Abstract and Applied AnalysisVolume 2011 (2011), Article ID 591254, 34 pagesdoi:10.1155/2011/591254

Research Article

Nonoscillation of Second-Order Dynamic Equations with Several Delays

Elena Braverman¹ and Başak Karpuz²

¹Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada, T2N 1N4²Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 30 December 2010; Accepted 13 February 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Elena Braverman and Başak Karpuz. Elena Braverman and . 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.

Existence of nonoscillatory solutions for the second-order dynamic equation for is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case for and for second-order nondelay difference equations ( for ). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.

1. Introduction

This paper deals with second-order linear delay dynamic equations on time scales. Differential equations of the second order have important applications and were extensively studied; see, for example, the monographs of Agarwal et al. [1], Erbe et al. [2], Győri and Ladas [3], Ladde et al. [4], Myškis [5], Norkin [6], Swanson [7], and references therein. Difference equations of the second order describe finite difference approximations of second-order differential equations, and they also have numerous applications.

We study nonoscillation properties of these two types of equations and some of their generalizations. The main result of the paper is that under some natural assumptions for a delay dynamic equation the following four assertions are equivalent: nonoscillation of solutions of the equation on time scales and of the corresponding dynamic inequality, positivity of the fundamental function, and the existence of a nonnegative solution for a generalized Riccati inequality. The equivalence of oscillation properties of the differential equation and the corresponding differential inequality can be applied to obtain new explicit nonoscillation and oscillation conditions and also to prove some well-known results in a different way. A generalized Riccati inequality is used to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the well-known Sturm-Picone comparison theorem for a second-order ordinary differential equation; see [7, Section 1.1]. Applying positivity of the fundamental function, positive solutions of two equations can be compared. There are many results of this kind for delay differential equations of the first-order and only a few for second-order equations. Myškis [5] obtained one of the first comparison theorems for second-order differential equations. The results presented here are generalizations of known nonoscillation tests even for delay differential equations (when the time scale is the real line).

The paper also contains conditions on the initial function and initial values which imply that the corresponding solution is positive. Such conditions are well known for first-order delay differential equations; however, to the best of our knowledge, the only paper concerning second-order equations is [8].

From now on, we will without furthermore mentioning suppose that the time scale is unbounded from above. The purpose of the present paper is to study nonoscillation of the delay dynamic equation where , , is the forcing term, , and for all , is the coefficient corresponding to the function , where on .

In this paper, we follow the method employed in [8] for second-order delay differential equations. The method can also be regarded as an application of that used in [9] for first-order dynamic equations.

As a special case, the results of the present paper allow to deduce nonoscillation criteria for the delay differential equation in the case for , they coincide with theorems in [8]. The case of a “quickly growing” function when the integral of its reciprocal can converge is treated separately.

Let us recall some known nonoscillation and oscillation results for the ordinary differential equations where is nonnegative, which are particular cases of (1.2) with , , and for all .

In [10], Leighton proved the following well-known oscillation test for (1.4); see [10, 11].

Theorem A (see [10]). Assume that then (1.3) is oscillatory.

This result for (1.4) was obtained by Wintner in [12] without imposing any sign condition on the coefficient .

In [13], Kneser proved the following result.

Theorem B (see [13]). Equation (1.4) is nonoscillatory if for all , while oscillatory if for all and some .

In [14], Hille proved the following result, which improves the one due to Kneser; see also [14–16].

Theorem C (see [14]). Equation (1.4) is nonoscillatory if while it is oscillatory if

Another particular case of (1.1) is the second-order delay difference equation to the best of our knowledge, there are very few nonoscillation results for this equation; see, for example, [17]. However, nonoscillation properties of the nondelay equations have been extensively studied in [1, 18–22]; see also [23]. In particular, the following result is valid.

Theorem D. Assume that then (1.10) is oscillatory.

The following theorem can be regarded as the discrete analogue of the nonoscillation result due to Kneser.

Theorem E. Assume that for all , then (1.10) is nonoscillatory.

