International Journal of Differential Equations
Volume 2015 (2015), Article ID 213935, 12 pages
http://dx.doi.org/10.1155/2015/213935

Research Article

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

A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, and O. A. Adesina

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

Academic Editor: Nikolai N. Leonenko

Copyright © 2015 A. T. Ademola et al. 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.

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.

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

Furthermore, Ademola [9] discussed existence and uniqueness of a periodic solution to the third-order differential equation

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 and the function ) 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

where , , , and are continuous functions in their respective arguments on , , , and , respectively, with and . The dots indicate differentiation with respect to the independent variable . Equation (3) is equivalent to the system of first-order delay differential equation

where , is a constant to be determined latter, and the derivatives and for all exist and are continuous for all and with . 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 and the forcing term 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

where is a continuous mapping and for all and for some positive constant . We assume that takes closed bounded sets into bounded sets in . is the Banach space of continuous function with supremum norm, ; for , we define by , and   is the open -ball in , .

Definition 1 (see [2]). A continuous function with , if and strictly increasing is a wedge. (We denote wedges by or , where is an integer.)

Definition 2 (see [2]). The zero solution of (5) is asymptotically stable if it is stable and if for each there is such that implies that

Definition 3 (see [1]). An element is in the -limit set of , say , if is defined on and there is a sequence , as , with as , where for .

Definition 4 (see [30]). A set is an invariant set if, for any , the solution of (5) is defined on and for .

Lemma 5 (see [6]). Suppose that and is periodic in of period , , and consequently for any there exists such that implies . Suppose that a continuous Lyapunov functional exists, defined on , , is the set of such that ( may be large), and satisfies the following conditions:

(i), where and are continuous, increasing, and positive for and as ;(ii), where is continuous and positive for .

Suppose that there exists , , such that

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

Lemma 6 (see [6]). Suppose that is defined and continuous on , , and there exists a continuous Lyapunov functional defined on , , and which satisfy the following conditions:

(i) if ;(ii) if ;(iii) for the associated system

 we have , where, for or , we understand that the condition is satisfied in case can be defined.

Then, for given initial value , , there exists a unique solution of (5).

Lemma 7 (see [6]). Suppose that a continuous Lyapunov functional exists, defined on , , and which satisfies the following conditions:

(i), where and are continuous, increasing, and positive;(ii) , where is continuous and positive for ;

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

Lemma 8 (see [1]). Let be continuous and locally Lipschitz in If

(i);(ii) , for some , where are wedges,

then of (5) is uniformly bounded and uniformly ultimately bounded for bound

3. Main Results

We will give the following notations before we state our main results. Let

For the first case of consideration set , system (4) reduces to

where , , , and are functions defined in Section 1. Let be any solution of (10); the continuously differentiable functional employed in the proof of our results is defined as

where and are fixed positive constants satisfying

with

, , and 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 , , , and , suppose that, for all  ,  , , , , and are positive constants and for all :

(i) for all ;(ii) for all ;(iii) for all ;(iv) for all and ;(v) , , ; and if

where

then the trivial solution of (10) is uniformly asymptotically stable.

Remark 11. (i) If , , , and , (10) reduces to the system considered in [29] and some of our hypotheses agree with the hypotheses obtained therein.
(ii) When , the functions , , , and the above result includes that discussed in [24].
(iii) Whenever , , , and , (10) specializes to that studied in [12]. Thus, the result of Theorem 10 coincides with results in [12] if .
(iv) When , , , and , (3) reduces to linear constant coefficients differential equations and conditions (i) to (v) of Theorem 10 specialize to the corresponding Routh-Hurwitz criteria , , and .
(v) When , , , and , (10) specializes to that discussed in [28]. Theorem 10 coincides with the stability result in [28].
(vi) When , , , and , then (3) reduces to the ordinary differential equation studied in [31].
(vii) If and then (3) coincides with (2) discussed in [27]; hence our hypotheses coincide with that of Tunç in [27].
(viii) Whenever , , , , and , (10) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for which is replaced by a more general condition.
(ix) Finally, the functions and 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 , , and such that for all

