Abstract and Applied Analysis

Volume 2012 (2012), Article ID 172939, 30 pages

doi:10.1155/2012/172939

Research Article

Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,¹ Wei Sun,¹ Jeong Sheok Ume,² and Shin Min Kang³

¹Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China

²Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

³Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 15 August 2012; Accepted 6 November 2012

Academic Editor: Norio Yoshida

Copyright © 2012 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.

Abstract.

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation . By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

1. Introduction

It is well known that the oscillation, nonoscillation, asymptotic behavior, and existence of solutions for second-order difference equations with delays have been widely studied in many papers over the last 20 years, see, for example, [1–9] and the references cited therein.

Recently, Cheng [5] considered the second-order neutral delay linear difference equation with positive and negative coefficients

and investigated the existence of a nonoscillatory solution of (1.1) under the condition by using the Banach fixed point theorem. M. Migda and J. Migda [9] and Luo and Bainov [8] discussed the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima

and the second-order neutral difference equation

Cheng and Chu [2] got sufficient and necessary conditions of the oscillatory solutions for the second-order difference equation

Li and Yeh [6] established some oscillation criteria of the second-order delay difference equation

Using the Leray-Schauder nonlinear alternative theorem, Agarwal et al. [1] studied the existence of nonoscillatory solutions for the discrete equation

under the condition . Very recently, Liu et al. [7] utilized the Banach contraction principle to establish the global existence and multiplicity of bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation

Motivated by the results in [1–9], in this paper, we discuss the solvability of the second-order nonlinear neutral delay difference equation

where , ,  , , ,   and

It is clear that (1.1)–(1.7) are special cases of (1.8). By utilizing the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and a few new techniques, we prove the existence of uncountably many bounded positive solutions of (1.8). Five examples are constructed to illuminate our results which extend essentially the corresponding results in [1, 7].

2. Preliminaries

Throughout this paper, we assume that is the forward difference operator defined by ,  ,  ,  ,   and stand for the sets of all integers, positive integers, and nonnegative integers, respectively,

denotes the Banach space of all bounded sequences with the norm

For any , put

It is easy to see that is a closed convex subset of ,   is a bounded open subset of and is a bounded open convex subset of and is a bounded closed and convex subset of .

By a solution of (1.8), we mean a sequence with a positive integer such that (1.8) is satisfied for all .

The following Lemmas play important roles in this paper.

Lemma 2.1 (Discrete Arzela-Ascoli’s Theorem [3]). A bounded, uniformly Cauchy subset of is relatively compact.

Lemma 2.2 (Rothe Fixed Point Theorem [10]). Let be a bounded convex open subset of a Banach space and be a continuous, condensing mapping, and . Then has a fixed point in .

Lemma 2.3 (Leray-Schauder Nonlinear Alternative Theorem [1]). Let be an open subset of a closed convex set in a Banach space with . Let be a continuous, condensing mapping with bounded. Then either

(a)has a fixed point in ; or(b)there exist an and a such that .

Lemma 2.4 (Krasnoselskill Fixed Point Theorem [5]). Let be a nonempty bounded closed convex subset of a Banach space and be mappings from into such that for every pair . If is a contraction mapping and is completely continuous, then the equation has at least one solution in .

3. Main Results

Now we use the Rothe fixed point theorem to show the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.1. Assume that there exist two constants and with and two positive sequences and satisfying

Then (1.8) has uncountably many bounded positive solutions in .

Proof. Let . First of all, we show that there exists a mapping with such that has a fixed point , which is also a bounded positive solution of (1.8).

It follows from (3.2) and (3.3) that there exists satisfying

Define a mapping as follows:

for each . On account of (3.1), (3.5), and (3.6), we conclude that for every and

which means that

that is, .

Now we assert that is a continuous and condensing mapping in . Let for each and with . Let . It follows from (3.2) and the continuity of and that there exist with and satisfying

In view of (3.1) and (3.6)–(3.10), we deduce that for any

which gives that , that is, is continuous in .

In light of (3.1), (3.5), and (3.6), we get that for any

which implies that is uniformly bounded.

