Abstract
The behaviour of solutions to certain second order nonlinear delay differential equations with variable deviating arguments is discussed. The main procedure lies in the properties of a complete Lyapunov functional which is used to obtain suitable criteria to guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results are new and also complement related ones that have appeared in the literature. Moreover, examples are given to illustrate the feasibility and correctness of the main results.
1. Introduction
It is well known that the existence of boundedness, stability, and periodic solutions are among the most attractive topics in the qualitative behaviour of solutions to ordinary and functional differential equations. There are many articles dedicated so far to the qualitative properties of solutions of second order differential equations. Various authors have discussed these properties of solutions under numerous assumptions through different approaches. Some of these contributions include but are not limited to the works of Burton [1–3], Hale [4], and Yoshizawa [5–7] which contain general results on the subject matter and the expository papers of Ademola [8], Alaba and Ogundare [9], Cahlon and Schmidt [10], Domoshnitsky [11], Grigoryan [12], Jin and Zengrong [13], Kroopnick [14], Ogundare and Afuwape [15], Ogundare and Okecha [16], Tunç [17–21], Wang and Zhu [22], Xianfeng and Wei [23], and Yeniçerioğlu [24, 25] and the rich references cited therein.
In an interesting article, Yeniçerioğlu [24] considered the behavior of solutions of second order delay differential equationwhere are real numbers and is positive real number and obtained criterion for stability of the considered equation. Furthermore, Yeniçerioğlu [25] discussed the stability properties of second order delay integrodifferential equations where is a real number, is positive real number, is a continuous real-valued function on the interval , and is a given continuously differentiable initial function on the interval .
In their contribution, Xianfeng and Wei [23] investigated the stability and boundedness of a retarded Lienard-type equation where is a nonnegative constant, , , and are continuous functions on , and is differentiable on .
For the case where there are no delay terms, we can mention some of the recent works of Alaba and Ogundare [9] and Ademola [8]. Alaba and Ogundare [9] studied the second order nonautonomous damped and forced nonlinear ordinary differential equation of the form where the functions , and depend only on the arguments displayed explicitly. Another work worthy of mention in this regard is the article of Ademola [8] where the problem of stability, boundedness, and existence of periodic solutions of second order nonlinear nonautonomous ordinary differential equation where , and , was considered.
Motivated by the above discussions, in this paper, we consider the problem of uniform asymptotic stability, uniform ultimate boundedness, and existence of unique periodic solutions to a second order delay differential equation whose nonlinear functions contain variable deviating arguments. In effect, we considerwhere , and are continuous functions in their respective arguments on , , and , respectively, with and . The continuity of these functions is sufficient for the existence of the solutions of (6). Furthermore, it is assumed that the functions , , and in (6) satisfy a Lipschitz condition in their respective arguments. The dots denote differentiation with respect to the independent variable . If , then (6) is equivalent to the systemwhere , is a constant to be determined later, and the derivative of the function (i.e., ) exists and is continuous for all . To the best of our knowledge from the relevant literature, we are yet to come across any work with the above framework despite the practical importance of (6). This may not be unconnected with the practical difficulties associated with the construction of suitable complete Lyapunov functional. Thus, the prime purpose of this paper is to fill the vacuum. An equally interesting problem is the situation where only the function in (6) is with a constant deviating argument. This has already been considered by us and the results arising in this direction will be advertized through another outlet.
The obtained results in the present paper are completely new and they extend previously known results in [8, 9] on second order ordinary differential equations and the works in [23–25] where the deviating arguments are constants. Some mathematical tools that will be needed in the sequel are discussed in Section 2. The main results of this paper are presented in Section 3 while examples are given in the last section to validate our results.
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 ; is the open -ball in , .
Definition 1 (see [3]). A continuous function with , if , and strictly increasing is a wedge. (We denote wedges by or , where is an integer.)
Definition 2 (see [3]). The zero solution of (8) is asymptotically stable if it is stable and if for each there is an such that implies that
Definition 3 (see [2]). 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 [26]). A set is an invariant set if for any , the solution of the system (8) is defined on , and for .
Lemma 5 (see [7]). Suppose that and is periodic in of period , , and consequently for any there exists an 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 an , , 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 (8) of period . In particular, inequality (10) can always be satisfied if is sufficiently small.
Lemma 6 (see [7]). Suppose that is defined and continuous on , and that there exists a continuous Lyapunov functional defined on , which satisfy the following conditions: (i) if ;(ii) if ;(iii)for the associated system one has , where for or , one understands that the condition is satisfied in the case can be defined.Then, for given initial value , , there exists a unique solution of system (8).
Lemma 7 (see [7]). Suppose that a continuous Lyapunov functional exists, defined on , , 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 the system (8) is uniformly asymptotically stable.
Lemma 8 (see [2]). Let be continuous and locally Lipschitz in . If(i),(ii), for some where are wedges,then of system (8) 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 , , and Let be any solution of system (7); the main tool to proofs of our results is the continuously differentiable functional defined aswhere , , are constants to be determined. Next, we present the main results of this work.
Theorem 9. In addition to the assumptions on the functions , and , suppose that are positive constants and that for all (i) for all and ;(ii) for all ;(iii) for all ;(iv) for all ;(v), , , where(vi), .Then the solutions of the system (7) are uniformly bounded and uniformly ultimately bounded.
Remark 10. Below are some observations from previous results which the current results are addressing: (i)When , and , then system (7) reduces to linear constant coefficients differential equation , and conditions (i) to (vi) of Theorem 9 specialize to the corresponding Routh Hurwitz criteria and .(ii)When , , and , system (7) specializes to ordinary differential equation discussed in [22]. Our result includes and extends the results in [22].(iii)If (a constant), , and , system (7) is trimmed down to that discussed in [10, 24]. Thus our results improve the results in [10, 24].(iv)If , , , and then system (7) reduces to that considered in [17]. Thus the result in [17] is a special case of Theorem 9.(v)Whenever , (6) reduces to the cases discussed in [8, 9].In what follows, our purpose now is to state and prove the following result that would be used in the proofs of Theorem 9 and the subsequent ones.
