#### Abstract

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.

#### 1. Introduction

Let be three nonnegative parameters and let , , , , and be continuous functions in all their variables satisfying some conditions determined in Section 2. We consider the following large class of second-order functional differential equations: where as usual the functional terms and satisfy , , and . Three cases are studied simultaneously: delay (, ), advanced (, ), and delay-advanced (, ), and both and are increasing functions.

A continuous function is called nonoscillatory if there is a point such that for all . Otherwise, is an oscillatory function. A function , , is called the (extendable) solution of (1) if it satisfies equality in (1) for all . Equation (1) is oscillatory if all its solutions are oscillatory.

In the paper, we investigate conditions on , , and under which (1) is oscillatory provided at least one of parameters , , and is large enough. It is shortly called the parametrically excited oscillations of (1). In Section 4 we discuss some known oscillation criteria published in [1–9] which allow the parametrically excited oscillations but only in the form of examples. On various problems concerning the functional differential equations we refer the reader to [10–15] and the references therein.

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 [16–37], and with time delay, see [38–48] 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.

#### 2. Main Assumptions and Results

Let . For the functions and both appearing in the second-order differential operator of (1), we suppose the following: and is odd, increasing, and For instance, it is simple to check that for , hypothesis (3) is fulfilled in the next three most important cases of : the linear operator if and , the quasilinear -Laplacian operator if and , and the quasilinear mean curvature operator if and . Some details about the number which depends on the function given in (3) are presented in Section 6.

The damped term satisfies the strong condition 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 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.

Lemma 1. *Let assumptions (2), (3), and (4) hold. Let and be two disjoint open intervals such that . Let the functions , , and , , on , be such that
**
for both and , for all , , and , and for some , where , , , and are constants defined, respectively, in (2) and (3). Let be a solution of (1) satisfying the next two statements:
**
Then has at least one zero point in .*

This lemma simultaneously holds for all three types of functional arguments: delay, advanced, and delay-advanced. It will be proved in Section 6. In Corollaries 9, 10, and 11 below, we give some simple conditions on two functions and two numbers , such that all solutions of (1) with the functions and satisfy required statements (7) and (8) with respect to some intervals and the explicitly given functions and satisfying (6), where satisfies a basic assumption.

In what follows, , , denotes a sequence of disjoint open intervals such that , and as . Now we present several variations of Lemma 1 in which essential inequality (6) is relaxed with some asymptotic assumptions that are simpler to be verified in several applications.

Lemma 2. *Let assumptions (2), (3), and (4) hold. Let the continuous function and the sequence of functions , , on , satisfy the next two inequalities: there are constants , , and such that
**
where the numbers , , , and are, respectively, from (2) and (3), and
**
If is a solution of (1) that satisfies (7) and (8) with , , , , and , then has at least one zero point in , .*

In the following slightly simpler version of Lemma 2, inequality (9) is replaced with an asymptotic condition and, at the same time, the limit in (10) is relaxed with the limit inferior. Thus, conditions (9) and (10) are replaced with more practical ones.

Lemma 3. *Let assumptions (2), (3), and (4) hold. Let the continuous function and the sequence of functions , , , on satisfy, respectively,
**
If is a solution of (1) that satisfies (7) and (8) with , , , , and , then has at least one zero point in , , and for some .*

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.

Lemma 4. *Let assumptions (2), (3), and (4) hold. Let the continuous function and the sequence of functions , , , on satisfy, respectively, (11) and, for some ,
**
and some . If is a solution of (1) that satisfies (7) and (8) with , , , , and , then has at least one zero point in , .*

Next, we suppose that the coefficient additionally satisfies and the forcing term satisfies We remark that remains arbitrary function outside the set .

The first result of the paper deals with delay equation (1).

Theorem 5. *Let assumptions (2), (3), (4), (14), and (15) hold, , , , , and let satisfy
**
where , number is from (3), sequence is from (15), and is a periodic function with period such that
**Then (1) is oscillatory in the next two cases: and parameter is large enough; , , , and at least one of parameters and is large enough.*

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).

Theorem 6. *Let assumptions (2), (3), (4), (14), and (15) hold, , , , , and let satisfy
**
where , number is from (3), sequence is from (15), and is a periodic function with period such that (17) is fulfilled. Then (1) is oscillatory in the next two cases: and parameter is large enough; , , , and at least one of parameters and is large enough.*

The proof of Theorem 6 is based on Lemma 4 (see Section 6), where

The third result deals with delay-advanced equation (1).