Hille's result in [14] also has a counterpart in the discrete case. In [22], Zhou and Zhang proved the nonoscillation part, and in [24], Zhang and Cheng justified the oscillation part which generalizes Theorem E.

Theorem F (see [22, 24]). Equation (1.10) is nonoscillatory if while is oscillatory if

In [23], Tang et al. studied nonoscillation and oscillation of the equation where is a sequence of nonnegative reals and obtained the following theorem.

Theorem G (see [23]). Equation (1.14) is nonoscillatory if (1.12) holds, while is it oscillatory if (1.13) holds.

These results together with some remarks on the -difference equations will be discussed in Section 7. The readers can find some nonoscillation results for second-order nondelay dynamic equations in the papers [20, 25–29], some of which generalize some of those mentioned above.

The paper is organized as follows. In Section 2, some auxiliary results are presented. In Section 3, the equivalence of the four above-mentioned properties is established. Section 4 is dedicated to comparison results. Section 5 includes some explicit nonoscillation and oscillation conditions. A sufficient condition for existence of a positive solution is given in Section 6. Section 7 involves some discussion and states open problems. Section 7 as an appendix contains a short account on the fundamentals of the time scales theory.

2. Preliminary Results

Consider the following delay dynamic equation: where , is a time scale unbounded above, , are the initial values, is the initial function, such that has a finite left-sided limit at the initial point provided that it is left dense, is the forcing term, , and for all , is the coefficient corresponding to the function , which satisfies for all and . Here, we denoted then is finite, since asymptotically tends to infinity.

Definition 2.1. A function with and a derivative satisfying is called a solution of (2.1) if it satisfies the equation in the first line of (2.1) identically on and also the initial conditions in the second line of (2.1).

For a given function with a finite left-sided limit at the initial point provided that it is left-dense and , problem (2.1) admits a unique solution satisfying on with and (see [30] and [31, Theorem 3.1]).

Definition 2.2. A solution of (2.1) is called eventually positive if there exists such that on , and if is eventually positive, then is called eventually negative. If (2.1) has a solution which is either eventually positive or eventually negative, then it is called nonoscillatory. A solution, which is neither eventually positive nor eventually negative, is called oscillatory, and (2.1) is said to be oscillatory provided that every solution of (2.1) is oscillatory.

For convenience in the notation and simplicity in the proofs, we suppose that functions vanish out of their specified domains, that is, let be defined for some , then it is always understood that for , where is the characteristic function of the set defined by for and for .

Definition 2.3. Let and . The solutions and of the problems which satisfy , are called the first fundamental solution and the second fundamental solution of (2.1), respectively.

The following lemma plays the major role in this paper; it presents a representation formula to solutions of (2.1) by the means of the fundamental solutions and .

Lemma 2.4. Let be a solution of (2.1), then can be written in the following form: for .

Proof. For , let We recall that and solve (2.3) and (2.4), respectively. To complete the proof, let us demonstrate that solves This will imply that the function defined by on is a solution of (2.1). Combining this with the uniqueness result in [31, Theorem 3.1] will complete the proof. For all , we can compute that Therefore, , , and on , that is, satisfies the initial conditions in (2.7). Differentiating after multiplying by and using the properties of the first fundamental solution , we get for all . For , set and . Making some arrangements, for all , we find and thus which proves that satisfies (2.7) on since and for each . The proof is therefore completed.

Next, we present a result from [9] which will be used in the proof of the main result.

Lemma 2.5 (see [9, Lemma 2.5]). Let and assume that is a nonnegative -integrable function defined on . If satisfy then for all implies for all .

3. Nonoscillation Criteria

Consider the delay dynamic equation and its corresponding inequalities

We now prove the following result, which plays a major role throughout the paper.