Given . Clearly (3.2) ensures that there exists satisfying

which together with (3.1) and (3.6) implies that for all and

which yields that is uniformly Cauchy. Thus Lemma 2.1 means that is relatively compact. Consequently is condensing in .

It follows from Lemma 2.2 that has a fixed point , that is,

which yields that

which together with (3.3) implies that

that is, (1.8) has a bounded positive solution .

Next we show that (1.8) has uncountably many bounded positive solutions in . Let and . For every , we infer similarly that there exist a constant and a mapping satisfying (3.4)–(3.6), where ,  , and are replaced by , and , respectively, and the mapping has a fixed point , which is a bounded positive solution of (1.8) in , that is,

Equation (3.2) ensures that there exists satisfying

In order to show that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to prove that . It follows from (3.1), (3.18), and (3.19) that for all

that is, . This completes the proof.

Theorem 3.2. Assume that there exist two constants and with and two positive sequences and satisfying (3.1) and

Then (1.8) has uncountably many bounded positive solutions.

Proof. Let . Firstly, we show that there exists a mapping with such that has a fixed point , which is also a bounded positive solution of (1.8). In view of (3.21) and (3.22), we choose a sufficiently large integer such that

Define a mapping as follows:

for each . It follows from (3.1), (3.24), and (3.25) that for every and

which means that

that is, .

Now we prove that is a continuous and condensing mapping in . Put for each and with . Let . Using (3.21) and the continuity of and , we conclude that there exist four positive integers ,  ,  , and with ,    satisfying

By virtue of (3.1) and (3.25)–(3.29), we infer that for each

which implies that , that is, is continuous in .

From (3.1), (3.24), and (3.25), we infer that for any

which implies that is uniformly bounded.

Let . It follows from (3.21) that there exists satisfying

which together with (3.1) and (3.25) yields that for all and

which gives that is uniformly Cauchy. Hence Lemma 2.1implies that is relatively compact, that is, is condensing in .

It is clear that Lemma 2.2 means that possesses a fixed point , that is,

which lead to

which together with (3.23) yields that

that is, (1.8) has a bounded positive solution in .

Next we show that (1.8) has uncountably many bounded positive solutions in . Let and . Similarly we infer that for each , there exist a constant and a mapping satisfying (3.23)–(3.25), where ,   and are replaced by , and , respectively, and the mapping has a fixed point , which is a bounded positive solution of (1.8) in , that is,

It follows from (3.21) that there exists such that

In order to show that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to prove that . By means of (3.1), (3.37) and (3.38), we infer that for each

that is, . This completes the proof.

Next we use the Leray-Schauder nonlinear alternative theorem to show the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.3. Assume that there exist four constants ,  ,  , and and two positive sequences and satisfying (3.1), (3.2) and

Then (1.8) has uncountably many bounded positive solutions in .

Proof. Let . Now we prove that there exists a mapping such that it has a fixed point , which is also a bounded positive solution of (1.8). It follows from (3.2), (3.40) and that there exists a sufficiently large number satisfying

Put , where is enough small and

Obviously, . Define a mapping by

for each , where the mappings are defined by

It follows from (3.1), (3.41), and (3.43)–(3.46) that for any and

which yields that .

Next we show that is a continuous and relatively compact mapping. Let and with . By virtue of (3.2) and the continuity of and , we infer that there exist with satisfying

It follows from (3.1) and (3.46)–(3.49) that for each

which yields that , that is, is continuous in .

In light of (3.1) and (3.43)–(3.46), we deduce that for all

which means that and are bounded.

Let . Notice that (3.2) ensures that there exists satisfying

which together with (3.1) and (3.46) implies that for all and

which means that is uniformly Cauchy. Thus is relatively compact.

By virtue of (3.41) and (3.45), we infer that for all and

which yields that

that is, is a contraction mapping in . It follows that is a continuous and condensing mapping.

Put

It is easy to see that . Suppose that there exist and with

Now we consider two possible cases as follows.

Case  1. Let . Obviously (3.41), (3.43)–(3.46), (3.56), and (3.58) guarantee that

which implies that

which is a contradiction.