Theorem 7. *Let assumptions (2), (3), (4), (14), and (15) hold, , , , , and and satisfy
**
where, additionally, are three periodic functions having a common period such that (17) is fulfilled, where is the forcing term in (1). Then (1) is oscillatory provided one of the next two cases is fulfilled, where the number is from (3): (in superlinear delay-advanced case) , , , and either parameter is large enough or at least one of and is large enough; (in supersublinear delay-advanced case) , , , , and at least one of parameters , , and is large enough.*

The proof of Theorem 7 is based on Lemma 4 (see Section 6), where where we denote Here, the numbers are chosen such that and . Let us mention that if , and , and , then and satisfy previous two equalities. About the existence of such -tuple in a general case, we refer to [49].

*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.

Now, we study an important class of second-order functional differential equations as a particular case of (1): where , , and and . Using previous theorems, we are able to state the main consequences showing the parametrically excited oscillations in (27).

Corollary 9 (delay equation). *Let assumptions (2), (3), (4), (14), and (15) hold. Let , , and on , where is a periodic function with period satisfying (16). Equation (27) is oscillatory in the following two cases: and parameter is large enough; , , , and at least one of parameters and is large enough.*

Corollary 10 (advanced equation). *Let assumptions (2), (3), (4), (14), and (15) hold. Let , , and on , where is a periodic function with period satisfying (17). Then (27) is oscillatory in the following two cases: and parameter is large enough; , , , and at least one of parameters and is large enough.*

Corollary 11 (delay-advanced equation). *Let assumptions (2), (3), (4), (14), and (15) hold, and and satisfy
**
where, additionally, are three periodic functions having a common period such that (17) is fulfilled, where is the forcing term in (1). Then (27) is oscillatory in the next two cases, where the number is from (3): (1) (superlinear case) , , , and either parameter is large enough or at least one of and is large enough; (2) (supersublinear case) , , , , and at least one of parameters , , and is large enough.*

According to previous corollaries, we can derive the following examples.

*Example 12 (delay case). *Let , , and be fixed and , . With the help of Corollary 9, the following two different classes of quasilinear delay differential equations:
are oscillatory provided at least one of and is large enough (the case is possible if ). It is because, for all , we have
where is the common period of the functions and . Thus, in order to apply Corollary 9, we can choose , , , and .

*Example 13 (advanced case). *Let , , and be fixed and , . With the help of Corollary 10, the following two classes of quasilinear advanced differential equations:
are oscillatory provided at least one of and is large enough (the case is possible if ). In order to apply Corollary 10, we can choose , , , and .

*Example 14 (delay-advanced case). *Let , , , and be fixed and and , . With the help of Corollary 11, the following class of quasilinear delay-advanced differential equations:
is oscillatory provided either is large enough or at least one of and is large enough. In order to apply Corollary 11, we can choose , , , and .

#### 3. Application to Duffing Equations with Time Delay Feedback

Let denote the control gain parameter (often called “displacement feedback coefficient”), the time delay, and and the amplitude and frequency of the external force, respectively. Let the function that will appear in the delay feedback term satisfy the general condition For instance, , , or more general, , , .

In this section, we consider the following large class of undamped possible nonautonomous and nonconservative Duffing equations without or with the general time delay feedback : where is the natural frequency, is the density of the nonlinear potential (or rigidity coefficient), and , , , are nonnegative constants, and .

When , , and ., (34) contains many most important classes of undamped autonomous Duffing oscillators such as the following:(i)the* strongly nonlinear Duffing oscillator with smooth odd nonlinearity* is given in (34) provided and ; let us recall some of its known particular cases:(a)the classic Duffing oscillator has been recently studied in the searching of solitary wave solutions of classic and generalized Zakharov equations of plasma physics (see [16]) and of nonlinear Schrödinger equation (see [17]); also, it is strongly connected with the Jacobi elliptic equation (see [18]);(b)the cubic-quintic oscillator is used as a model for the nonlinear dynamics of a slender elastica (see [19]) in nonlinear wave systems (see [20]) for the propagation of a short electromagnetic pulse in a nonlinear medium (see [21]) and in the unimodal Duffing temporal problem (see [22]);(c)the cubic truly nonlinear oscillator models the motion of a ball bearing that oscillates in a glass tube that is bent into a curve (see [23]) as well as the motion of a mass attached to identical stretched elastic wires (see [24]);(d)the nonhomogeneous Duffing oscillator describes various forced vibrations of beams, springs with nonlinear stiffness, cables, plates, shells, and optical fibres in electrical circuits, in nonlinear isolators, and so forth (see, for instance, [25, 26]);(ii)the* general Duffing-harmonic oscillator* (with rational or irrational nonlinear restoring-force) is given in (30) if , , and ; the most known subclasses of these oscillators are(a)the classic Duffing-harmonic oscillator which models many conservative nonlinear oscillatory systems; see [27];(b)the relativistic harmonic oscillator ; see [28];(c)the nonlinear oscillator , , which is typified as a mass attached to a stretched elastic wire; see [29, 30];(d)the nonlinear oscillator which presents nonlinear oscillations of a punctual charge in the electric field of charged ring; see [31].Finding several explicit forms of periodic approximate solutions for these oscillators has been intensively studied last years by many authors; see, for instance, [28, 30, 32–37] and also the references therein.