Theorem 3.1. Suppose that the following conditions hold: , for , , for , satisfies for all and , then the following conditions are equivalent: (i)the second-order dynamic equation (3.1) has a nonoscillatory solution,(ii)the second-order dynamic inequality (3.2) has an eventually positive solution and/or (3.3) has an eventually negative solution,(iii)there exist a sufficiently large and a function with satisfying the first-order dynamic Riccati inequality (iv)the first fundamental solution of (3.1) is eventually positive, that is, there exists a sufficiently large such that for all and all .

Proof. The proof follows the scheme: (i)(ii)(iii)(iv)(i).(i)(ii) This part is trivial, since any eventually positive solution of (3.1) satisfies (3.2) too, which indicates that its negative satisfies (3.3).(ii)(iii) Let be an eventually positive solution of (3.2), then there exists such that for all . We may assume without loss of generality that (otherwise, we may proceed with the function , which is also a solution since (3.2) is linear). Let then evidently and which proves that . This implies that the exponential function is well defined and is positive on the entire time scale ; see [32, Theorem 2.48]. From (3.5), we see that satisfies the ordinary dynamic equation whose unique solution is see [32, Theorem 2.77]. Hence, using (3.8), for all , we get which gives by substituting into (3.2) and using [32, Theorem 2.36] that for all . Since the expression in the brackets is the same as the left-hand side of (3.4) and on , the function is a solution of (3.4).(iii)(iv) Consider the initial value problem Denote where is any solution of (3.11) and is a solution of (3.4). From (3.12), we have whose unique solution is see [32, Theorem 2.77]. Now, for all , we compute that and similarly for . From (3.12) and (3.15), we have for all . We substitute (3.14), (3.15), (3.16), and (3.17) into (3.11) and find that for all . Then, (3.18) can be rewritten as for all , where for . We now show that on . Indeed, by using (3.4) and the simple useful formula (A.2), we get for all . On the other hand, from (3.11) and (3.12), we see that . Then, by [32, Theorem 2.77], we can write (3.19) in the equivalent form where, for , we have defined Note that implies (indeed, we have on ), and thus the exponential function is also well defined and positive on the entire time scale , see [32, Exercise 2.28]. Thus, on implies on . For simplicity of notation, for , we let Using the change of integration order formula in [33, Lemma 1], for all , we obtain and similarly Therefore, we can rewrite (3.23) in the equivalent form of the integral operator whose kernel is nonnegative. Consequently, using (3.22), (3.24), and (3.28), we obtain that on implies on ; this and Lemma 2.5 yield that on . Therefore, from (3.14), we infer that if on , then on too. On the other hand, by Lemma 2.4, has the following representation: Since is eventually nonnegative for any eventually nonnegative function , we infer that the kernel of the integral on the right-hand side of (3.29) is eventually nonnegative. Indeed, assume to the contrary that on but is not nonnegative, then we may pick and find such that . Then, letting for , we are led to the contradiction , where is defined by (3.29). To prove that is eventually positive, set for , where , to see that and on , which implies is nonincreasing on . So that, we may let so large that (i.e., ) is of fixed sign on . The initial condition and (A1) together with imply that on . Consequently, we have for all .(iv)⇒(i) Clearly, is an eventually positive solution of (3.1).The proof is completed.

Let us introduce the following condition: with

Remark 3.2. It is well known that (A4) ensures existence of such that for all , for any nonoscillatory solution of (3.1). This fact follows from the formula for all , obtained by integrating (3.1) twice, where . In the case when (A4) holds, (iii) of Theorem 3.1 can be assumed to hold with , which means that any positive (negative) solution is nondecreasing (nonincreasing).

Remark 3.3. Let (A4) hold and exist and the function satisfying inequality (3.4), then the assertions (i), (iii), and (iv) of Theorem 3.1 are also valid on .

Remark 3.4. It should be noted that (3.4) is also equivalent to the inequality see (3.20) and compare with [26, 28, 29, 34].

Example 3.5. For , (3.4) has the form see [8] for the case , , and [35] for , , .