Furthermore, there exists a constant such that

Proof. Let be any solution of (10); since , (11) can be recast in the form

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 such that

for all , , and , where

Estimate (19) establishes the lower inequality in (16) with , respectively. Moreover, from inequality (19) we find that if and only if and if and only if , and it follows that for all , ,

Furthermore, since , for all , for all , and inequality , there exists positive constants , such that

where

Estimate (22) establishes the upper inequality in (16) with and , 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 , along a solution of system (10), is simplified to give

where

Now from the assumptions of Theorem 10 we find that

for all , , and and

for all , , . Using estimates and in (24), we find that

for all , , , where

Since , , and

for all , , , we find

for all and . Moreover, using the estimate

in we obtain

for all and . Inserting estimates and in inequality (28) with hypothesis of Theorem 10, choosing , , and , we have

for all , , .
Now in view of the inequalities in (12) there exists a positive constant such that

where

Inequality (35) establishes (17) with , respectively. This completes the proof of Lemma 12.

Proof of Theorem 10. Let 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 is uniformly asymptotically stable. This completes the proof of Theorem 10.

Next, we will consider the case of , and we have the following result.

Theorem 13. If hypotheses (i)–(v) and the inequality in (14) of Theorem 10 hold and

for all   , , and , where and are continuous functions satisfying

and there exists an such that

then

(i)the solutions of system (4) are uniformly bounded and uniformly ultimately bounded;(ii) equation (4) has a unique periodic solution of period

Remark 14. (i) Whenever , , , , and , system (4) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for which is replaced by a more general condition in ours.
(ii) If , , , , and , system (4) reduces to that considered in [16].
(iii) When , the functions , , , and , the above result includes that discussed in [24].
(iv) Whenever , , , , and , (4) specializes to that studied in [12].
(v) When , , , , , and , system (4) reduces to that considered in [28]. Theorem 13 coincides with the boundedness result in [28].
(vi) If , and in inequality (37), our result specializes to that studied in [9, 27].
(vii) Whenever , 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 be any solution of system (4); the time derivative of the functional defined in system (11) along a solution of system (4) is

Using inequality (35), the above relation becomes

where . Applying inequality (37), we find that

From estimates (38) and (39) and on choosing , there exist constants and such that

where and .
The inequalities in (19), (22), and (43) establish the hypotheses of Lemma 8. Hence by Lemma 8, the solution of system (4) is uniformly bounded and uniformly ultimately bounded.
(ii) From estimate (42), using the inequalities in (38) and (39), we have

Choosing sufficiently small such that and we have

provided that where . 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 . This completes the proof of Theorem 13.

Next, if in system (4) is replaced by , we have

where , , and are the functions defined in Section 1, and , we have the following result.

Theorem 15. If hypotheses (i)–(v) and estimate (14) of Theorem 10 hold, and

then for any given finite constants , , there exists a constant such that any solution of system (46) determined by , , , for , satisfies

for all .

Remark 16. if , , , , and , (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 be any solution of system (46). In view of the hypotheses (i)–(v) and estimate (14), inequality (19) holds. The derivative of the functional defined in system (11) with respect to the independent variable along a solution of system (46) is

By inequality (35), for all , and from the fact that , it follows that

for all , , and . Also from inequality (19), the above inequality becomes

Solving this first-order differential inequality by multiplying each side by

integrating from to , and employing inequality (47), we find that

where .
Engaging inequality (19), we have

for all , where

Equating , 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

Reducing (56) to system of first-order delay differential equations by setting and we obtain

Comparing system (10) with system (57), we have the following relations:

(1)The function . It is clear from the above equation that

(2) The function . It is not difficult to show that

(3)The function , from where we obtain the following estimates:

(a),  ;(b), .

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

(a); we choose ;(b) or , and we choose ;(c), and we choose ;(d).

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:

where

As usual, system (60) is equivalent to

where

Comparing systems (4) and (62) we observe that the function

Substituting appropriately for and , the above equation can be recast in the form

or

From inequalities (37) and (66) we obtain the following periodic functions (see Figures 1 and 2):

It is not difficult to show that

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 1: Periodic function .

Figure 2: Periodic function .

Conflict of Interests

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

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