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.