Case  2. Let . It follows from (3.41), (3.43)–(3.46), (3.57), and (3.58) that

which is absurd. Thus Lemma 2.3 ensures that there exists satisfying , that is,

which means that

which yields that

that is, is a bounded positive solution of (1.8).

Finally we show that (1.8) has uncountably many bounded positive solutions in . Let and . Similarly we infer that for each , there exists a mapping satisfying (3.41)–(3.46), where ,  ,  ,  ,  , and are replaced by ,  ,  ,  ,   and , respectively, and the mapping has a fixed point , which is a bounded positive solution of (1.8) in , that is,

It follows from (3.2) that there exists satisfying

In order to prove that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to verify that . In terms of (3.1), (3.65), and (3.66), we deduce that for

which means that

that is, . This completes the proof.

Theorem 3.4. Assume that there exist four constants ,  ,  , and and two positive sequences and satisfying (3.1), (3.2) and

Then (1.8) has uncountably many bounded positive solutions in .

Proof. Let . Now we show that there exists a mapping such that it has a fixed point , which is also a bounded positive solution of (1.8). It follows from (3.2) and (3.69) that there exists satisfying

Let , where is enough small and

Obviously, . Define a mapping by (3.44), where the mappings are defined by

for each . By virtue of (3.1), (3.70), and (3.72)–(3.74), we get that for any and

which gives that . The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof.

Now we employ the Krasnoselskii fixed point theorem to prove the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.5. Assume that there exist four constants ,  ,   and and two positive sequences ,   satisfying (3.1), (3.2) and

Then (1.8) has uncountably many bounded positive solutions in .

Proof. Let . Now we show that there exist two mappings such that the equation has a solution , which is also a bounded positive solution of (1.8). It follows from (3.2) and (3.76) that there exists satisfying

Define two mappings and by (3.73) and (3.74), respectively. It follows from (3.1), (3.73), (3.74), (3.77), and (3.78) that for any , and

which yield that

As in the proof of Theorem 3.3, we infer similarly that is continuous in and is relatively compact. Thus is completely continuous, which together with (3.77), (3.80), and Lemma 2.4, ensures that the equation has a solution , which is also a bounded positive solution of (1.8) in . The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof.

Remark 3.6. Theorems 3.1–3.5 extend and improve Theorem  2.1 in [1] and Theorems 2.1–2.7 in [7], respectively. The examples in Section 4 show that our results are indeed generalizations of the corresponding results in [1, 7].

4. Examples

Now we construct five examples to show the applications of the results presented in Section 3. Note that none of the known results can be applied to the five examples.

Example 4.1. Consider the second-order nonlinear neutral delay difference equation

where is fixed. Let ,  ,  ,  ,  ,   and

It is easy to verify that (3.1)–(3.3) hold. Thus Theorem 3.1 guarantees that (4.1) has uncountably many bounded positive solutions in . But the results in [1, 7] are not applicable for (4.1).

Example 4.2. Consider the second-order nonlinear neutral delay difference equation

where is fixed. Let ,  ,  ,  ,  ,   and

It is clear that (3.1), (3.21), and (3.22) hold. Hence Theorem 3.2 ensures that (4.3) has uncountably many bounded positive solutions in . But the results in [1, 7] are not valid for (4.3).

Example 4.3. Consider the second-order nonlinear neutral delay difference equation

where is fixed. Let ,   and

It is clear that (3.1), (3.2) and (3.40) are satisfied. Hence Theorem 3.3 implies that (4.5) has uncountably many bounded positive solutions in . But the results in [1, 7] are unapplicable for (4.5).

Example 4.4. Consider the second-order nonlinear neutral delay difference equation

where is fixed. Let and

It is easy to verify that (3.1), (3.2), and (3.69) hold. Hence Theorem 3.4 implies that (4.7) has uncountably many bounded positive solutions in . But the results in [1, 7] are not valid for (4.7).

Example 4.5. Consider the second-order nonlinear neutral delay difference equation

where is fixed. Let and

It is easy to see that (3.1), (3.2), and (3.76) hold. Hence Theorem 3.5 guarantees that (4.9) possesses uncountably many bounded positive solutions in . But the results in [1, 7] are inapplicable for (4.9).

Acknowledgments

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).

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