Stability, Boundedness, and Existence of Periodic Solutions to Certain Third-Order Delay Differential Equations with Multiple Deviating Arguments

Abstract

The behaviour of solutions for certain third-order nonlinear differential equations with multiple deviating arguments is considered. By employing Lyapunov's second method, a complete Lyapunov functional is constructed and used to establish sufficient conditions that guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results not only are new but also include many outstanding results in the literature. Finally, the correctness and effectiveness of the results are justified with examples.

Academic Editor:Nikolai N. Leonenko

Research Group in Differential Equations and Applications (RGDEA), Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria

Received 13 July 2015; Accepted 31 August 2015; 16 September 2015

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.

1. Introduction

Differential equations of second and third order with and without delay are essential tools in scientific modeling of problems from many fields of sciences and technologies, such as biology, chemistry, physics, mechanics, electronics, engineering, economy, control theory, medicine, atomic energy, and information theory. Many authors have proposed different methods, in the literature, to discuss qualitative bahaviour of solutions to nonlinear differential equations. Here, we will single out two methods. In this direction, we can mention Lyapunov's second method which demands the construction of a suitable positive definite function (or functional) whose derivative is negative definite; that is, it involves finding the system of closed surfaces that contained the origin and are converging to it. The second method is the frequency domain technique which involves the study of position of the characteristics polynomial roots in the complex plane to obtain certain matrix inequalities which must be positive.

The qualitative behaviors of solutions of differential equations of third order have been discussed extensively and are still receiving attention of authors because of their practical applications. In this regard, we can mention the works of Burton [1, 2], Driver [3], Hale [4], and Yoshizawa [5, 6] which contain general results on the subject matters and expository papers of Abou-El-Ela et al. [7], Ademola et al. [8-10], Adesina [11], Afuwape and Omeike [12], Chukwu [13], Gui [14], Omeike [15, 16], Sadek [17], Tejumola and Tchegnani [18], Tunç et al. [19-28], Yao and Wang [29], and Zhu [30] and the references cited therein.

Recently, Tunç [27] employed Lyapunov's second method to prove two results on stability and boundedness of nonautonomous differential equations with constant delay [figure omitted; refer to PDF] Furthermore, Ademola [9] discussed existence and uniqueness of a periodic solution to the third-order differential equation [figure omitted; refer to PDF] Unfortunately, the problem of uniform asymptotic stability, uniform ultimate boundedness, and existence and uniqueness of periodic solutions of the third-order delay differential equation (3), where all the nonlinear terms (specifically, the forcing term [figure omitted; refer to PDF] and the function [figure omitted; refer to PDF] ) are sum of multiple deviating arguments, is yet to be investigated. This is not unconnected with the difficulties associated with the construction of suitable complete Lyapunov functional. The aim of this paper is to fill this gap. We will consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are continuous functions in their respective arguments on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively, with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The dots indicate differentiation with respect to the independent variable [figure omitted; refer to PDF] . Equation (3) is equivalent to the system of first-order delay differential equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is a constant to be determined latter, and the derivatives [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] exist and are continuous for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . This work is motivated by the recent works in [9, 27]. Our results are new; in fact according to our observation from relevant literature, this is the first paper where both the functions [figure omitted; refer to PDF] and the forcing term [figure omitted; refer to PDF] contain sum of multiple deviating arguments. For the next section, for easy references, we recall the main mathematical tools that will be used in the sequel. Our main results are stated and proved in Section 3 while in the last section, examples are given.

2. Preliminary Results

Consider the following general nonlinear nonautonomous delay differential equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a continuous mapping and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and for some positive constant [figure omitted; refer to PDF] . We assume that [figure omitted; refer to PDF] takes closed bounded sets into bounded sets in [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the Banach space of continuous function [figure omitted; refer to PDF] with supremum norm, [figure omitted; refer to PDF] ; for [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the open [figure omitted; refer to PDF] -ball in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Definition 1 (see [2]).

A continuous function [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] strictly increasing is a wedge. (We denote wedges by [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is an integer.)

