Nonoscillation of Second-Order Dynamic Equations with Several Delays

and Applied Analysis 3 In 10 , Leighton proved the following well-known oscillation test for 1.4 ; see 10, 11 . Theorem A see 10 . Assume that ∫∞ t0 1 A0 ( η )dη ∞, ∫∞ t0 A1 ( η ) dη ∞, 1.5 then 1.3 is oscillatory. This result for 1.4 was obtained by Wintner in 12 without imposing any sign condition on the coefficient A1. In 13 , Kneser proved the following result. Theorem B see 13 . Equation 1.4 is nonoscillatory if tA1 t ≤ 1/4 for all t ∈ t0,∞ R, while oscillatory if tA1 t > λ0/4 for all t ∈ t0,∞ R and some λ0 ∈ 1,∞ T. 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


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 Swanson 7 , and references therein.Difference equations of the second order describe finite difference approximations of secondorder 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 secondorder 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 firstorder 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 T is unbounded from above.The purpose of the present paper is to study nonoscillation of the delay dynamic equation where n ∈ N, t 0 ∈ T, f ∈ C rd t 0 , ∞ T , R is the forcing term, A 0 ∈ C rd t 0 , ∞ T , R , and for all i ∈ 1, n N , A i ∈ C rd t 0 , ∞ T , R is the coefficient corresponding to the function α i , where α i ≤ σ on t 0 , ∞ T .
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 A 0 t ≡ 1 for t ∈ t 0 , ∞ R , they coincide with theorems in 8 .The case of a "quickly growing" function A 0 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 A 1 is nonnegative, which are particular cases of 1.2 with n 1, α 1 t t, and A 0 t ≡ 1 for all t ∈ t 0 , ∞ R .
In 10 , Leighton proved the following well-known oscillation test for 1.4 ; see 10, 11 .Theorem A see 10 .Assume that This result for 1.4 was obtained by Wintner in 12 without imposing any sign condition on the coefficient A 1 .
In 13 , Kneser proved the following result.
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.
then 1.10 is oscillatory.
The following theorem can be regarded as the discrete analogue of the nonoscillation result due to Kneser.
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.
In 23 , Tang et al. studied nonoscillation and oscillation of the equation where {A 1 k } 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 q-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.

Preliminary Results
Consider the following delay dynamic equation: where n ∈ N, T is a time scale unbounded above, t 0 ∈ T, x 1 , x 2 ∈ R are the initial values, ϕ ∈ C rd t −1 , t 0 T , R is the initial function, such that ϕ has a finite left-sided limit at the initial point Here, we denoted then t −1 is finite, since α min asymptotically tends to infinity.
1 if it satisfies the equation in the first line of 2.1 identically on t 0 , ∞ T and also the initial conditions in the second line of 2.1 .
For a given function ϕ ∈ C rd t −1 , t 0 T , R with a finite left-sided limit at the initial point t 0 provided that it is left-dense and x 1 , x 2 ∈ R, problem 2.1 admits a unique solution satisfying x ϕ on t −1 , t 0 T with x t 0 x 1 and x Δ t 0 x 2 see 30 and 31, Theorem 3.1 .
Definition 2.2.A solution of 2.1 is called eventually positive if there exists s ∈ t 0 , ∞ T such that x > 0 on s, ∞ T , and if −x is eventually positive, then x 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 f : D → R be defined for some D ⊂ R, then it is always understood that f t χ D t f t for t ∈ R, where χ D is the characteristic function of the set D ⊂ R defined by χ D t ≡ 1 for t ∈ D and χ D t ≡ 0 for t / ∈ D.
Definition 2.3.Let s ∈ T and s −1 : inf t∈ s,∞ T {α min t }.The solutions X 1 X 1 •, s and X 2 X 2 •, s of the problems , 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 X 1 and X 2 .
Lemma 2.4.Let x be a solution of 2.1 , then x can be written in the following form:

2.6
We recall that X 1 •, t 0 and X 2 •, t 0 solve 2.3 and 2.4 , respectively.To complete the proof, let us demonstrate that y solves

2.7
This will imply that the function z defined by z : x 2 X 1 •, t 0 x 1 X 2 •, t 0 y on t 0 , ∞ T is a solution of 2.1 .Combining this with the uniqueness result in 31, Theorem 3.1 will complete the proof.For all t ∈ t 0 , ∞ T , we can compute that

