Research Article | Open Access

Zeqing Liu, Ming Jia, Jeong Sheok Ume, Shin Min Kang, "Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation", *Abstract and Applied Analysis*, vol. 2011, Article ID 328956, 16 pages, 2011. https://doi.org/10.1155/2011/328956

# Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

**Academic Editor:**Sergey V. Zelik

#### Abstract

This paper deals with the higher-order nonlinear neutral delay differential equation , , where , , , , , and , . By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.

#### 1. Introduction and Preliminaries

This paper is concerned with the higher-order nonlinear neutral delay differential equation: where ,,,,, and

Theory of neutral delay differential equations has undergone a rapid development in the last over thirty years. We refer the readers to [1β8] and the references therein for a wealth of reference materials on the subject. The authors [1β8] and others discussed the oscillation, nonoscillation, and existence of a nonoscillatoy solution for some special cases of (1.1) under various conditions. By using the Banach fixed point theorem, Zhang et al. [4] and KulenoviΔ and HadΕΎiomerspahiΔ [1] studied, respectively, the existence of a nonoscillatory solution for the first-order neutral delay differential equation: where , , , and , and the second-order neutral delay differential equation with positive and negative coefficients: where , and . Zhang et al. [6] considered the second-order nonlinear neutral differential equation with positive and negative terms: and its corresponding equation with forced term: where , , , and for . Lin [2] investigated sufficient conditions of oscillation and nonoscillation for the second-order nonlinear neutral differential equation: where , , with for all . Liu and Huang [3] used the coincidence degree theory to establish the existence and uniqueness of -periodic solutions for the second-order neutral functional differential equation of the form where are continuous functions, , , are constants, and are -periodic, , and . Zhou and Zhang [8] extended the results in [1] to the higher-order neutral functional differential equation with positive and negative coefficients: where , and . Zhou et al. [7] used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: where , for and . Zhang et al. [5] obtained some sufficient conditions for the oscillation of all solutions of the even order nonlinear neutral differential equations with variable coefficients: where is an even number, with , for all , and .

The purpose of this paper is to investigate the solvability of (1.1). By constructing appropriate mappings and using the Laray-Schauder nonlinear alternative theorem, we establish a few sufficient conditions which ensure the existence of uncountably many bounded positive solutions for (1.1). Our results improve and generalize some corresponding results in [1, 2, 4, 6β8]. Three examples are given to illustrate the advantages of the results presented in this paper.

Throughout this paper, we assume that , , and denote the sets of all real numbers, nonnegative numbers, and positive integers, respectively, and Let stand for the Banach space of all continuous and bounded functions in with norm for all and where with . Clearly, is a nonempty closed convex subset of and is an open subset of .

By a solution of (1.1), we mean a function with some such that is times continuously differentiable in and is times continuously differentiable in and (1.1) holds for .

Lemma 1.1 (the Leray-Schauder nonlinear alterative theorem [9]). *Let be a closed convex subset of a Banach space and let be an open subset of with . Also, is a continuous, condensing mapping with bounded, where denotes the closure of Then,*

()* has a fixed point in , or*()* there are and with .*

#### 2. Main Results

Now, we apply the Leray-Schauder nonlinear alterative theorem to investigate the existence of uncountably many bounded positive solutions of (1.1) under certain conditions.

Theorem 2.1. *Assume that there exist constants and functions satisfying
**
Then, (1.1) has uncountably many bounded positive solutions in .*

*Proof. *Let . It follows from (2.3) and (2.4) that there exists a constant satisfying
Choose with
Put . Clearly, . Define two mappings by
for all . It is clear to see that and are continuous for each . Let . In view of (2.1), (2.2), and (2.4)β(2.8), we get that
which gives that .

Now, we show that is continuous and compact. Let be an arbitrary sequence and with
Since is closed, it follows that . For any , put
It follows from (2.1), (2.2), and (2.11) that
which together with (2.8)β(2.11), the continuity of for , and the Lebesgue dominated convergence theorem yields that
which means that is continuous in . It follow from (2.1), (2.2), (2.6), and (2.8) that
which yields that is uniformly bounded in .

Let be an arbitrary positive number. Equation (2.3) ensures that there exists satisfying
Set
For any and with , we consider the following three cases.*Case 1 (). *In view of (2.1), (2.2), (2.8), and (2.15), we deduce that
*Case 2 (). *Suppose that . It follows from (2.1), (2.2), (2.6), and (2.8) that
Suppose that . It follows from the mean value theorem that, for each , there exists satisfying
which together with (2.1), (2.2), (2.6), and (2.8) yields that
*Case 3 (). *Equation (2.8) gives that
Thus, is equicontinuous in . Hence, is a relatively compact subset of . That is, is a compact mapping.

Note that for any and which implies that
which together with (2.4) gives that is a contraction mapping. It follows that is a continuous and condensing mapping. Let for all . Notice that . Thus, (2.4)β(2.7), (2.14), and (2.23) yield that
that is, is uniformly bounded in .

Put
It is easy to verify that .

Next, we show that () in Lemma 1.1 does not hold. Otherwise, there exist and satisfying . We have to discuss the following possible cases. *Case 1. *Let . By means of (2.1), (2.2), and (2.4)β(2.8), we get that, for ,
which implies that
which is a contradiction.*Case 2. *Let . It follows from (2.1), (2.2), and (2.4)β(2.8) that
which is absurd.

Thus, Lemma 1.1 ensures that has a fixed point ; that is,
which yields that
which means that is a bounded positive solution of (1.1).

Let with . Similarly, we can prove that, for each , there exist a constant and two mappings satisfying (2.6)β(2.8), where , and are replaced by , and , respectively, and has a fixed point , which is a bounded positive solution of (1.1) in . In order to prove that (1.1) possesses uncountably many bounded positive solutions in , we need only to prove that . By means of (2.1)β(2.3), we know that there exists satisfying
It follows from (2.1), (2.2), (2.4), (2.7), (2.8), and (2.31) that for which implies that
that is, . This completes the proof.

Theorem 2.2. *Assume that there exist constants and functions satisfying (2.1)β(2.3) and
**
Then, (1.1) has uncountably many bounded positive solutions in .*

*Proof. *Let . It follows from (2.3) and (2.34) that there exists a constant satisfying
Take such that
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.

Theorem 2.3. *Assume that there exist constants and functions satisfying (2.1)β(2.3) and
**
Then, (1.1) has uncountably many bounded positive solutions in .*

*Proof. *Let . It follows from (2.3) and (2.37) that there exists a constant satisfying
Choose such that
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.

*Remark 2.4. *Theorems 2.1β2.3 extend, improve, and unify the theorem in [1], Theoremββ2.2 in [2], Theoremββ1 in [4], Theoremsββ2.1 and 2.3 in [6], Theoremsββ1 and 3 in [7], and Theoremsββ1 and 3 in [8].

#### 3. Examples and Applications

Now, we construct three nontrivial examples to show the superiority and applications of Theorems 2.1β2.3, respectively.

*Example 3.1. *Consider the higher-order nonlinear neutral delay differential equation:
Let ,,,,,,,