Definition 2 (see [2]).

The zero solution of (5) is asymptotically stable if it is stable and if for each [figure omitted; refer to PDF] there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF]

Definition 3 (see [1]).

An element [figure omitted; refer to PDF] is in the [figure omitted; refer to PDF] -limit set of [figure omitted; refer to PDF] , say [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] is defined on [figure omitted; refer to PDF] and there is a sequence [figure omitted; refer to PDF] , [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Definition 4 (see [30]).

A set [figure omitted; refer to PDF] is an invariant set if, for any [figure omitted; refer to PDF] , the solution [figure omitted; refer to PDF] of (5) is defined on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Lemma 5 (see [6]).

Suppose that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is periodic in [figure omitted; refer to PDF] of period [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and consequently for any [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] . Suppose that a continuous Lyapunov functional [figure omitted; refer to PDF] exists, defined on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] may be large), and [figure omitted; refer to PDF] satisfies the following conditions:

(i) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous, increasing, and positive for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is continuous and positive for [figure omitted; refer to PDF] .

Suppose that there exists [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a constant which is determined in the following way: By the condition on [figure omitted; refer to PDF] there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] . Under the above conditions, there exists a periodic solution of (5) of period [figure omitted; refer to PDF] . In particular, relation (7) can always be satisfied if [figure omitted; refer to PDF] is sufficiently small.

Lemma 6 (see [6]).

Suppose that [figure omitted; refer to PDF] is defined and continuous on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and there exists a continuous Lyapunov functional [figure omitted; refer to PDF] defined on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] which satisfy the following conditions:

(i) [figure omitted; refer to PDF] if [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] if [figure omitted; refer to PDF] ;

(iii) : for the associated system [figure omitted; refer to PDF]

: we have [figure omitted; refer to PDF] , where, for [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , we understand that the condition [figure omitted; refer to PDF] is satisfied in case [figure omitted; refer to PDF] can be defined.

Then, for given initial value [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , there exists a unique solution of (5).

Lemma 7 (see [6]).

Suppose that a continuous Lyapunov functional [figure omitted; refer to PDF] exists, defined on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] which satisfies the following conditions:

(i) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous, increasing, and positive;

(ii) [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is continuous and positive for [figure omitted; refer to PDF] ;

then the zero solution of (5) is uniformly asymptotically stable.

Lemma 8 (see [1]).

Let [figure omitted; refer to PDF] be continuous and locally Lipschitz in [figure omitted; refer to PDF] If

(i) [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] are wedges,

then [figure omitted; refer to PDF] of (5) is uniformly bounded and uniformly ultimately bounded for bound [figure omitted; refer to PDF]

3. Main Results

We will give the following notations before we state our main results. Let [figure omitted; refer to PDF] For the first case of consideration set [figure omitted; refer to PDF] , system (4) reduces to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are functions defined in Section 1. Let [figure omitted; refer to PDF] be any solution of (10); the continuously differentiable functional employed in the proof of our results is [figure omitted; refer to PDF] defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are fixed positive constants satisfying [figure omitted; refer to PDF] with [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are nonnegative constants which will be determined later.

Remark 9.

The Lyapunov functional defined in (11) is an improvement on the one used in [9].

At last, we now state our main results and give their proofs.

Theorem 10.

Further to the assumptions on the functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , suppose that, for all [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive constants and for all [figure omitted; refer to PDF] :

(i) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(iii) : [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(iv) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

(v) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; and if [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] then the trivial solution of (10) is uniformly asymptotically stable.

Remark 11.

(i) If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (10) reduces to the system considered in [29] and some of our hypotheses agree with the hypotheses obtained therein.

(ii) When [figure omitted; refer to PDF] , the functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] the above result includes that discussed in [24].

(iii) Whenever [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (10) specializes to that studied in [12]. Thus, the result of Theorem 10 coincides with results in [12] if [figure omitted; refer to PDF] .

(iv) When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (3) reduces to linear constant coefficients differential equations and conditions (i) to (v) of Theorem 10 specialize to the corresponding Routh-Hurwitz criteria [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

(v) When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (10) specializes to that discussed in [28]. Theorem 10 coincides with the stability result in [28].