2.8
Therefore, y t 0 0, y Δ t 0 0, and y ϕ on t −1 , t 0 T , that is, y satisfies the initial conditions in 2.7 .Differentiating y Δ after multiplying by A 0 and using the properties of the first fundamental solution X 1 , we get 1}. Making some arrangements, for all t ∈ t 0 , ∞ T , we find

and thus
which proves that y satisfies 2.
Next, we present a result from 9 which will be used in the proof of the main result. 12

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: 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 iv the first fundamental solution X 1 of 3.1 is eventually positive, that is, there exists a sufficiently large 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 x be an eventually positive solution of 3.2 , then there exists t 1 ∈ t 0 , ∞ T such that x t > 0 for all t ∈ t 1 , ∞ T .We may assume without loss of generality that x t 1 1 otherwise, we may proceed with the function x/x t 1 , which is also a solution since 3.2 is linear .Let This implies that the exponential function e Λ/A 0 •, t 1 is well defined and is positive on the entire time scale t 1 , ∞ T ; see 32, Theorem 2.48 .From 3.5 , we see that Λ satisfies the ordinary dynamic equation

whose unique solution is
x t e Λ/A 0 t, t 1 ∀t ∈ t 1 , ∞ T , 3.8 see 32, Theorem 2.77 .Hence, using 3.8 , for all t ∈ t 1 , ∞ T , we get 3.9 which gives by substituting into 3.2 and using 32, Theorem 2.36 that for all t ∈ t 1 , ∞ T .Since the expression in the brackets is the same as the left-hand side of 3.4 and e Λ/A 0 •, iii ⇒ iv Consider the initial value problem

3.11
Denote where x is any solution of 3.11 and Λ is a solution of 3.4 .From 3.12 , we have x t 1 0,

3.13
whose unique solution is 3.15 and similarly for i ∈ 1, n N .From 3.12 and 3.15 , we have for all t ∈ t 1 , ∞ T .We substitute 3.14 , 3.15 , 3.16 , and 3.17 into 3.11 and find that for all t ∈ t 1 , ∞ T .Then, 3.18 can be rewritten as for all t ∈ t 1 , ∞ T , where for t ∈ t 1 , ∞ T .We now show that Υ ≥ 0 on t 1 , ∞ T .Indeed, by using 3.4 and the simple useful formula A.2 , we get for all t ∈ t 1 , ∞ T .On the other hand, from 3.11 and 3.12 , we see that g t 1 0.Then, by 32, Theorem 2.77 , we can write 3.19 in the equivalent form where, for t ∈ t 1 , ∞ T , we have defined and thus the exponential function e Λ/A 0 •, t 1 is also well defined and positive on the entire time scale t 1 , ∞ T , see 32, Exercise 2.28 .Thus, f ≥ 0 on t 1 , ∞ T implies h ≥ 0 on t 1 , ∞ T .For simplicity of notation, for s, t ∈ t 1 , ∞ T , we let

3.25
Using the change of integration order formula in 33, Lemma 1 , for all t ∈ t 1 , ∞ T , we obtain

14
Abstract and Applied Analysis and similarly

3.27
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 f ≥ 0 on t 1 , ∞ T implies h ≥ 0 on t 1 , ∞ T ; this and Lemma 2.5 yield that g ≥ 0 on t 1 , ∞ T .Therefore, from 3.14 , we infer that if f ≥ 0 on t 1 , ∞ T , then x ≥ 0 on t 1 , ∞ T too.On the other hand, by Lemma 2.4, x has the following representation: Since x is eventually nonnegative for any eventually nonnegative function f, we infer that the kernel X 1 of the integral on the right-hand side of 3.29 is eventually nonnegative.Indeed, assume to the contrary that x ≥ 0 on t 1 , ∞ T but X 1 is not nonnegative, then we may pick t 2 ∈ t 1 , ∞ T and find s ∈ t 1 , t 2 T such that X 1 t 2 , σ s < 0.Then, letting f t : − min{X 1 t 2 , σ t , 0} ≥ 0 for t ∈ t 1 , ∞ T , we are led to the contradiction x t 2 < 0, where x is defined by 3.29 .To prove that X 1 is eventually positive, set x t : X 1 t, s for t ∈ t 0 , ∞ T , where s ∈ t 1 , ∞ T , to see that x ≥ 0 and A 0 x Δ Δ ≤ 0 on s, ∞ T , which implies A 0 x Δ is nonincreasing on s, ∞ T .So that, we may let t 1 ∈ t 0 , ∞ T so large that x Δ i.e., A 0 x Δ is of fixed sign on s, ∞ T ⊂ t 1 , ∞ T .The initial condition and A1 together with x Δ s 1/A 0 s > 0 imply that x Δ > 0 on s, ∞ T .Consequently, we have x t The proof is completed.
Let us introduce the following condition:

