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,

(1.1)

(see [14]),

(1.2)

(see [11]),

(1.3)

(see [6]),

(1.4)

(see [10]),

(1.5)

(see [9]),

(1.6)

(see [8]),

(1.7)

(see [15]),

(1.8)

(see [3, 4, 12, 13]),

(1.9)

(see [16]),

(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:

(1.11)

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

(1.12)

whenever for any in .

Lemma 1.4 (discrete Arzela-Ascoli's theorem [5]).

A bounded, uniformly Cauchy subset of is relatively compact.

Let

(1.13)

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

(1.14)

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 ,

(2.1)
(2.2)
(2.3)
(2.4)

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

(2.5)
(2.6)

Define a mapping by

(2.7)

for all .

  1. (i)

    It is claimed that , for all .

In fact, for every and , it follows from (2.3) and (2.6) that

(2.8)

That is, .

  1. (ii)

    It is declared that is continuous.

Let and be any sequence such that as . For , (2.2) guarantees that

(2.9)

This inequality and (2.4) imply that is continuous.

  1. (iii)

    It can be asserted that is relatively compact.

By (2.4), for any , take large enough so that

(2.10)

Then, for any and , (2.10) ensures that

(2.11)

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 ,

(2.12)

which derives that

(2.13)

That is,

(2.14)

by which it follows that

(2.15)

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 ,

(2.16)

Then,

(2.17)

that is, .

Theorem 2.3.

Assume that there exist constants and with and sequences , , , , satisfying (2.2)–(2.4) and

(2.18)

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

Choose . By (2.18) and (2.4), an integer can be chosen such that

(2.19)

Define a mapping by

(2.20)

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 ,

(2.21)

by which it follows that

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

(2.23)

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

Choose . By (2.23) and (2.4), an integer can be chosen such that

(2.24)

Define two mappings by

(2.25)

for all .

  1. (i)

    It is claimed that , for all .

In fact, for every and , it follows from (2.3), (2.24) that

(2.26)

That is, .

  1. (ii)

    It is declared that is a contraction mapping on .

In reality, for any and , it is easy to derive that

(2.27)

which implies that

(2.28)

Then, ensures that is a contraction mapping on .

  1. (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

(2.29)

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

Choose . By (2.29) and (2.4), an integer can be chosen such that

(2.30)

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

(2.31)

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

(2.32)

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

Take sufficiently small satisfying

(2.33)

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

(2.34)

Define two mappings by

(2.35)

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

(2.36)

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:

(3.1)

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:

(3.2)

where

(3.3)

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).