Example 3.6. For , (3.4) becomes where the product over the empty set is assumed to be equal to one; see [1, 18] (or (1.10)) for , , , and [20] for , , , . It should be mentioned that in the literature all the results relating difference equations with discrete Riccati equations consider only the nondelay case. This result in the discrete case is therefore new.

Example 3.7. For with , under the same assumption on the product as in the previous example, condition (3.4) reduces to the inequality for all .

4. Comparison Theorems

Theorem 3.1 can be employed to obtain comparison nonoscillation results. To this end, together with (3.1), we consider the second-order dynamic equation where for .

The following theorem establishes the relation between the first fundamental solution of the model equation with positive coefficients and comparison (4.1) with coefficients of arbitrary signs.

Theorem 4.1. Suppose that (A2), (A3), (A4), and the following condition hold: (A5)for , with for all . Assume further that (3.4) admits a solution for some , then the first fundamental solution of (4.1) satisfies for all and all , where denotes the first fundamental solution of (3.1).

Proof. We consider the initial value problem where . Let , and define the function as By the Leibnitz rule (see [32, Theorem 1.117]), for all , we have Substituting (4.3) and (4.5) into (4.2), we get where in the last step, we have used the fact that for all and all . Therefore, we obtain the operator equation where whose kernel is nonnegative. An application of Lemma 2.5 shows that nonnegativity of implies the same for , and thus is nonnegative by (4.3). On the other hand, by Lemma 2.4, has the representation Proceeding as in the proof of the part (iii)(iv) of Theorem 3.1, we conclude that the first fundamental solution of (4.1) satisfies for all and all . To complete the proof, we have to show that for all and all . Clearly, for any fixed and all , we have which by the solution representation formula yields that for all . This completes the proof since the first fundamental solution satisfies for all and all by Remark 3.3.

Corollary 4.2. Suppose that (A1), (A2), (A3), and (A5) hold, and (3.1) has a nonoscillatory solution on , then (4.1) admits a nonoscillatory solution on .

Corollary 4.3. Assume that (A2) and (A3) hold. (i)If (A1) holds and the dynamic inequality where for and , has a positive solution on , then (3.1) also admits a positive solution on . (ii)If (A4) holds and there exist a sufficiently large and a function satisfying the inequality then the first fundamental solution of (3.1) satisfies for all and all .

Proof. Consider the dynamic equation Theorem 3.1 implies that for this equation the assertions (i) and (ii) hold. Since for all , we have for all , the application of Corollary 4.2 and Theorem 4.1 completes the proof.

Now, let us compare the solutions of problem (2.1) and the following initial value problem: where are the initial values, is the initial function such that has a finite left-sided limit at the initial point provided that it is left dense, is the forcing term.

Theorem 4.4. Suppose that (A2), (A3), (A4), (A5), and the following condition hold: and satisfy Moreover, let (2.1) have a positive solution on , , and , then the solution of (4.15) satisfies for all .

Proof. By Theorem 3.1 and Remark 3.3, we can assume that is a solution of the dynamic Riccati inequality (3.4), then by , the function is also a solution of the dynamic Riccati inequality which is associated with (4.15). Hence, by Theorem 3.1 and Remark 3.3, the first fundamental solution of (4.15) satisfies for all and all . Rewriting (2.1) in the form applying Lemma 2.4, and using , we have for all . This completes the proof.

Remark 4.5. If for , for all and for all , then holds.

The following example illustrates Theorem 4.4 for the quantum time scale .

Example 4.6. Let , and consider the following initial value problems: where is the identity function on , that is, for , and Denoting by and the solutions of (4.20) and (4.22), respectively, we obtain for all by Theorem 4.4. For the graph of the first 10 iterates, see Figure 1.

Figure 1: The graph of 10 iterates for the solutions of (4.20) and (4.22) illustrates the result of Theorem 4.4, here for all .

As an immediate consequence of Theorem 4.4, we obtain the following corollary.

Corollary 4.7. Suppose that (A1), (A2), and (A3) hold and that (3.1) is nonoscillatory, then, for , the dynamic equation is also nonoscillatory.