3.30
Remark 3.2.It is well known that A4 ensures existence of t 1 ∈ t 0 , ∞ T such that x t x Δ t ≥ 0 for all t ∈ t 1 , ∞ T , for any nonoscillatory solution x of 3.1 .This fact follows from the formula for all t ∈ t 0 , ∞ T , obtained by integrating 3.1 twice, where s ∈ t 0 , ∞ T .In the case when A4 holds, iii of Theorem 3.1 can be assumed to hold with Λ ∈ C 1 rd t 1 , ∞ T , R 0 , which means that any positive negative solution is nondecreasing nonincreasing .Remark 3.3.Let A4 hold and exist t 1 ∈ t 0 , ∞ T and the function Λ ∈ C 1 rd t 1 , ∞ T , R 0 satisfying inequality 3.4 , then the assertions i , iii , and iv of Theorem 3.1 are also valid on t 1 , ∞ T .Remark 3.4.It should be noted that 3.4 is also equivalent to the inequality 3.32 see 3.20 and compare with 26, 28, 29, 34 .
Example 3.5.For T R, 3.4 has the form Example 3.6.For T N, 3.4 becomes where the product over the empty set is assumed to be equal to one; see 1, 18 or 1.10 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 T q Z with q ∈ 1, ∞ R , under the same assumption on the product as in the previous example, condition 3.4 reduces to the inequality

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 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: Assume further that 3.4 admits a solution Λ ∈ C 1 rd t 1 , ∞ T , R 0 for some t 1 ∈ t 0 , ∞ T , then the first fundamental solution Y 1 of 4.1 satisfies Y 1 t, s ≥ X 1 t, s > 0 for all t ∈ s, ∞ T and all s ∈ t 1 , ∞ T , where X 1 denotes the first fundamental solution of 3.1 .

Proof. We consider the initial value problem
where in the last step, we have used the fact that X 1 t, σ s ≡ 0 for all t ∈ t 1 , ∞ T and all s ∈ t, ∞ T .Therefore, we obtain the operator equation where Hg t : whose kernel is nonnegative.An application of Lemma 2.5 shows that nonnegativity of f implies the same for g, and thus x is nonnegative by 4.3 .On the other hand, by Lemma 2.4, x has the representation Proceeding as in the proof of the part iii ⇒ iv of Theorem 3.1, we conclude that the first fundamental solution Y 1 of 4.1 satisfies Y 1 t, s ≥ 0 for all t ∈ s, ∞ T and all s ∈ t 1 , ∞ T .
To complete the proof, we have to show that Y 1 t, s ≥ X 1 t, s > 0 for all t ∈ s, ∞ T and all s ∈ t 1 , ∞ T .Clearly, for any fixed s ∈ t 1 , ∞ T and all t ∈ s, ∞ T , we have i If (A1) holds and the dynamic inequality where A i t : max{A i t , 0} for t ∈ t 0 , ∞ T and i ∈ 1, n N , has a positive solution on t 0 , ∞ T , then 3.1 also admits a positive solution on t 1 , ∞ T ⊂ t 0 , ∞ T .
ii If (A4) holds and there exist a sufficiently large then the first fundamental solution X 1 of 3.1 satisfies X 1 t, s > 0 for all t ∈ s, ∞ T and all s ∈ t 1 , ∞ T .
Proof.Consider the dynamic equation Theorem 3.1 implies that for this equation the assertions i and ii hold.Since for all i ∈ 1, n N , we have A i t ≤ A i t for all t ∈ t 0 , ∞ T , 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:

4.15
where y 1 , y 2 ∈ R are the initial values, ψ ∈ C rd t −1 , t 0 T , R is the initial function such that ψ has a finite left-sided limit at the initial point t 0 provided that it is left dense, g ∈ C rd t 0 , ∞ T , R is the forcing term.
Proof.By Theorem 3.1 and Remark 3.3, we can assume that Λ ∈ C rd t 0 , ∞ T , R 0 is a solution of the dynamic Riccati inequality 3.4 , then by A5 , the function Λ is also a solution of the dynamic Riccati inequality applying Lemma 2.4, and using A6 , we have for all t ∈ t 0 , ∞ T .This completes the proof.Remark 4.5.
The following example illustrates Theorem 4.4 for the quantum time scale T 2 Z .
Example 4.6.Let 2 Z : {2 k : k ∈ Z} ∪ {0}, and consider the following initial value problems: where Id 2 Z is the identity function on 2 Z , that is, Id 2 Z t t for t ∈ 2 Z , and

