Research Article | Open Access

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

**Academic Editor:**Norio Yoshida

#### 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, [1â€“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 [1â€“9], in this paper, we discuss the solvability of the second-order nonlinear neutral delay difference equation where , ,â€‰â€‰, , ,â€‰â€‰ and It is clear that (1.1)â€“(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 ,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰ 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 ,â€‰â€‰ 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-Ascoliâ€™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)â€“(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)â€“(3.6), where ,â€‰â€‰, 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 ,â€‰â€‰,â€‰â€‰, and with ,â€‰â€‰â€‰ satisfying
By virtue of (3.1) and (3.25)â€“(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