We now consider the following dynamic equation: where the parameters are the same as in (4.15).

We obtain the most complete result if we compare solutions of (2.1) and (4.24) by omitting the condition and assuming that the solution of (2.1) is positive.

Corollary 4.8. Suppose that (A3), (A4), and the following condition hold: and satisfy If is a positive solution of (2.1) on with and , then for the solution of (4.24), one has for all .

Proof. Corollary 4.3 and Remark 3.3 imply that the first fundamental solution associated with (2.1) (and (4.24)) satisfies for all and all . Hence, the claim follows from the solution representation formula.

Remark 4.9. If at least one of the inequalities in the statements of Theorem 4.4 and Corollary 4.8 is strict, then the conclusions hold with the strict inequality too.

Let us compare equations with different coefficients and delays. Now, we consider

Theorem 4.10. Suppose that (A2), (A4), (A5), and the following condition hold: for , satisfies for all and . Assume further that the first-order dynamic Riccati inequality (3.4) has a solution for some , then the first fundamental solution of (4.26) satisfies for all and all .

Proof. Note that implies for all and , then we have for all . The reference to Corollary 4.3 (ii) concludes the proof.

Remark 4.11. If the condition in Theorem 4.1, Theorem 4.4, Corollary 4.8, and Theorem 4.10 is replaced with , then the claims of the theorems are valid eventually.

Let us introduce the function

Corollary 4.12. Suppose that (A1), (A2), (A3), and (A5) hold. If is nonoscillatory, then (4.1) is also nonoscillatory.

Remark 4.13. The claim of Corollary 4.12 is also true when is replaced by .

5. Explicit Nonoscillation and Oscillation Results

Theorem 5.1. Suppose that (A1), (A2), and (A3) hold and that where and is the identity function on , then (3.1) is nonoscillatory.

Proof. The statement of the theorem yields that for is a positive solution of the Riccati inequality (3.32).

Next, let us apply Theorem 5.1 to delay differential equations.

Corollary 5.2. Let , for , , and such that for all and . If for some , then (1.2) is nonoscillatory.

Now, let us proceed with the discrete case.

Corollary 5.3. Let be a positive sequence, for , let be a nonnegative sequence, and let be a divergent sequence such that for all . If for some , then (1.8) is nonoscillatory.

Let us introduce the function

Theorem 5.4. Suppose that (A1), (A2), and (A3) hold, and for every , the dynamic equation is oscillatory, where satisfies for all , then (3.1) is also oscillatory.

Proof. Assume to the contrary that (3.1) is nonoscillatory, then there exists a solution of (3.1) such that , on . This implies that is nonincreasing on , then it follows that or simply by using (5.4), Now, let By the quotient rule, (5.4) and (5.7), we have proving that is nonincreasing on . Therefore, for all , we obtain where satisfies for all . Using (5.10) in (3.1), we see that solves which shows that (5.5) is also nonoscillatory by Theorem 3.1. This is a contradiction, and the proof is completed.

The following theorem can be regarded as the dynamic generalization of Leighton's result (Theorem A).

Theorem 5.5. Suppose that (A2), (A3), and (A4) hold and that where , then every solution of (3.1) is oscillatory.

Proof. Assume to the contrary that (3.1) is nonoscillatory. It follows from Theorem 3.1 and Remark 3.2 that (3.4) has a solution . Using (3.5) and (5.7), we see that which together with (3.4) implies that Integrating the last inequality, we get which is in a contradiction with (5.12). This completes the proof.

We conclude this section with applications of Theorem 5.5 to delay differential equations and difference equations.

Corollary 5.6. Let , for , , and such that for all and . If where then (1.2) is oscillatory.

Corollary 5.7. Let be a positive sequence, for , let be a nonnegative sequence and let be a divergent sequence such that for all . If where then (1.8) is oscillatory.

