Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

We study oscillatory behaviour of a large class of second-order functional differential equations with three freedom real nonnegative parameters. According to a new oscillation criterion, we show that if at least one of these three parameters is large enough, then the main equation must be oscillatory. As an application, we study a class of Duffing type quasilinear equations with nonlinear time delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term. Finally, some open questions and comments are given for the purpose of further study on this topic.

In Section 2 we state a fundamental lemma proposing a new oscillation criterion that plays a crucial role in the formulation of the main results illustrated on some suitable chosen examples.In Section 3 we consider an application of the main results to the Duffing type quasilinear equations with time delayed feedback, taking into account the known results in applied sciences concerning such kind of nonlinear oscillators without time delay, see , and with time delay, see [38][39][40][41][42][43][44][45][46][47][48] and the references therein.In Section 5 we present some open questions and comments for further study that can follow our main results.And in Section 6, we describe the method for proving the main results of the paper.
We hope that (4) can be relaxed with some weaker condition, which is commented as an open problem in Section 5 below.In the following fundamental lemma which plays a crucial role in the proof of the main results, we are working with such solutions () of (1) that satisfies the inequality [ (,  ( ())) +  (,  ( ())) −  ()] × (| () | −1  ()) −1 ≥  (, , )  () , for all  ∈  and some interval , where the functions (, , ) and () do not depend on () but only on , ,  and they are determined in the process below.The functions (, , ) and () present the key point in the parametrically excited oscillations.
In some concrete cases, we use the next version of Lemmas 2 and 3, where condition (10) or ( 12) is replaced with appropriate one that appears in (1) with periodic coefficients.
The first result of the paper deals with delay equation (1).
The proof of Theorem 5 is presented in Section 6 and it is based on Lemma 4, where The second result deals with advanced equation (1).
The third result deals with delay-advanced equation (1).
Remark 8.A difference between assumptions of Theorems 5, 6, and 7 is that () in Theorems 5 and 6 is not necessarily periodic or bounded function as it is supposed in Theorem 7.
According to previous corollaries, we can derive the following examples.
Example 13 (advanced case).Let  ≥ 1,  ≥ 0, and  ∈ N be fixed and   ∈ R, 0 ≤   < /(4).With the help of Corollary 10, the following two classes of quasilinear advanced differential equations: are oscillatory provided at least one of  > 0 and  > 0 is large enough (the case  = 0 is possible if  = 0).In order to apply Corollary
When  ̸ = 0 and linear time delay feedback Φ(, (−)) = ( − ), the following topics have been studied for various types of Duffing oscillators with time delayed feedback: in [38] authors constructed a low-order approximate solution under weak feedback gain parameter; about the low-and high-order approximations see also [39]; in [40] with  = 0, the Hopf bifurcation diagrams have been explored for the approximate periodic solutions (amplitude versus time delay  and feedback gain  versus time delay ); moreover, in [41] authors made an analysis on the effect of the control gain and time delay parameters on the amplitude of approximate period solution from the theoretical and numerical points of view; see also [42]; in [43] authors studied the chaotic behaviour with respect to gains and time delay parameters; see also [44].
Equations under time delay control such as (34) (especially with damped term) are used as a model for various controlled physical, mechanical, and engineering systems with time delays; see, for instance, [39,[45][46][47][48] and the references therein.
Here, (34) contains very general nonlinear time delay feedback Φ(, ( − )) with Φ satisfying (33) and the linear time delay feedback ( − ) is only a particular case of it, and, to the best of our knowledge, the previous topics are not considered for (34), as yet.Moreover, with such an Φ, the oscillations of (34) can be taken under a doubt even with the linear time delay feedback (see the nature of the approximations given in [38,39]).Hence, we can pose the following question: under what conditions on equation's parameters, ( 34) is a nonlinear oscillator, that is, possesses only oscillatory solutions?An answer is given in the next result as an easy consequence of the parametrically excited oscillations by Theorem 5.
Remark 16.Even in the linear forced case (() ̸ ≡ 0), it is not easy to establish the oscillations of all solutions, since the oscillation and nonoscillation can occur simultaneously.The most simple and important example for the coincidence of oscillation and nonoscillation is the following linear forced differential equation:   + (2/)  +  = 2/,  > 0, that allows an oscillatory solution  1 () = (3 sin )/ + 2/ and a nonoscillatory solution  2 () = 2/.This is not possible in the linear case with () ≡ 0, because of Sturm's separation theorem.