4.22
Denoting by x and y the solutions of 4.20 and 4.22 , respectively, we obtain y t ≥ x t for all t ∈ 1, ∞ 2 Z by Theorem 4.4.For the graph of the first 10 iterates, see Figure 1.
As an immediate consequence of Theorem 4.4, we obtain the following corollary.
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 A2 and assuming that the solution of 2.1 is positive.Corollary 4.8.Suppose that (A3), (A4), and the following condition hold: If x is a positive solution of 2.1 on t 0 , ∞ T with x 1 y 1 and y 2 ≥ x 2 , then for the solution y of 4.24 , one has y t ≥ x t for all t ∈ t 0 , ∞ T .
Proof.Corollary 4.3 and Remark 3.3 imply that the first fundamental solution X 1 associated with 2.1 and 4.24 satisfies X 1 t, s > 0 for all t ∈ s, ∞ T and all s ∈ t 0 , ∞ T .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: Assume further that the first-order dynamic Riccati inequality 3.4 has a solution Now, let us proceed with the discrete case.
Corollary 5.3.Let {A 0 k } be a positive sequence, for i ∈ 1, n N , let {A i k } be a nonnegative sequence, and let {α i k } be a divergent sequence such that Let us introduce the function A t, s : Theorem 5.4.Suppose that (A1), (A2), and (A3) hold, and for every t 1 ∈ t 0 , ∞ T , the dynamic equation is oscillatory, where t 2 ∈ t 1 , ∞ T satisfies α min t > t 1 for all t ∈ t 2 , ∞ T , then 3.1 is also oscillatory.
Proof.Assume to the contrary that 3.1 is nonoscillatory, then there exists a solution or simply by using 5.4 , Now, let By the quotient rule, 5.4 and 5.7 , we have proving that ψ is nonincreasing on t 1 , ∞ T .Therefore, for all i ∈ 1, n N , we obtain where t 2 ∈ t 1 , ∞ T satisfies α min t > t 1 for all t ∈ t 2 , ∞ T .Using 5.10 in 3.1 , we see that x 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 .
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 Λ ∈ C rd t 0 , ∞ T , R 0 .Using 3.5 and 5.7 , we see that which together with 3.4 implies that

5.14
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.
where A k, l : then 1.8 is oscillatory.

Existence of a Positive Solution
Theorem 6.1.Suppose that (A2), (A3), and (A4) hold, f ∈ C rd t 0 , ∞ T , R 0 , and the first-order dynamic Riccati inequality 3.4 has a solution Λ ∈ C 1 rd t 0 , ∞ T , R 0 .Moreover, suppose that there exist x 1 , x 2 ∈ R such that ϕ t ≤ x 1 for all t ∈ t −1 , t 0 T and x 2 ≥ Λ t 0 x 1 /A 0 t 0 , then 2.1 admits a positive solution x such that x t ≥ x 1 for all t ∈ t 0 , ∞ T .
Proof.First assume that y is the solution of the following initial value problem: then, by following similar arguments to those in the proof of the part ii ⇒ iii of Theorem 3.1, we obtain for all t ∈ t 0 , ∞ T .So z is a solution to

6.4
Theorem 4.4 implies that y t ≥ z t ≥ x 1 > 0 for all t ∈ t 0 , ∞ T .By the hypothesis of the theorem, Theorem 4.4, and Corollary 4.8, we have x t ≥ y t ≥ x 1 > 0 for all t ∈ t 0 , ∞ T .This completes the proof for the case f ≡ 0 and g ≡ 0 on t 0 , ∞ T .The general case where f / ≡ 0 on t 0 , ∞ T 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 N 0 : { √ k : k ∈ N 0 }, and consider the following delay dynamic equation: x Δ 1 2, x t ≡ 2 for t ∈ 0, 1 √ N 0 , 6.5 then 5.1 takes the form Φ t ≤ 1 for all t ∈ 1, ∞ √ N 0 , where the function Φ is defined by and is decreasing on 1, ∞ R and thus is not greater than Φ 1 ≈ 0.79, that is, Theorem 5.1 holds.Theorem 6.1 therefore ensures that the solution is positive on 1, ∞ √ N 0 .For the graph of 15 iterates, see Figure 2.