6. Existence of a Positive Solution

Theorem 6.1. Suppose that (A2), (A3), and (A4) hold, , and the first-order dynamic Riccati inequality (3.4) has a solution . Moreover, suppose that there exist such that for all and , then (2.1) admits a positive solution such that for all .

Proof. First assume that is the solution of the following initial value problem: Denote then, by following similar arguments to those in the proof of the part (ii)(iii) of Theorem 3.1, we obtain for all . So is a solution to Theorem 4.4 implies that for all . By the hypothesis of the theorem, Theorem 4.4, and Corollary 4.8, we have for all . This completes the proof for the case and on .The general case where on is also a consequence of Theorem 4.4.

Let us illustrate the result of Theorem 6.1 with the following example.

Example 6.2. Let , and consider the following delay dynamic equation: then (5.1) takes the form for all , where the function is defined by and is decreasing on and thus is not greater than , that is, Theorem 5.1 holds. Theorem 6.1 therefore ensures that the solution is positive on . For the graph of 15 iterates, see Figure 2.

Figure 2: The graph of 15 iterates for the solution of (6.5) illustrates the result of Theorem 6.1.

7. Discussion and Open Problems

We start this section with discussion of explicit nonoscillation conditions for delay differential and difference equations. Let us first consider the continuous case. Corollary 5.6 with and for reduces to Theorem A. Nonoscillation part of Kneser's result for (1.4) follows from Corollary 5.2 by letting , , and for . Theorem E is obtained by applying Corollary 5.3 to (1.10).

Known nonoscillation tests for difference equations can also be deduced from the results of the present paper. In [18, Lemma 1.2], Chen and Erbe proved that (1.9) is nonoscillatory if and only if there exists a sequence with for all and some satisfying Since this result is a necessary and sufficient condition, the conclusion of Theorem F could be deduced from which is a particular case of (7.1) with for . We present below a short proof for the nonoscillation part only. Assuming (1.12) and letting we get and this yields That is, the discrete Riccati inequality (7.2) has a positive solution implying that (1.10) is nonoscillatory. It is not hard to prove that (1.13) implies nonexistence of a sequence satisfying the discrete Riccati inequality (7.2) (see the proof of [23, Lemma 3]). Thus, oscillation/nonoscillation results for (1.10) in [21] can be deduced from nonexistence/existence of a solution for the discrete Riccati inequality (7.2); see also [20].

An application of Theorem 3.1 with for and implies the following result for quantum scales.

Example 7.1. Let with . If there exist and such that then the delay -difference equation is nonoscillatory.

In [36], Bohner and Ünal studied nonoscillation and oscillation of the -difference equation where , and proved that (7.7) is nonoscillatory if and only if For the above -difference equation, (7.6) reduces to the algebraic inequality whose discriminant is . The discriminant is nonnegative if and only if If the latter one holds, then the inequality (7.6) holds with an equality for the value It is easy to check that this value is not less than , that is, the solution is nonnegative. This gives us the nonoscillation part of [36, Theorem 3].

Let us also outline connections to some known results in the theory of second-order ordinary differential equations. For example, the Sturm-Picone comparison theorem is an immediate corollary of Theorem 4.10 if we remark that a solution of the inequality (3.32) satisfying is also a solution of (3.32) with instead of for .

Proposition 7.2 (see [28, 32, 36]). Suppose that , , and for all , then nonoscillation of implies nonoscillation of

The following result can also be regarded as another generalization of the Sturm-Picone comparison theorem. It is easily deduced that there is a solution of the inequality (3.4).

Proposition 7.3. Suppose that (A4) and the conditions of Proposition 7.2 are fulfilled, then nonoscillation of implies the same for

Finally, let us present some open problems. To this end, we will need the following definition.

Definition 7.4. A solution of (3.1) is said to be slowly oscillating if for every there exist with for all and such that , , for all .