Parametrically Excited Oscillations and Well-Known Oscillation Criteria
In this section, we would like to draw the reader's attention to the fact that the parametrically excited oscillations have been already appearing in some published papers on the oscillation of functional differential equations, but only in some examples illustrating certain main oscillation criteria.However, with the help of our main results in which the parametrically excited oscillations are studied in a general setting, the equations from these examples are replaced with general ones also having parameters  and .
In the linear case (analogously for the superlinear case see [1, Theorem 2]), the author proved the following oscillation criterion.In what follows, we denote then (36) with  = 1 is oscillatory.
Previous criterion has been applied on the following particular equation: where  ≥ 0 and  = 1.Applying Theorem 17 to (39), the author proved that (39) is oscillatory provided the following inequality: holds for sufficiently large .Thus, the oscillation of ( 39) is excited by the large enough parameter .However, according to Theorems 5 and 6, we are able to show that the next parametric equation that corresponds to general equation ( 36) is oscillatory provided  is large enough, where  1 =  2 = ,  = 0, and  = 1.Next, in [5] (see also [6][7][8]), the authors consider the oscillation of the following class of second-order differential equations with delay and advanced arguments: where  1 ,  or for  = 1, 2, then (42) with  1 =  2 = 1 is oscillatory.
As a consequence of this result, it has been concluded that the particular equation ( ()   ()) +  sin ()  ( −  12 ) is oscillatory provided either  or  is large enough.However, by following Theorems 5 and 6, one can obtain the same conclusion for the following general equation associated with (42): Related observation can be done with [8, Example 3.3] and [9, Example 2.1], where the quasilinear second-order functional differential equations have been considered.It is left to the reader.

Some Open Questions and Comments
In this section, we discuss some problems related to our main results that are not studied here.
(1) Quasiperiodic Case.In the theory of nonlinear oscillators, a particularly important case occurs when the periodic coefficients in the oscillator do not have any common period.It is called the quasiperiodic (or two-frequency) nonlinear oscillator and studied, for instance, in [50][51][52].Since in Theorems 5, 6, and 7 we assume that the corresponding periodic functions have a common period, it is natural to pose the next question.
Open Question 1.Is it possible to derive sufficient conditions for the oscillation of ( 27) in the case when () and () (resp., (), (), and ℎ()) are two (resp., three) periodic functions not having a common period?
Comment.We suggest the reader to enlarge the main results of this paper to ( 48) and ( 49).
(3) Damped Duffing Equation.In the application, the Duffing equation ( 34) is often appearing with the linear damped term   (); that is, ) where  0 is the damped coefficient which can, in an active way, influence various behaviours of (50).Since (, (),   ()) =  0   () does not satisfy the required assumption (4), we are not able to apply our main results to (50).Hence we pose the following question.
Open Question 2. Is it possible to obtain the parametrically excited oscillation for (1) in the case when the damped term (, , V) satisfies a larger condition than (4) in which the linear damped term   () is especially included?(4) Functional Argument in Damped Term.In a class of Duffing equations, we have two time delayed feedback and, hence, besides the control gain parameter  1 another parameter  2 appears, the so-called velocity gain parameter.Hence, instead of (34) one can consider Therefore, we suggest the following problem for further study.

Proofs of Main Results
The proof of Lemma 1 is based on the following three steps: two working forms of condition (6) (see Lemmas 19 and 20), the existence of an explosive solution of a suitable Riccati differential inequality (see Proposition 22), and a comparison principle (see Proposition 24).
In conclusion, according to previous two lemmas, we see that supposed condition ( 6) implies (59), which plays an important role in the proof of the main results.
The second step in the proof of Lemma 1 is to prove the existence of a function () which blows up in the finite time and satisfies a generalized Riccati differential lower inequality; we briefly present the existence and properties of the so-called generalized tangent type function.In what follows, let  * be a positive real number defined in (3).Let us remark that () =   ,  > 1, implies  * = (2)/( sin(/)), see, for instance, [54], and obviously for  = 2 we have  * = .
Proof of Lemma 2. From assumption (10), we obtain the existence of an  0 ∈ N such that ())   ()) Now from (9) and previous inequality we deduce that for large enough , , , and which shows (6).Thus, all assumptions of Lemma 1 are fulfilled and, hence, Lemma 2 immediately follows from Lemma 1.
Proof of Lemma 3. Obviously assumption (11) is a particular case of assumption (9).Hence, this proof is very similar to the proof of Lemma 2 and so it is left to the reader.
Proof of Lemma 4. It is clear that from assumption (13) we obtain Thus, hypothesis ( 12) is fulfilled and, therefore, Lemma 3 proves this lemma.
The proof that the function   () given in (19), (22), or (25) satisfies the second claim in (13).Without loss of generality, we prove this claim only in case (i), since for other cases the proof follows analogously.In this sense, let   () = ()  ().Since Next, to the end of this proof, let () be a solution of (1).In particular, it implies that (()(  ())) (96) Now we need the following lemma.