(vi) When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , then (3) reduces to the ordinary differential equation studied in [31].

(vii) If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] then (3) coincides with (2) discussed in [27]; hence our hypotheses coincide with that of Tunç in [27].

(viii) Whenever [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (10) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for [figure omitted; refer to PDF] which is replaced by a more general condition.

(ix) Finally, the functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] used in this paper extend the works in [7-10, 12, 24, 27-29, 31].

In what follows, we will state and prove a result that would be useful in the proof of Theorem 10 and subsequent ones.

Lemma 12.

Under the hypotheses of Theorem 10 there exist positive constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that for all [figure omitted; refer to PDF] [figure omitted; refer to PDF] Furthermore, there exists a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Proof.

Let [figure omitted; refer to PDF] be any solution of (10); since [figure omitted; refer to PDF] , (11) can be recast in the form [figure omitted; refer to PDF] From hypotheses (ii), (iii), and (iv) and the fact that the double integrals appearing in inequality (18) are nonnegative, it follows that there exists a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] Estimate (19) establishes the lower inequality in (16) with [figure omitted; refer to PDF] , respectively. Moreover, from inequality (19) we find that [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] , and it follows that for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] Furthermore, since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , and inequality [figure omitted; refer to PDF] , there exists positive constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Estimate (22) establishes the upper inequality in (16) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. Hence, from inequalities (19) and (22) estimate (16) of Lemma 12 is established.

Next, the time derivative of the functional defined in inequality (11) with respect to the independent variable [figure omitted; refer to PDF] , along a solution of system (10), is simplified to give [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Now from the assumptions of Theorem 10 we find that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Using estimates [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in (24), we find that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we find [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Moreover, using the estimate [figure omitted; refer to PDF] in [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Inserting estimates [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in inequality (28) with hypothesis [figure omitted; refer to PDF] of Theorem 10, choosing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Now in view of the inequalities in (12) there exists a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Inequality (35) establishes (17) with [figure omitted; refer to PDF] , respectively. This completes the proof of Lemma 12.

Proof of Theorem 10.

Let [figure omitted; refer to PDF] be any solution of system (10), in view of the inequalities in (19), (22), and (35); all the assumptions of Lemma 7 hold. Thus by Lemma 7 the trivial solution of system (10) or (3) for [figure omitted; refer to PDF] is uniformly asymptotically stable. This completes the proof of Theorem 10.

Next, we will consider the case of [figure omitted; refer to PDF] , and we have the following result.

Theorem 13.

If hypotheses (i)-(v) and the inequality in (14) of Theorem 10 hold and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous functions satisfying [figure omitted; refer to PDF] and there exists an [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] then

(i) the solutions of system (4) are uniformly bounded and uniformly ultimately bounded;

(ii) equation (4) has a unique periodic solution of period [figure omitted; refer to PDF]

Remark 14.

(i) Whenever [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , system (4) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for [figure omitted; refer to PDF] which is replaced by a more general condition in ours.

(ii) If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , system (4) reduces to that considered in [16].

(iii) When [figure omitted; refer to PDF] , the functions [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the above result includes that discussed in [24].

(iv) Whenever [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (4) specializes to that studied in [12].

(v) When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , system (4) reduces to that considered in [28]. Theorem 13 coincides with the boundedness result in [28].

(vi) If [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] in inequality (37), our result specializes to that studied in [9, 27].

(vii) Whenever [figure omitted; refer to PDF] , in inequality (37) the result in Theorem 13 reduces to that discussed in [8].

Hence, Theorem 13 includes and improves the results in [7-9, 12, 16, 24, 27, 28].

Proof of Theorem 13.