Following the method of [8, Theorem 10], we can demonstrate that if (A1), (A2) with positive coefficients and (A3) hold, then the existence of a slowly oscillating solution of (3.1) which has infinitely many zeros implies oscillation of all solutions.(P1)Generally, will existence of a slowly oscillating solution imply oscillation of all solutions? To the best of our knowledge, slowly oscillating solutions have not been studied for difference equations yet, the only known result is [9, Proposition 5.2].

All the results of the present paper are obtained under the assumptions that all coefficients of (3.1) are nonnegative, and if some of them are negative, it is supposed that the equation with the negative terms omitted has a positive solution.(P2)Obtain sufficient nonoscillation conditions for (3.1) with coefficients of an arbitrary sign, not assuming that all solutions of the equation with negative terms omitted are nonoscillatory. In particular, consider the equation with one oscillatory coefficient. (P3)Describe the asymptotic and the global properties of nonoscillatory solutions. (P4)Deduce nonoscillation conditions for linear second-order impulsive equations on time scales, where both the solution and its derivative are subject to the change at impulse points (and these changes can be matched or not). The results of this type for second-order delay differential equations were obtained in [37]. (P5)Consider the same equation on different time scales. In particular, under which conditions will nonoscillation of (1.8) imply nonoscillation of (1.2)? (P6)Obtain nonoscillation conditions for neutral delay second-order equations. In particular, for difference equations some results of this type (a necessary oscillation conditions) can be found in [17]. (P7)In the present paper, all parameters of the equation are rd-continuous which corresponds to continuous delays and coefficients for differential equations. However, in [8], nonoscillation of second-order equations is studied under a more general assumption that delays and coefficients are Lebesgue measurable functions. Can the restrictions of rd-continuity of the parameters be relaxed to involve, for example, discontinuous coefficients which arise in the theory of impulsive equations?

Appendix

Time Scales Essentials

A time scale, which inherits the standard topology on , is a nonempty closed subset of reals. Here, and later throughout this paper, a time scale will be denoted by the symbol , and the intervals with a subscript are used to denote the intersection of the usual interval with . For , we define the forward jump operator by while the backward jump operator is defined by , and the graininess function is defined to be . A point is called right dense if and/or equivalently holds; otherwise, it is called right scattered, and similarly left dense and left scattered points are defined with respect to the backward jump operator. For and , the -derivative of at the point is defined to be the number, provided it exists, with the property that, for any , there is a neighborhood of such that where on . We mean the -derivative of a function when we only say derivative unless otherwise is specified. A function is called rd-continuous provided that it is continuous at right-dense points in and has a finite limit at left-dense points, and the set of rd-continuous functions is denoted by . The set of functions includes the functions whose derivative is in too. For a function , the so-called simple useful formula holds where if and satisfies ; otherwise, . For and a function , the -integral of is defined by where is an antiderivative of , that is, on . Table 1 gives the explicit forms of the forward jump, graininess, -derivative, and -integral on the well-known time scales of reals, integers, and the quantum set, respectively.

Table 1: Forward jump, -derivative, and -integral.

,

A function is called regressive if on , and positively regressive if on . The set of regressive functions and the set of positively regressive functions are denoted by and , respectively, and is defined similarly.

Let , then the exponential function on a time scale is defined to be the unique solution of the initial value problem for some fixed . For , set , , and . For , we define the cylinder transformation by for , then the exponential function can also be written in the form Table 2 illustrates the explicit forms of the exponential function on some well-known time scales.

Table 2: The exponential function.

,

The exponential function is strictly positive on if , while alternates in sign at right-scattered points of the interval provided that . For , let , the circle plus and the circle minus are defined by and , respectively. Further throughout the paper, we will abbreviate the operations and simply by and , respectively. It is also known that is a subgroup of , that is, , implies and , where on .

The readers are referred to [32] for further interesting details in the time scale theory.

Acknowledgment

E. Braverman is partially supported by NSERC Research grant. This work is completed while B. Karpuz is visiting the Department of Statistics and Mathematics, University of Calgary, Canada, in the framework of Doctoral Research Scholarship of the Council of Higher Education of Turkey.

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