Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 875020, 17 pages

http://dx.doi.org/10.1155/2014/875020

Research Article

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

Mervan Pašić

Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia

Received 8 December 2013; Accepted 12 February 2014; Published 16 April 2014

Academic Editor: Zhengqiu Zhang

Copyright © 2014 Mervan Pašić. 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.

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 . Thus, if we set , then previous two statements and (67) prove this lemma.

Next, we prove the main result of this section.

Proposition 22. Let (2) and (6) hold, where . Let be a real number and let , , be a continuous function, both obtained in Lemma 20. Let be from (3) and from (59), and let be an arbitrary real number. If is the generalized tangens function defined in (63) and is a function defined by

then there is a such that

Moreover, for a function defined by

one has , , and

where the numbers and are from (3) and the functions and are from (6).

Proof. Under assumptions (2) and (6) and because of Lemmas 19 and 20, we obtain and , , satisfying inequality (59).

Next, since (see Lemma 21), from (70) we directly obtain

Since we obtain , and from (74) we observe that there exist numbers such that . Also, gives , which proves statement (71). Moreover, it together with Lemma 21 and (72) proves that

Next, according to (59), (63), and (72), we make the following calculation on the interval :

Thus, all assertions of this proposition are proved.

Remark 23. In the proof of the main result, the number is determined by , where denotes a function associated with a nonoscillatory solution and it is given by (84) below.

The third step in the proof of Lemma 1 is to show the following pointwise comparison principle for the functions and satisfying, respectively, the lower and upper differential inequalities (73) and

Proposition 24. Let be an arbitrary interval. One supposes that all coefficients of Riccati differential inequalities (73) and (77) are continuous and strictly positive functions. Let be two functions satisfying, respectively, (73) and (77) on the interval . Then

Proof. Let be a function defined by

Let and be arbitrary. For any two , ,  , let be an interval defined by . Since is a -function on , we know by the Lagrange mean value theorem applied on that there is a such that

since . Hence for any and , , , we have

Thus, the function from (79) satisfies required condition of [55, Lemma 19] and, applying it to (73) and (77), we prove this proposition.

Proof of Lemma 1. On the contrary, let be a solution of (1) such that

that is, on or on since is a continuous function on . Let, for instance,

Another case can be analogously treated; let us see the comment at the end of this proof. In particular from (83) we have on which implies (since and are increasing functions) for all , which yields , , and on . Hence, by assumption (7), we may use inequality (5) on the interval .

Firstly, we show that the following classic Riccati transformation of :

satisfies upper Riccati differential inequality (77). Let us remark that from (1) we have in particular

Taking the first derivative on both sides of (84) and using assumptions (3), (4), and (5) as well as equality (85) and , we obtain

Thus, according to inequality (5), it is shown that if is a solution of (1) which satisfies (83), then the function defined by (84) satisfies the Riccati differential inequality (77) and . On the other hand, let be a real number defined by . According to (6) and Lemma 19 we obtain (54) which together with Lemma 20 ensures that we may use Proposition 22 for such chosen real number . Hence, we obtain a function defined by (72) which satisfies the lower Riccati differential inequality (73) on , , such that and . Therefore, by and Proposition 24, we conclude that too, which is a contradiction with the above conclusion saying that . Thus, hypothesis (82) is not true and, consequently, Lemma 1 is shown.

For the analogous case on , we also have on which implies (since and are increasing functions)

which yields , , and on . Now we can repeat the preceding procedure but on interval and using (8) instead of and (7).

Proof of Lemma 2. From assumption (10), we obtain the existence of an such that

that is,

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.

Proof of Theorems 5, 6, and 7. This proof is based on Lemma 4. In order to simplify notation, in many places in this proof we set and . Since assumptions (2), (3), and (4) have been already supposed in Theorems 5, 6, and 7, in order to prove these theorems by Lemma 4, we are going to show that the functions and explicitly given, respectively, in (18), (21), or (24) and (19), (22), or (25) satisfy required conditions (11) and (13), respectively, and that every solution of (27) satisfies conditions (7) and (8) with respect to functions and , where , , , and .

The proof that the function given in (18), (21), or (24) satisfies (11). Passing to the limit in (18), (21), or (24) it is very simple to show (11).

The proof that the function given in (19), (22), or (25) satisfies the first claim in (13). From (25) we immediately obtain

Next, by assumptions of this corollary we can conclude that there are three positive constants , , such that and on in cases (i) and (ii) and on in cases (iii) and (iv). Putting previous inequalities into (19), (22), or (25), for all and , it holds that

which shows the first claim in (13).

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 , , , and , where is the period of the function , we have and , . Hence

which proves that the integral on the left hand side does not depend on ; that is, the second claim in (13) is shown on . This claim follows in the same way on . Thus, the second claim in (13) is proved on .

Next, to the end of this proof, let be a solution of (1). In particular, it implies that . It together with assumptions (15), (16), (20), and (23) easily gives the next two statements:

Now we need the following lemma.

Lemma 25. Let and be defined by

and let be an arbitrary function. If for all or for all , then

Since is supposed to be odd and increasing function, just before (3), and satisfies (14), the proof of Lemma 25 in the first case, that is, for all , is the same as the proof of [9, Corollaries 17 and 18]. But in the second case, that is, for all , the proof is as follows: if previous inequality holds, then for all and, therefore, to the function one can apply the first case of this lemma and consequently one obtains

which proves this lemma in the second case.

Now, combining statements (95), (96), and (98), one easily obtains

where and are defined in (26).

The proof that satisfies (7) and (8). In this proof, we frequently use assumptions (16), (20), and (23) and statements (100) and (101). Also, because of (15) and , , in both cases (100) and (101), we can simultaneously use

where in the case of (100) and in the case of (101).

(i) Delay or Advanced Case with . Since , we obtain

where the functions and are defined in (26).

(ii) Delay Case with . In this part we use the next elementary inequality:

Since and using (104) especially for

for all we obtain

where the function is from (18).

(iii) Advanced Case with . Using the same line of arguments as in the proof of the previous case, for all we obtain

where the function is from (21).

(iv) Superlinear Delay-Advanced Case. Since , for all we obtain

Now, just the same as in the proofs of previous delay and advanced cases with and with the help of (104) in particular for

we have

where the function is from (24). Analogously, we show that

Summarizing previous calculation, we obtain

where the function is from (24).

(v) Supersublinear Delay-Advanced Case. Since and the following well-known elementary inequality holds

from , and , we obtain for all , for all we obtain

where and are given, respectively, in (24) and (25). Thus, it is shown that required condition (5) in the cases (i)–(iv) is fulfilled with respect to and determined by (18), (21), or (24) and (19), (22), or (25).

In conclusion, according to the previous observation, we see that all assumptions of Lemma 4 are fulfilled and, hence, Lemma 4 proves Theorems 5, 6, and 7.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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