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Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation
Advances in Difference Equations volume 2010, Article number: 767620 (2010)
Abstract
The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation ,
, where
,
,
, and
are integers,
and
are real sequences,
, and
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
. 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,

(see [14]),

(see [11]),

(see [6]),

(see [10]),

(see [9]),

(see [8]),

(see [15]),


(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:

where ,
,
, and
are integers,
and
are real sequences,
, and
is a mapping. Clearly, difference 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 be a nonempty closed convex subset of a Banach space
. Let
be a continuous mapping such that
is a relatively compact subset of
. Then
has at least one fixed point in
.
Lemma 1.2 (Krasnoselskii fixed point theorem).
Let be a bounded closed convex subset of a Banach space
, and let
satisfy
for each
. If
is a contraction mapping and
is a completely continuous mapping, then the equation
has at least one solution in
.
The forward difference is defined as usual, that is,
. The higher-order difference for a positive integer
is defined as
,
. Throughout this paper, assume that
,
and
stand for the sets of all positive integers and integers, respectively,
,
,
,
, and
denotes the set of real sequences defined on the set of positive integers lager than
where any individual sequence is bounded with respect to the usual supremum norm
for
. It is well known that
is a Banach space under the supremum norm. A subset
of a Banach space
is relatively compact if every sequence in
has a subsequence converging to an element of
.
Definition 1.3 (see [5]).
A set of sequences in
is uniformly Cauchy (or equi-Cauchy) if, for every
, there exists an integer
such that

whenever for any
in
.
Lemma 1.4 (discrete Arzela-Ascoli's theorem [5]).
A bounded, uniformly Cauchy subset of
is relatively compact.
Let

Obviously, is a bounded closed and convex subset of
. Put

By a solution of (1.11), we mean a sequence with a positive integer
such that (1.11) is satisfied for all
. As is customary, a solution of (1.11) is said to be oscillatory about zero, or simply oscillatory, if the terms
of the sequence
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 of (1.11) are given.
Theorem 2.1.
Assume that there exist constants and
with
and sequences
,
,
, and
such that, for
,




Then (1.11) has a bounded nonoscillatory solution in .
Proof.
Choose . By (2.1), (2.4), and the definition of convergence of series, an integer
can be chosen such that


Define a mapping by

for all .
-
(i)
It is claimed that
, for all
.
In fact, for every and
, it follows from (2.3) and (2.6) that

That is, .
-
(ii)
It is declared that
is continuous.
Let and
be any sequence such that
as
. For
, (2.2) guarantees that

This inequality and (2.4) imply that is continuous.
-
(iii)
It can be asserted that
is relatively compact.
By (2.4), for any , take
large enough so that

Then, for any and
, (2.10) ensures that

which means that is uniformly Cauchy. Therefore, by Lemma 1.4,
is relatively compact.
By Lemma 1.1, there exists such that
, which is a bounded nonoscillatory solution of (1.11). In fact, for
,

which derives that

That is,

by which it follows that

Therefore, is a bounded nonoscillatory solution of (1.11). 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 with
. For
and
, as the preceding proof in Theorem 2.1, there exist integers
and mappings
satisfying (2.5)–(2.7), where
are replaced by
,
and
,
, respectively, and
for some
. Then the mappings
and
have fixed points
, respectively, which are bounded nonoscillatory solutions of (1.11) in
. For the sake of proving that (1.11) possesses uncountably many bounded nonoscillatory solutions in
, it is only needed to show that
. In fact, by (2.7), we know that, for
,

Then,

that is, .
Theorem 2.3.
Assume that there exist constants and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
Proof.
Choose . By (2.18) and (2.4), an integer
can be chosen such that

Define a mapping by

for all .
The proof that has a fixed point
is analogous to that in Theorem 2.1. It is claimed that the fixed point
is a bounded nonoscillatory solution of (1.11). In fact, for
,

by which it follows that

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 ,
, and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
Proof.
Choose . By (2.23) and (2.4), an integer
can be chosen such that

Define two mappings by

for all .
-
(i)
It is claimed that
, for all
.
In fact, for every and
, it follows from (2.3), (2.24) that

That is, .
-
(ii)
It is declared that
is a contraction mapping on
.
In reality, for any and
, it is easy to derive that

which implies that

Then, ensures that
is a contraction mapping on
.
-
(iii)
Similar to (ii) and (iii) in the proof of Theorem 2.1, it can be showed that
is completely continuous.
By Lemma 1.2, there exists such that
, which is a bounded nonoscillatory solution of (1.11). This completes the proof.
Theorem 2.5.
Assume that there exist constants and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
Proof.
Choose . By (2.29) and (2.4), an integer
can be chosen such that

Define two mappings 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 and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
Theorem 2.7.
Assume that there exist constants and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
Proof.
Take sufficiently small satisfying

Choose . By (2.33), an integer
can be chosen such that

Define two mappings by

for all . 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 and
with
and sequences
,
,
,
, satisfying (2.2)–(2.4) and

Then (1.11) has a bounded nonoscillatory solution in .
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 of Cheng [6], Theorems
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:

Choose and
. 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
. 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:

where

Choose and
. 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
. However, the results in [3, 4, 6, 8–17] are unapplicable for (3.2).
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Liu, M., Guo, Z. Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation. Adv Differ Equ 2010, 767620 (2010). https://doi.org/10.1155/2010/767620
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DOI: https://doi.org/10.1155/2010/767620
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation