Research Article | Open Access

# Existence of Nonoscillatory Solutions for a Third-Order Nonlinear Neutral Delay Differential Equation

**Academic Editor:**Elena Braverman

#### Abstract

The aim of this paper is to study the solvability of a third-order nonlinear neutral delay differential equation of the form , . By using the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we demonstrate the existence of uncountably many bounded nonoscillatory solutions for the above differential equation. Several nontrivial examples are given to illustrate our results.

#### 1. Introduction and Preliminaries

In recent years, the study of the oscillation, nonoscillation, asymptotic behaviors and existence of solutions for various kinds of first- and second-order neutral delay differential equations and systems of differential equations have attracted much attention, for example, see [1–12] and the references therein. Dorociaková and Olach [2] discussed the existence of nonoscillatory solutions and asymptotic behaviors for the first-order delay differential equation Elbert [3] and Huang [5] established a few oscillation and nonoscillation criteria for the second-order linear differential equation where . Tang and Liu [10] studied the existence of bounded oscillation for the second-order linear delay differential equation of unstable type where and on any interval of length . In view of the Banach fixed point theorem, Kulenovic and Hadžiomerspahic [7] deduced the existence of a nonoscillatory solution for the second-order linear neutral delay differential equation with positive and negative coefficients where . Qin et al. [9] and Yang et al. [11] developed several oscillation criteria for the second-order differential equation where and are nonnegative constants, , and . By utilizing the Krasnoselskii’s fixed point theorem, Zhou [12] discussed the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation where is an integer, , and for .

However, the existence of nonoscillatory solutions of third-order neutral differential equations received much less attention, moreover, the results in [7, 11, 12] only figured out the existence of a nonoscillatory solution of (1.3)–(1.5), respectively.

Motivated by the papers mentioned above, in this paper, we investigate the following third-order nonlinear neutral delay differential equation where is an integer, , and . By applying the Krasnoselskii's fixed point theorem and the Schauder's fixed point theorem, we obtain some sufficient conditions for the existence of uncountably many bounded nonoscillatory solutions of (1.7).

Throughout this paper, we assume that denotes the Banach space of all continuous and bounded functions on with the norm for each and It is easy to see that is a bounded closed and convex subset of .

By a solution of (1.7), we mean a function for some such that and are continuously differentiable in and such that (1.7) is satisfied for . As is customary, a solution of (1.7) is called oscillatory if it has arbitrarily large zeros, and otherwise it is said to be * nonoscillatory*.

*Definition 1.1 (see [6]). *A family of functions in is* equicontinuous* on if for any , the interval can be decomposed into a finite number of subintervals such that

Lemma 1.2 (see Krasnoselskii’s fixed point theorem, [4]). *Let be a Banach space. Let be a bounded closed convex subset of and and mappings from into such that for every pair . If is a contraction and is completely continuous, then the equation has at least one solution in .*

Lemma 1.3 (see Schauder’s fixed point theorem, [4]). *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 .*

#### 2. Main Results

Now we study those conditions under which (1.7) possesses uncountably many bounded nonoscillatory solutions.

Theorem 2.1. *Assume that there exist constants and a function satisfying
**
Then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . From (2.4), we pick up such that
Define two mappings and by
for .

Firstly, we prove that . By (2.1)–(2.3) and (2.5)–(2.7), we get that
which infer that for any .

Secondly, we show that is a contraction mapping. By (2.1), (2.2), and (2.6), we deduce that
for and , which gives that

Thirdly, we show that is completely continuous. Let and with . By (2.7), we obtain that
Using (2.3) and (2.4), we conclude that
In light of (2.11)–(2.12),
and the Lebesgue dominated convergence theorem, we conclude that
which means that is continuous in .

Now we show that is completely continuous. By virtue of (2.3), (2.4), and (2.7), we get that
That is, is uniform bounded. It follows from (2.4) that for each , there exists such that
In view of (2.3), (2.7), and (2.16), we infer that
From (2.3) and (2.7), we get that
which together with (2.4) ensures that there exists satisfying
Clearly,
That is, is equicontinuous on . Consequently, is completely continuous. By Lemma 1.2, there is such that , which is a bounded nonoscillatory solution of (1.7).

Lastly, we demonstrate that (1.7) possesses uncountably many bounded nonoscillatory solutions in . Let and . For each , we choose and the mappings and satisfying (2.5)–(2.7) with and replaced by and , respectively, and some such that
Obviously, there are such that and , respectively. That is, and are two bounded nonoscillatory solutions of (1.7) in . In order to prove that (1.7) possesses uncountably many bounded nonoscillatory solutions in , we prove only that . In fact, by (2.6) and (2.7), we gain that for
which together with (2.1)–(2.3) imply that
which means that
by (2.1) and (2.21). That is, . This completes the proof.

Theorem 2.2. *Assume that there exist constants and a function satisfying (2.3), (2.4) and
**
Then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . It follows from (2.4) that there exists such that
Define two mappings and by (2.6) and (2.7), respectively. By virtue of (2.3), (2.6), (2.7), (2.26), and (2.27), we obtain that
which yield that .

By a similar argument used in the proof of Theorem 2.1, we gain that is a contraction mapping is completely continuous, and (1.7) possesses uncountably many nonoscillatory solutions. This completes the proof.

Theorem 2.3. *Assume that there exist constants and a function satisfying (2.3), (2.4), (2.25), and
**
Then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . It follows from (2.4) that there exists satisfying
Let and be defined by (2.6) and (2.7), respectively. In view of (2.3), (2.6), (2.7), (2.30), and (2.31), we obtain that
which mean that .

The rest of the proof is similar to the proof of Theorem 2.1 and is omitted. This completes the proof.

Theorem 2.4. *Assume that there exist constants and with and a function satisfying (2.3) and (2.4). If for each , then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . It follows from (2.4) that there exists satisfying
Define a mapping by
for .

Firstly, we prove that . By virtue of (2.3), (2.33) and (2.34), we obtain that
which imply that .

Secondly, we show that is continuous in . Let and with . By (2.34), we get that
In view of (2.12), (2.13), (2.36), and the Lebesgue dominated convergence theorem, we deduce that
which means that is continuous in .

Thirdly, we show that is relatively compact. From (2.3), (2.33), and (2.34), we gain that
which means that is uniform bounded.

Let . It follows from (2.4) that there exist such that
By (2.3), (2.34), and (2.39), we deduce that
Choose an integer with , and put
Equation (2.4) means that . It follows from (2.3), (2.34), and (2.40)–(2.42) that for
It is not difficult to verify that
Therefore is equicontinuous on , and consequently is relatively compact. By Lemma 1.3, there is such that , which together with (2.34) yields that for
which mean that
It is easy to show that is a bounded nonoscillatory solution of (1.7).

Finally, we demonstrate that (1.7) possesses uncountably many bounded nonoscillatory solutions in . Let and . For each , we pick up a positive integer and the mapping satisfying (2.33) and (2.34), where and are replaced by and , respectively, and some satisfying (2.21). Clearly, there are and such that and , respectively. That is, and are bounded nonoscillatory solutions of (1.7) in . By (2.34) we get that for
which together with (2.3) and (2.21) yield that
which implies that . This completes the proof.

Theorem 2.5. *Assume that there exist constants and with and a function satisfying (2.3) and
**
If for each , then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . It follows from (2.49) that there exists satisfying
Define a mapping by
for .

Firstly, we prove that . By (2.3), (2.50), and (2.51), we obtain that
which imply that .

Secondly, we show that is continuous in . Let and with . By (2.51), we get that
Equation (2.53) together with (2.3), (2.49) and the Lebesgue dominated convergence theorem yields that
that is, is continuous in .

Thirdly, we show that is relatively compact. From (2.3), (2.50), and (2.51), we obtain that
which means that is uniform bounded. It follows from (2.49) that, for each , there exists such that
Notice that (2.3), (2.51), and (2.56) yield that
Choose a positive integer satisfying . Put
By (2.3), (2.51), and (2.58), we gain that
Clearly,
That is, is equicontinuous on , and is relatively compact. The rest argument is similar to the proof of Theorem 2.4 and is omitted. This completes the proof.

Theorem 2.6. *Assume that there exist constants and a function satisfying (2.3), (2.4), and
**
Then (1.7) possesses uncountably many bounded nonoscillatory solutions in .*

*Proof. *Set . It follows from (2.4) that there exists such that
Define two mappings and by
for .

Firstly, we show that . From (2.3), (2.61)–(2.65), we get that
That is, .

Secondly, by (2.61), (2.62), and (2.64), we conclude that
which implies that is a contraction mapping in .

Thirdly, we show that is completely continuous. Let and be such that as . By (2.61), (2.62), and (2.65), we gain that
In view of (2.12), (2.13), (2.68), and the Lebesgue dominated convergence theorem, we obtain that
that is, is continuous in .

For each , it follows from (2.4) that there exists satisfying
From (2.3), (2.61), (2.62), (2.65), and (2.70), we gain that
By (2.3), (2.61), (2.62), (2.65), and (2.70), we obtain that, for ,