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 n 1 and α 1 t t for t ∈ t 0 , ∞ R reduces to Theorem A. Nonoscillation part of Kneser's result for 1.4 follows from Corollary 5.2 by letting n 1, A 0 t ≡ 1, and α 1 t t for t ∈ t 0 , ∞ R .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 Since this result is a necessary and sufficient condition, the conclusion of Theorem F could be deduced from where a ∈ R 0 , and proved that 7.7 is nonoscillatory if and only if a ≤ 1 √ q 1 2 .7.9 For the above q-difference equation, 7.6 reduces to the algebraic inequality 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 2/ √ q 1 2 , 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 Λ ∈ C 1 rd t 1 , ∞ T , R of the inequality 3.32 satisfying Λ/A 0 ∈ R t 1 , ∞ T , R is also a solution of 3.32 with B i instead of A i for i 0, 1. Proposition 7.2 see 28, 32, 36 .Suppose that B 0 t ≥ A 0 t > 0, A 1 t ≥ 0, and A 1 t ≥ B 1 t for all t ∈ t 0 , ∞ T , then 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 Λ ∈ C 1 rd t 1 , ∞ T , R 0 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 x of 3.1 is said to be slowly oscillating if for every t 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?η s f η t s f η d q η : q − 1 log q t/q η log q s f q η q η Table 2: The exponential function.
T R Z q Z , q > 1 e f t, s exp{ t s f η dη} t−1 η s 1 f η log q t/q η log q s 1 q − 1 q η f q η Let f ∈ R T, R , then the exponential function e f •, s on a time scale T is defined to be the unique solution of the initial value problem Table 2 illustrates the explicit forms of the exponential function on some well-known time scales.
The exponential function e f •, s is strictly positive on s, ∞ T if f ∈ R s, ∞ T , R , while e f •, s alternates in sign at right-scattered points of the interval s, ∞ T provided that f ∈ R − s, ∞ T , R .For h ∈ R 0 , let z, w ∈ C h , the circle plus ⊕ h and the circle minus h are defined by z ⊕ h w : z w hzw and z h w : z − w / 1 hw , respectively.Further throughout the paper, we will abbreviate the operations ⊕ μ and μ simply by ⊕ and , respectively.It is also known that R T, R is a subgroup of R T, R , that is, 0 ∈ R T, R , f, g ∈ R T, R implies f⊕ μ g ∈ R T, R and μ f ∈ R T, R , where μ f : 0 μ f on T.
The readers are referred to 32 for further interesting details in the time scale theory.

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 y t > x t for all t ∈ 1, ∞ 2 Z .

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

s 1 A. 4 for
x Δ t f t x t for t ∈ T κ , x some fixed s ∈ T. For h ∈ R , set C h : {z ∈ C : z / − 1/h}, Z h : {z ∈ C : −π/h < Im z ≤ π/h}, and C 0 : Z 0 : C. For h ∈ R 0 , we define the cylinder transformation ξ h : C h → Z h by ξ h z : z ∈ C h , then the exponential function can also be written in the form e f t, s : exp t s ξ μ η f η Δη for s, t ∈ T. A.6 al. 1 , Erbe et al. 2 , Győri and Ladas 3 , Ladde et al. 4 , Myškis 5 , Norkin 6 , 3.14 see 32, Theorem 2.77 .Now, for all t ∈ t 1 , ∞ T , we compute that By the Leibnitz rule see 32, Theorem 1.117 , for all t ∈ t 1 , ∞ T , we have R , and define the function x asx t t t 1 X 1 t, σ η g η Δη ∀t ∈ t 1 , ∞ T .4.3 ∞ T .This completes the proof since the first fundamental solution X 1 satisfies X 1 t, s > 0 for all t ∈ s, ∞ T and all s ∈ t 1 , ∞ T by Remark 3.3., ∞ T ⊂ t 0 , ∞ T , then 4.1 admits a nonoscillatory solution on t 2 , ∞ T ⊂ t 1 , ∞ T .
Let {A 0 k } be a positive sequence, for i ∈ 1, n N , let {A i k } be a nonnegative sequence and let {α i k } be a divergent sequence such that α