 jd t2 t1
i1 aiti ∞
∞
≤
∞
∞
∞
∞
···
j1 t1 N3 jd t2 t1
∞
∞
∞
∞ ∞
qt
k
tk tk−1 ttk i1 aiti ···
j1 t1 N3 jd t2 t1
<
2.11
∞ ∞
qt
k
tk tk−1 ttk a
i1 iti ε ε
ε,
2 2
which means that TL AM, N is uniformly Cauchy. Therefore, by Lemma 1.4, TL AM, N is
relatively compact.
By Lemma 1.1, there exists x {xn } ∈ AM, N such that TL x x, which is a bounded
nonoscillatory solution of 1.11. In fact, for n ≥ N0 d,
xn L − −1k
∞
∞
∞
···
j1 t1 njd t2 t1
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
k
tk tk−1 ttk
i1 aiti
2.12
xn−d
∞
k
∞
∞
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst L − −1
···
,
k
tk tk−1 ttk
j1 t1 nj−1d t2 t1
i1 aiti
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which derives that
xn − xn−d −1k
∞
njd−1
∞
···
j1 t1 nj−1d t2 t1
Δxn − xn−d −1k
∞
njd
∞
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
k
tk tk−1 ttk
i1 aiti
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst k
tk tk−1 ttk
i1 aiti
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst k
tk tk−1 ttk
i1 aiti
j1 t1 n1j−1d t2 t1
− −1k
∞
njd−1
∞
j1 t1 nj−1d t2 t1
∞
k
2.13
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst −−1
···
a1nj−1d ki2 aiti
tk tk−1 ttk
j1 t2 nj−1d t3 t2
−1k
∞
∞
∞
∞
∞
···
j1 t2 njd t3 t2
−1k−1
∞ ∞
···
t2 n t3 t2
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst a1njd ki2 aiti
tk tk−1 ttk
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst .
a1n ki2 aiti
tk tk−1 ttk
That is,
a1n Δxn − xn−d −1k−1
∞ ∞
···
t2 n t3 t2
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
k
tk tk−1 ttk
i2 aiti
2.14
by which it follows that
∞ ∞
Δa1n Δxn − xn−d −1k−1
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst k
tk tk−1 ttk
i2 aiti
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst k
tk tk−1 ttk
i2 aiti
t2 n1 t3 t2
−−1k−1
∞ ∞
t2 n t3 t2
−1k−2
∞ ∞
t3 n t4 t3
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
a2n ki3 aiti
tk tk−1 ttk
..
.
Δakn · · · Δa2n Δa1n Δxn bn xn−d −1k−k1 fn, xn−r1n , xn−r2n , . . . , xn−rsn −fn, xn−r1n , xn−r2n , . . . , xn−rsn .
2.15
Therefore, x is a bounded nonoscillatory solution of 1.11. This completes the proof.
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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.
L2 . For L1 and L2 , as the preceding proof
In fact, let L1 , L2 ∈ M, N with L1 /
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,
∞
k
∞
∞
∞ ∞
and
j1
t2 t1 · · ·
tk tk−1
ttk ht /|
i1 aiti | < |L1 − L2 |/2N for some N4 ≥
t1 N4 jd
max{N1 , N2 }. Then the mappings TL1 and TL2 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, it is
only needed to show that x /
y. In fact, by 2.7, we know that, for n ≥ N4 ,
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti
∞
∞
∞
∞ ∞ f t, y
t−r1t , yt−r2t , . . . , yt−rst
yn L2 − −1k
···
.
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti
xn L1 − −1k
∞
∞
∞
···
2.16
Then,
xn − yn ≥ |L1 − L2 |
∞ ∞ ft, x
t−r1t , xt−r2t , . . . , xt−rst − f t, yt−r1t , yt−r2t , . . . , yt−rst
···
−
k
tk tk−1 ttk
j1 t1 njd t2 t1
i1 aiti ∞
∞
∞
∞
≥ |L1 − L2 | − x − y
∞
∞
j1 t1 N4 jd t2 t1
≥ |L1 − L2 | − 2N
∞
∞
∞
j1 t1 N4 jd t2 t1
> 0,
···
···
∞ ∞
h
t k
tk tk−1 ttk i1 aiti ∞ ∞
h
t k
tk tk−1 ttk i1 aiti 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 , satisfying 2.2–2.4 and
bn ≡ 1,
eventually.
Then 1.11 has a bounded nonoscillatory solution in AM, N.
