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 delayFurthermore, Ademola [9] discussed existence and uniqueness of a periodic solution to the third-order differential equationUnfortunately, 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 considerwhere , , , 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 equationwhere , 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 equationwhere 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 thatwhere 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. LetFor the first case of consideration set , system (4) reduces towhere , , , 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 aswhere and are fixed positive constants satisfyingwith, , 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 ifwherethen 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 formFrom 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 thatfor all , , and , whereEstimate (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 thatwhereEstimate (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 givewhereNow from the assumptions of Theorem 10 we find thatfor all , , and andfor all , , . Using estimates and in (24), we find thatfor all , , , whereSince , , andfor all , , , we findfor all and . Moreover, using the estimatein we obtainfor all and . Inserting estimates and in inequality (28) with hypothesis of Theorem 10, choosing , , and , we havefor all , , .
Now in view of the inequalities in (12) there exists a positive constant such thatwhereInequality (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 andfor all , , and , where and are continuous functions satisfyingand there exists an such thatthen(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) isUsing inequality (35), the above relation becomeswhere . Applying inequality (37), we find thatFrom estimates (38) and (39) and on choosing , there exist constants and such thatwhere 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 haveChoosing sufficiently small such that and we haveprovided 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 havewhere , , 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, andthen for any given finite constants , , there exists a constant such that any solution of system (46) determined by , , , for , satisfiesfor 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) isBy inequality (35), for all , and from the fact that , it follows thatfor all , , and . Also from inequality (19), the above inequality becomesSolving this first-order differential inequality by multiplying each side by integrating from to , and employing inequality (47), we find thatwhere .
Engaging inequality (19), we havefor all , whereEquating , 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 equationReducing (56) to system of first-order delay differential equations by setting and we obtainComparing 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:whereAs usual, system (60) is equivalent towhereComparing systems (4) and (62) we observe that the function Substituting appropriately for and , the above equation can be recast in the formorFrom inequalities (37) and (66) we obtain the following periodic functions (see Figures 1 and 2):It is not difficult to show thatThe 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 .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.