(i) Let [figure omitted; refer to PDF] be any solution of system (4); the time derivative of the functional [figure omitted; refer to PDF] defined in system (11) along a solution of system (4) is [figure omitted; refer to PDF] Using inequality (35), the above relation becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Applying inequality (37), we find that [figure omitted; refer to PDF] From estimates (38) and (39) and on choosing [figure omitted; refer to PDF] , there exist constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

The inequalities in (19), (22), and (43) establish the hypotheses of Lemma 8. Hence by Lemma 8, the solution [figure omitted; refer to PDF] of system (4) is uniformly bounded and uniformly ultimately bounded.

(ii) From estimate (42), using the inequalities in (38) and (39), we have [figure omitted; refer to PDF] Choosing [figure omitted; refer to PDF] sufficiently small such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] provided that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . In view of (19), (21), (22), and (45) all assumptions of Lemmas 5 and 6 are met. Hence by Lemmas 5 and 6, system (4) has a unique periodic solution of period [figure omitted; refer to PDF] . This completes the proof of Theorem 13.

Next, if [figure omitted; refer to PDF] in system (4) is replaced by [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the functions defined in Section 1, and [figure omitted; refer to PDF] , we have the following result.

Theorem 15.

If hypotheses (i)-(v) and estimate (14) of Theorem 10 hold, and [figure omitted; refer to PDF] then for any given finite constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] there exists a constant [figure omitted; refer to PDF] such that any solution [figure omitted; refer to PDF] of system (46) determined by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , satisfies [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Remark 16.

if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (46) reduces to that considered in [19]. Our results are quite different from this because of the non-Liapunov approach used in [19].

Proof of Theorem 15.

Let [figure omitted; refer to PDF] be any solution of system (46). In view of the hypotheses (i)-(v) and estimate (14), inequality (19) holds. The derivative of the functional [figure omitted; refer to PDF] defined in system (11) with respect to the independent variable [figure omitted; refer to PDF] along a solution of system (46) is [figure omitted; refer to PDF] By inequality (35), [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , and from the fact that [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Also from inequality (19), the above inequality becomes [figure omitted; refer to PDF] Solving this first-order differential inequality by multiplying each side by [figure omitted; refer to PDF] integrating from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , and employing inequality (47), we find that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Engaging inequality (19), we have [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] Equating [figure omitted; refer to PDF] , the inequalities in (48) are satisfied. This completes the proof of Theorem 15.

4. Examples

Example 1.

Consider the homogeneous third-order scalar delay differential equation [figure omitted; refer to PDF] Reducing (56) to system of first-order delay differential equations by setting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] Comparing system (10) with system (57), we have the following relations:

(1) The function [figure omitted; refer to PDF] . It is clear from the above equation that [figure omitted; refer to PDF]

(2) The function [figure omitted; refer to PDF] . It is not difficult to show that [figure omitted; refer to PDF]

(3) The function [figure omitted; refer to PDF] , from where we obtain the following estimates:

(a) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(b) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

(4) The calculation of the following constants also follows:

(a) [figure omitted; refer to PDF] ; we choose [figure omitted; refer to PDF] ;

(b) [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , and we choose [figure omitted; refer to PDF] ;

(c) [figure omitted; refer to PDF] , and we choose [figure omitted; refer to PDF] ;

(d) [figure omitted; refer to PDF] .

All the assumptions of Theorem 10 are satisfied. Hence by Theorem 10 trivial solution of system (57) is uniformly asymptotically stable.

Example 2.

Consider the nonhomogeneous third-order delay differential equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] As usual, system (60) is equivalent to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Comparing systems (4) and (62) we observe that the function [figure omitted; refer to PDF] Substituting appropriately for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the above equation can be recast in the form [figure omitted; refer to PDF] or [figure omitted; refer to PDF] From inequalities (37) and (66) we obtain the following periodic functions (see Figures 1 and 2): [figure omitted; refer to PDF] It is not difficult to show that [figure omitted; refer to PDF] The estimates in (68) and that of Example 1 satisfy the hypotheses of Theorem 13. Hence by Theorem 13,

(i) solutions of system (62) are uniformly bounded and uniformly ultimately bounded;

(ii) system (62) has a unique periodic solution of period [figure omitted; refer to PDF] .

Figure 1: Periodic function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 2: Periodic function [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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