2.18
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Proof. Choose L ∈ M, N. By 2.18 and 2.4, an integer N0 > n0 d |α| can be chosen such
that
bn ≡ 1,
∞
N0 2jd−1
∞
···
j1 t1 N0 2j−1d t2 t1
∀n ≥ N0 ,
∞ ∞
qt
≤ min{L − M, N − L}.
k
tk tk−1 ttk a
it
i
i1
2.19
Define a mapping TL : AM, N → X by
⎧
n2jd−1
∞
∞
⎪
⎪
k
⎪
⎪
L
−1
⎪
⎪
⎪
j1 t1 n2j−1d t2 t1
⎪
⎨
∞
∞
TL xn ft, xt−r1t , xt−r2t , . . . , xt−rst ⎪
⎪
···
, n ≥ N0 ,
k
⎪
⎪
⎪
tk tk−1 ttk
⎪
i1 aiti
⎪
⎪
⎩
β ≤ n < N0
TL xN0 ,
2.20
for all x ∈ 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 point x is a bounded nonoscillatory solution of 1.11.
In fact, for n ≥ N0 d,
xn L −1k
∞
n2jd−1
∞
···
j1 t1 n2j−1d t2 t1
xn−d
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ,
k
tk tk−1 ttk
i1 aiti
∞ n2j−1d−1
∞
k
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst L −1
···
,
k
tk tk−1 ttk
j1 t1 n2j−1d t2 t1
i1 aiti
2.21
by which it follows that
xn xn−d 2L −1k
∞
njd−1
∞
j1 t1 nj−1d t2 t1
···
∞ ∞
ft, xt−r1t , xt−r2t , . . . , xt−rst .
k
tk tk−1 ttk
i1 aiti
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
{ain }n≥n0 1 ≤ i ≤ k, {bn }n≥n0 , {hn }n≥n0 , {qn }n≥n0 , satisfying 2.2–2.4 and
|bn | ≤ b <
N−M
,
2N
eventually.
Then 1.11 has a bounded nonoscillatory solution in AM, N.
2.23
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Proof. Choose L ∈ M bN, N − bN. By 2.23 and 2.4, an integer N0 > n0 d |α| can be
chosen such that
|bn | ≤ b <
∞ ∞
t1 N0 t2 t1
···
N−M
,
2N
∀n ≥ N0 ,
∞ ∞
qt
≤ min{L − bN − M, N − bN − L}.
k
tk tk−1 ttk a
it
i
i1
2.24
Define two mappings T1L , T2L : AM, N → X by
T1L xn ⎧
⎨L − bn xn−d ,
n ≥ N0 ,
⎩T x ,
β ≤ n < N0 ,
1L N0
⎧
∞ ∞ ∞
∞
ft, xt−r1t , xt−r2t , . . . , xt−rst ⎪
⎨−1k
···
,
k
T2L xn aiti
t1 n t2 t1
tk tk−1 ttk
i1
⎪
⎩
T2L xN0 ,
2.25
n ≥ N0 ,
β ≤ n < N0
for all x ∈ AM, N.
i It is claimed that T1L x T2L y ∈ AM, N, for all x, y ∈ AM, N.
In fact, for every x, y ∈ AM, N and n ≥ N0 , it follows from 2.3, 2.24 that
T1L x T2L y
T1L x T2L y
n
n
≥ L − bN −
∞ ∞
···
t1 N0 t2 t1
≤ L bN ∞ ∞
t1 N0 t2 t1
···
∞ ∞
qt
≥ M,
k
tk tk−1 ttk a
i1 iti ∞ ∞
qt
k
tk tk−1 ttk 2.26
≤ N.
i1 aiti That is, T1L x T2L yAM, N ⊆ AM, N.
ii It is declared that T1L is a contraction mapping on AM, N.
In reality, for any x, y ∈ AM, N and n ≥ N0 , it is easy to derive that
T1L x − T1L y ≤ |bn |xn−d − yn−d ≤ bx − y,
n
n
2.27
T1L x − T1L y ≤ bx − y.
2.28
which implies that
Then, b < N − M/2N < 1 ensures that T1L 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 T2L is
completely continuous.
By Lemma 1.2, there exists x {xn } ∈ AM, N such that T1L x T2L x x, which is a
bounded nonoscillatory solution of 1.11. This completes the proof.
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Theorem 2.5. Assume that there exist constants M and N with N > 2 − b/1 − bM > 0 and
sequences {ain }n≥n0 1 ≤ i ≤ k, {bn }n≥n0 , {hn }n≥n0 , {qn }n≥n0 , satisfying 2.2–2.4 and
2.29
bn ≥ 0, eventually, and 0 ≤ b ≤ b < 1.
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Choose L ∈ M 1 b/2N, N b/2M. By 2.29 and 2.4, an integer N0 >
n0 d |α| can be chosen such that
b
1b
≤ bn ≤
, ∀n ≥ N0 ,
2
2
∞ ∞
∞ ∞
b
qt
1b
≤ min L − M −
N, N − L M .