When and linear time delay feedback , 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 , 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, 45–48] and the references therein.

Here, (34) contains very general nonlinear time delay feedback with 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 , 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.

Theorem 15. *Let and (33) hold. Equation (34) is oscillatory in the next two cases:*(i)* and is large enough;*(ii)*, , , and at least one of and is large enough.*

*Proof. *Let , , , , , and
It is easy to check that all assumptions of Theorem 5 are fulfilled with respect to the sequence and , where since it is supposed that . Hence, Theorem 5 proves this theorem.

*Remark 16. *Even in the linear forced case (), 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: , , that allows an oscillatory solution and a nonoscillatory solution . This is not possible in the linear case with , because of Sturm's separation theorem.

#### 4. 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 [1] (see also [2, Example 3.1] with , [3, Example 3.1], and [4, Section 3]), the author considers the oscillation of the second-order delay differential equation: in the linear case () and the superlinear (). 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

Theorem 17 ([1, Theorem 1]). *Suppose that, for any , there exist constants , , , such that , , and on , on , and on . If there exists , , such that
**
then (36) with is oscillatory.*

Previous criterion has been applied on the following particular equation: where and . 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 , , and .

Next, in [5] (see also [6–8]), the authors consider the oscillation of the following class of second-order differential equations with delay and advanced arguments: where . When the authors prove the following result (for other cases see [5, Theorems 3.2, 3.3, and 3.4].

Theorem 18 ([5, Theorem 3.1]). *Suppose that, for any , there exist intervals , , , and contained in such that , , , , and
**
and , , and , . If there exist and such that either
**
or
**
for , then (42) with is oscillatory.*

As a consequence of this result, it has been concluded that the particular equation 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.

#### 5. 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 [50–52]. 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 ) are two (resp., three) periodic functions not having a common period?

*(2) Equation with More Functional Arguments*. Next, regarding some second-order functional differential equations considered in the references of this paper, more than two nonlinear functional terms are appearing and, therefore, instead of main equation (1) and corresponding particular equation (27) considered in Theorems 5, 6, and 7, we suggest the following classes of equations:
where , , , , and
where and .

*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 is the damped coefficient which can, in an active way, influence various behaviours of (50). Since 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 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 another parameter appears, the so-called velocity gain parameter. Hence, instead of (34) one can consider
Therefore, we suggest the following problem for further study.

*Open Question **3*. Is it possible to obtain the parametrically excited oscillation for the following more general functional differential equation than (1) in which the functional argument appears in the damped term too, as follows:
or
About known oscillation criteria for the second-order functional differential equations having the functional argument in the damped term, we refer the reader to, for instance, [53] and the references therein.

#### 6. 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).

Lemma 19 (a necessary condition to (6)). *Let on . If assumption (6) is fulfilled, then there is a positive real number such that
**
for all , , and and some .*

*Proof. *Since for , we conclude that, for
it holds that , , and hence
On the other hand, from (6) we observe
which, together with (55) and (56), gives
for all , , , and . It proves this lemma.

Lemma 20 (an equivalent condition to (54)). *Assumption (54) is fulfilled if and only if there is a real number and a continuous function , , such that
**
for all , , , and and some .*

*Proof. *This proof is very elementary. Indeed, if (54) holds, then the function and number , defined by
obviously satisfy and which shows (59). Conversely, if (59) holds, then, integrating both sides of the second inequality in (59), we obtain
which shows (54).

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 , , implies , see, for instance, [54], and obviously for we have .

Lemma 21. *Let be a continuous function such that
**
Then there is a real number and a function , , such that
**
Moreover, is increasing and odd:
**
In particular, for , one can take and .*

*Proof. *Let , , be a function defined by
The function is well defined since is positive and continuous on , is increasing and odd function, and
Moreover, because of (62), there is a real number such that
Thus, and there exists an inverse function of the original function and . Also, from and on , we also derive that on its domain and
Putting for into (66) and using (68) we easily obtain
Moreover, from (67) we have