## Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations

View this Special IssueResearch Article | Open Access

# Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

**Academic Editor:**Jifeng Chu

#### Abstract

This paper is concerned with the higher order nonlinear neutral delay differential equation for all . Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

#### 1. Introduction and Preliminaries

In recent years, the existence problems of nonoscillatory solutions for neutral delay differential equations of first, second, third, and higher order have been studied intensively by using fixed point theorems; see, for example, [1â€“12] and the references therein.

Using the Banach, Schauder, and Krasnoselskii fixed point theorems, Zhang et al. [9] and Liu et al. [7] considered the existence of nonoscillatory solutions for the following first order neutral delay differential equations: where and . Making use of the Banach and Krasnoselskii fixed point theorems, KulenoviÄ‡ and HadÅ¾iomerspahiÄ‡ [2] and Zhou [10] studied the existence of a nonoscillatory solution for the following second order neutral differential equations: where and . Zhou and Zhang [11], Zhou et al. [12], and Liu et al. [4], respectively, investigated the existence of nonoscillatory solutions for the following higher order neutral delay differential equations: where , and . Candan [1] proved the existence of a bounded nonoscillatory solution for the higher order nonlinear neutral differential equation: where .

Motivated by the results in [1â€“12], in this paper we consider the following higher order nonlinear neutral delay differential equation: where and with , , , , and with It is clear that (5) includes (1)â€“(4) as special cases. Utilizing the Banach fixed point theorem, we prove several existence results of uncountably many positive solutions for (5), construct a few Mann iterative schemes, and discuss error estimates between the sequences generated by the Mann iterative schemes and the positive solutions. Nine examples are given to show that the results presented in this paper extend substantially the existing ones in [1, 2, 4, 5, 8, 9, 11].

Throughout this paper, we assume that , denotes the set of all positive integers, , stands for the Banach space of all continuous and bounded functions in with norm , and for any It is easy to check that , and are closed subsets of .

By a solution of (5), we mean a function for some , such that are times continuously differentiable in and such that (5) is satisfied for .

Lemma 1. *Let , and . Then*(a)*;
*(b)*;
*(c)*if , then
*(d)*if , then
*

*Proof. *Let denote the largest integral number not exceeding . Note that
Clearly (12) means that
Thus (a) follows from (11) and (13).

Assume that . As in the proof of (a), we infer that
that is, (c) holds.

Similar to the proofs of (a) and (c), we conclude that (b) and (d) hold. This completes the proof.

#### 2. Existence of Uncountably Many Positive Solutions and Mann Iterative Schemes

Now we show the existence of uncountably many positive solutions for (5) and discuss the convergence of the Mann iterative sequences to these positive solutions.

Theorem 2. *Assume that there exist three constants , , and and four functions satisfying
**
Then** for any , there exist and such that for each , the Mann iterative sequence generated by the following scheme
**
converges to a positive solution of (5) and has the following error estimate:
**
where is an arbitrary sequence in such that
** Equation (5) has uncountably many positive solutions in .*

*Proof. *Firstly, we prove that (a) holds. Set . From (15) and (18), we know that there exist and satisfying
Define a mapping by
It is obvious that is continuous for each . By means of (16), (22), (23), and (25), we deduce that for any and
which yields that
On the basis of (17), (22), (24), and (25), we acquire that for any and
which guarantee that . Consequently, (27) gives that is a contraction mapping in and it has a unique fixed point . It is easy to see that is a positive solution of (5).

It follows from (19), (25), and (27) that
which yields that
That is, (20) holds. Thus (20) and (21) ensure that .

Secondly, we show that (b) holds. Let with . In light of (15) and (18), we know that for each , there exist , and with and satisfying (22)â€“(24) and
where and are replaced by and , respectively. Let the mapping be defined by (25) with and replaced by and , respectively. As in the proof of (a), we deduce easily that the mapping possesses a unique fixed point , that is, is a positive solution of (5) in . In order to prove (b), we need only to show that . In fact, (25) means that for each and
It follows from (16), (22), (31), and (32) that for each
which implies that
that is, . This completes the proof.

Theorem 3. *Assume that there exist three constants , , and and four functions satisfying (16)â€“(18) and
**
Then*(a)*for any , there exist and such that for each , the Mann iterative sequence generated by (19) converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in satisfying (21);*(b)*Equation (5) has uncountably many positive solutions in .*

*Proof. *Let . Equations (18) and (36) ensure that there exist and satisfying (23),
Define a mapping by (25). Obviously, is continuous for every . Using (16), (23), (25), and (36), we conclude that for any and
which implies that (27) holds. In light of (17), (25), (36), and (37), we know that for any and
which mean that . Equation (27) guarantees that is a contraction mapping in and it possesses a unique fixed point . As in the proof of Theorem 2, we infer that is a positive solution of (5). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 4. *Assume that there exist three constants , , and and four functions satisfying (16)â€“(18) and
**
Then*(a)*for any , there exist and such that for each , the Mann iterative sequence generated by (19) converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in satisfying (21);*(b)*Equation (5) has uncountably many positive solutions in .*

*Proof. *Set . It follows from (18) and (40) that there exist and satisfying (23),
Define a mapping by (25). Distinctly, is continuous for each . In terms of (16), (23), (25), and (41), we reason that for any and
which means that (27) holds. Owing to (17), (25), (41), and (42), we earn that for any and
which yield that . Thus (27) ensures that is a contraction mapping in and it owns a unique fixed point . As in the proof of Theorem 2, we infer that is a positive solution of (5). The rest of the proof is parallel to that of Theorem 2, and hence is elided. This completes the proof.

Theorem 5. *Assume that there exist three constants , , and and four functions satisfying (18) and
*