···
k
2
2
tk tk−1 ttk a
t1 N0 t2 t1
it
i
i1
2.30
Define two mappings T1L , T2L : 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 {ain }n≥n0 1 ≤ i ≤ k, {bn }n≥n0 , {hn }n≥n0 , {qn }n≥n0 , satisfying 2.2–2.4 and
2.31
bn ≤ 0, eventually, and − 1 < b ≤ b ≤ 0.
Then 1.11 has a bounded nonoscillatory solution in AM, N.
2
Theorem 2.7. Assume that there exist constants M and N with N > bb − b/bb2 − bM > 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, 1 < b and b < b2 < ∞.
2.32
Then 1.11 has a bounded nonoscillatory solution in AM, N.
Proof. Take ε ∈ 0, b − 1 sufficiently small satisfying
2
1<b−ε <bε < b−ε ,
2 2 2 2
bε b−ε − bε
bε b−ε − b−ε
N>
M.
2.33
12
Advances in Difference Equations
Choose L ∈ b εM b ε/b − εN, b − εN b − ε/b εM. By 2.33, an integer
N0 > n0 d |α| can be chosen such that
b − ε < bn < b ε, ∀b ≥ N0 ,
∞ ∞
∞ ∞
b−ε
b−ε
qt
≤ min
L − b − ε M − N,
M b−ε N−L .
···
k
bε
bε
tk tk−1 ttk t1 N0 t2 t1
i1 aiti 2.34
Define two mappings T1L , T2L : AM, N → X by
⎧
xnd
L
⎪
⎪
−
, n ≥ N0 ,
⎨
bnd bnd
T1L xn ⎪
⎪
⎩
β ≤ n < N0 ,
T1L xN0 ,
⎧
∞ ∞
∞
∞ ft, xt−r1t , xt−r2t , . . . , xt−rst −1k ⎪
⎪
···
, n ≥ N0 ,
⎨
k
bnd t1 n t2 t1 tk tk−1 ttk
T2L xn i1 aiti
⎪
⎪
⎩
β ≤ n < N0
T2L xN0 ,
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 {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 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
2.1–2.7 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:
Δ4n Δ3n Δ2n Δxn − 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 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:
2n − 1
sin2xn−2 cos3xn−3 x
−
0,
Δ 2n − nΔ n2 − n 1 Δ xn n−4
n
3
n2
n3
n ≥ 5,
3.2
where
a1n n2 − n 1,
fn, u1 , u2 a2n 2n − n,
sin2u1 cos3u2 −
,
n2
n3
bn 2n − 1
,
3n
hn qn 2
.
n2
3.3
Choose M 1 and N 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.
References
1
R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd
edition, 2000.
2
R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation Theory for Difference and Functional Differential
Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
3
R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, “Oscillations of higher-order neutral difference
equations,” Applied Mathematics Letters, vol. 10, no. 1, pp. 71–78, 1997.
4
R. P. Agarwal and S. R. Grace, “Oscillation of higher-order nonlinear difference equations of neutral
type,” Applied Mathematics Letters, vol. 12, no. 8, pp. 77–83, 1999.
5
S. S. Cheng and W. T. Patula, “An existence theorem for a nonlinear difference equation,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 20, no. 3, pp. 193–203, 1993.
6
J. Cheng, “Existence of a nonoscillatory solution of a second-order linear neutral difference equation,”
Applied Mathematics Letters, vol. 20, no. 8, pp. 892–899, 2007.
7
I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical
Monographs, The Clarendon Press Oxford University Press, New York, NY, USA, 1991.
8
Z. Liu, Y. Xu, and S. M. Kang, “Global solvability for a second order nonlinear neutral delay difference
equation,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 587–595, 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. 211, no. 2, pp.
156–172, 2008.
10
M. Migda and J. Migda, “Asymptotic properties of solutions of second-order neutral difference
equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e789–e799, 2005.
14
Advances in Difference Equations
11
E. 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, no. 1, pp. 85–89, 2002.
13
B. G. Zhang and B. Yang, “Oscillation in higher-order nonlinear difference equations,” Chinese Annals
of Mathematics, vol. 20, no. 1, pp. 71–80, 1999.
14
Z. Zhang and Q. Li, “Oscillation theorems for second-order advanced functional difference
equations,” Computers & Mathematics with Applications, vol. 36, no. 6, pp. 11–18, 1998.
15
Y. Zhou, “Existence of nonoscillatory solutions of higher-order neutral difference equations with
general coefficients,” Applied Mathematics Letters, vol. 15, no. 7, pp. 785–791, 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, no. 1, pp. 63–76, 2003.
17
Y. Zhou and B. G. Zhang, “Existence of nonoscillatory solutions of higher-order neutral delay
difference equations with variable coefficients,” Computers & Mathematics with Applications, vol. 45,
no. 6–9, pp. 991–1000, 2003.