Lemma 11. Under the hypotheses of Theorem 9, there exist positive constants , , and such thatfor all , and .
Furthermore, there exist positive constants and such thatfor all , and .
Proof. Let be any solution of system (7). It is clear from (13) that Since the double integrals terms are nonnegative, for all , and for all and , then there exists a positive constant such that, and , where Thus the lower inequality in (15) holds with . Also, from inequality (17) we deduced thatFurthermore, since for all , and the fact that , then it follows from (13) that there exist positive constants and such thatfor all , and , where Inequality (22) establishes the upper inequality in (15) where and . Hence from inequalities (17) and (22), inequality (15) is established.
Next, let be any solution of the system (7). The time derivative of the function , defined in (13), along a solution path of the system (7) isNow, from the hypotheses of Theorem 9, for all , for all and , for all , for all , and the fact that , then from (24) it follows thatwhere Employing the inequalities for all and , respectively, it follows thatFrom inequality (28) and the fact that and for all , the inequality in (25) yieldsIf we choose and in view of inequality (14), there exist positive constants and such that inequality (29) becomesfor all , and , whereEstimate (31) establishes the inequality in (16) with and equal to and , respectively. This completes the proof of Lemma 11.
Next, we will give the proof of Theorem 9.
Proof of Theorem 9. Let be any solution of system (7), hypothesis (vi) of Theorem 9, and the fact that , and then there exist positive constants and such that inequality (31) yieldsfor all , and , where and . From inequalities (17), (22), and (33) the hypotheses of Lemma 8 hold. Thus by Lemma 8, the solution of the system (7) is uniformly bounded and uniformly ultimately bounded. This completes the proof of Theorem 9.
Next, suppose that and in (6) and system (7), respectively; we have the following delay differential equation:If , (34) is equivalent to the systemwhere the functions , and are as defined in Section 1. We have the following result for system (35).
Theorem 12. Suppose that hypotheses (i) to (v) of Theorem 9 hold; then the zero solution of system (35) is uniformly asymptotically stable.
Remark 13. (i) The stability results in [2, 8–10] are special cases of Theorem 12.
(ii) When and (34) reduces to the one discussed in [12].
(iii) Whenever and then (34) is a case studied in [22].
(iv) If and then (34) reduces to the one discussed in [25].
Proof of Theorem 12. Let be any solution of system (35), with hypotheses of Theorem 12; estimates (17), (21), and (22) hold. Furthermore, the derivative of the functional defined in (13) with respect to the independent variable along the solution path of the system (35) isfor all , and , where is defined after the inequality in (31). From inequalities (17), (21), (22), and (36) all hypotheses of Lemma 7 hold. Thus by Lemma 7 the zero solution of system (35) is uniformly asymptotically stable.
Theorem 14. If hypotheses (i) to (vi) of Theorem 9 hold, then there exists a unique periodic solution of period for the system (7).
Proof. Let be any solution of system (7). In view of the hypotheses of Theorem 14 (19) and inequality (20) hold. Moreover, from hypothesis (vi) and the fact that for all and , it follows from inequality (31) that , and . Choose and then there exists a positive constant such thatfor all and , where . Now from (19) and inequalities (20) and (40), all hypotheses of Lemma 7 hold. By Lemma 7, the existence of a unique solution for the system (7) is assured. Furthermore, from inequalities (17), (21), (22), and (40) all hypotheses of Lemma 6 hold. Hence by Lemma 6 the existing unique solution is periodic of period . This completes the proof of Theorem 14.
4. Examples
In this section, we present examples to check the validity and effectiveness of our results.
Example 1. Consider a second order delay differential equationEquation (41) is equivalent to system of first order delay differential equationComparing system (7) with (42), we obtain the following relations. (i) The function Since for all , it follows that for all . The behaviour of is shown in Figure 1. (ii) The function It is not difficult to show that for all , and . See the behaviour of in Figure 2. (iii) The function Since for all , then we assert that (iv) The function Clearly, is a bounded function; thus it is not difficult to show that The boundedness of and is shown in Figure 3. (v) The function Since for all , and , it follows that for all , and ; and if (vi) , and with the values of as above, then inequality (14) becomes All hypotheses of Theorems 9, 12, and 14 are satisfied; hence the conclusions of Theorems 9, 12, and 14 follow.
Example 2. Consider the second order delay differential equationwhere Equation (57) is equivalent to system of first order differential equationwhereComparing system (7) with system (59), we obtain the following relations. (i)The function Noting that for all , and , we assert that for all , and . The function is shown in Figure 4. (ii)The function Let Since for all , it follows that for all . (iii)Furthermore, from (ii) above we have It is not difficult to show that for all . The functions , , and are shown in Figure 5. (iv)The function Since for , it follows that for all . The graph of function is shown in Figure 6. (v)Moreover, choose and from (i) to (iv) of Example 2 we have so that (vi)The function Since for all and , it follows that for all , and .All hypotheses of Theorems 9 and 14 are satisfied; thus by Theorems 9 and 14 we have that (i)solutions of system (59) are uniformly bounded and uniformly ultimately bounded; (ii)there exists a unique periodic solution of period ,respectively. If and in (57) and system (59), respectively, and conditions (i) to (v) of Example 2 hold, then hypotheses of Theorem 12 hold; hence by Theorem 12 (iii)the trivial solution of the system (59) (when ) is uniformly asymptotically stable.
Competing Interests
The authors declare that they have no competing interests.