1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [1]). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale calculus; we refer also to the last book by Bohner and Peterson [4] for advances in dynamic equations on time scales. For the notions used below, we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson [3].

A time scale is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Not only does the new theory of the so-called dynamic equations unify the theories of differential equations and difference equations, but also it extends these classical cases to cases in between, for example, to the so-called difference equations when which has important applications in quantum theory and can be applied on different types of time scales like , and the space of the harmonic numbers.

Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models which are discrete in season (and may follow a difference scheme with variable step-size or often modeled by continuous dynamic systems), die out, say in winter, while their eggs are incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping population (see Bohner and Peterson [3]).

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to Akin-Bohner and Hoffacker [5, 6], Akin-Bohner et al. [7], Bohner and Saker [8], Erbe [9], Erbe et al. [10], Li et al. [11], and Saker [12, 13]. However, there are few results dealing with the oscillation of the solutions of delay dynamic equations on time scales [14–29].

Following this trend, in this paper, we consider a second-order nonlinear delay differential equation

(1.1)

For oscillation of the second-order delay dynamic equations, Agarwal et al. [14] considered the second-order delay dynamic equations on time scales

(1.2)

and established some sufficient conditions for oscillation of (1.2).

Zhang and Shanliang [29] studied the second-order nonlinear delay dynamic equations on time scales

(1.3)

and the second-order nonlinear dynamic equations on time scales

(1.4)

where and is continuous and nondecreasing , and for , and established the equivalence of the oscillation of (1.3) and (1.4). However, the results established in [29] are valid only when the graininess function is bounded which is a restrictive condition. Also the restriction is required.

Şahiner [23] considered the second-order nonlinear delay dynamic equations on time scales

(1.5)

where is continuous, for and , and and obtained some sufficient conditions for oscillation of (1.5).

Han et al. [17] investigated the second-order Emden-Fowler delay dynamic equations on time scales

(1.6)

established some sufficient conditions for oscillation of (1.6), and extended the results given in [14].

Erbe et al. [28] considered the general nonlinear delay dynamic equations on time scales

(1.7)

where and are positive, real-valued rd-continuous functions defined on , is rd-continuous, and as , and satisfies for some positive constant , for all nonzero , and extended the generalized Riccati transformation techniques in the time scales setting to obtain some new oscillation criteria which improve the results given by Şahiner [23] and Zhang and Shanliang [29].

Clearly, (1.2), (1.3), (1.5), (1.6), and (1.7) are different from (1.1). To develop the qualitative theory of delay dynamic equations on time scales, in this paper, we consider the second-order nonlinear delay dynamic equation on time scales (1.1).

As we are interested in oscillatory behavior, we assume throughout this paper that the given time scale is unbounded above, that is, it is a time scale interval of the form with .

We assume that is a quotient of odd positive integer, and are positive, real-valued rd-continuous functions defined on and , is a rd-continuous function such that and .

We shall also consider the two cases

(1.8)
(1.9)

By a solution of (1.1), we mean a nontrivial real-valued function satisfying (1.1) for . A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all solutions are oscillatory. Our attention is restricted to those solutions of (1.1) which exist on some half line with for any .

We note that if , then , and (1.1) becomes the second-order Emden-Fowler delay differential equation

(1.10)

If , then and (1.1) becomes the second-order Emden-Fowler delay difference equation

(1.11)

In the case of , (1.1) is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler superlinear dynamic equations, and if , then (1.1) is the prototype of dynamic equations called Emden-Fowler sublinear dynamic equations. It is interesting to study (1.1) because the continuous version, that is, (1.10), has several physical applications—see, for example, [1] —and when is a discrete variable as in (1.11), (1.1) also has important applications.

Numerous oscillation and nonoscillation criteria have been established for equations as (1.10) and (1.11); see, for example, [1, 30–36] and references therein.

In this paper, we intend to use the Riccati transformation technique for obtaining several oscillation criteria for (1.1). Our results in this paper not only extend the results given in Agarwal et al. [14] and Han et al. [17] but also unify the oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation. Applications to equations to which previously known criteria for oscillation are not applicable are given.

This paper is organized as follows: in Section 2, we present the basic definitions and the theory of calculus on time scales. In Section 3, we apply a simple consequence of Kellers chain rule, devoted to the proof of the sufficient conditions for oscillation of all solutions of (1.1). In Section 4, some applications and examples are considered to illustrate the main results.

2. Some Preliminaries

A time scale is an arbitrary nonempty closed subset of the real numbers . Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above, that is, it is a time scale interval of the form . On any time scale, we define the forward and backward jump operators by

(2.1)

A point is said to be left-dense if , right-dense if , left-scattered if , and right-scattered if . The graininess of the time scale is defined by .

For a function (the range of may actually be replaced by any Banach space), the (delta) derivative is defined by

(2.2)

if is continuous at and is right-scattered. If is not right-scattered, then the derivative is defined by

(2.3)

provided this limit exists.

A function is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit in all left-dense points. The set of rd-continuous functions is denoted by .

is said to be differentiable if its derivative exists. The set of functions that are differentiable and whose derivative is rd-continuous function is denoted by .

The derivative and the shift operator are related by the formula

(2.4)

Let be a real-valued function defined on an interval . We say that is increasing, decreasing, nondecreasing, and nonincreasing on if and imply , , , and , respectively. Let be a differentiable function on . Then is increasing, decreasing, nondecreasing, and nonincreasing on if , , , and for all , respectively.

We will make use of the following product and quotient rules for the derivative of the product and the quotient of two differentiable functions and

(2.5)
(2.6)

For and a differentiable function , the Cauchy integral of is defined by

(2.7)

The integration by parts formula reads

(2.8)

and infinite integrals are defined as

(2.9)

In case , we have

(2.10)

And in case , we have

(2.11)

3. Main Results

In this section, we give some new oscillation criteria for (1.1). Since we are interested in oscillatory behavior, we will suppose that the time scale under consideration is not bounded above, that is, it is a time scale interval of the form . In order to prove our main results, we will use the formula

(3.1)

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [3, Theorem ]). Also, we need the following auxiliary result.

Lemma 3.1 (Şahiner [23, Lemma ]).

Suppose that the following conditions hold:

  • (H1) , where for some ;

  • (H2) , and for .

Then, for each , there exists a constant , , such that

(3.2)

Lemma 3.2.

Assume that (1.8) holds. Furthermore, assume that , , and is an eventually positive solution of (1.1). Then, there exists a such that

(3.3)

Proof.

Suppose is an eventually positive solution of (1.1) such that and for all . In view of (1.1), we have

(3.4)

and so is an eventually decreasing function. We first show that is eventually positive. Indeed, the decreasing function is either eventually positive or eventually negative. Suppose that there exists an integer such that , then from (3.4) we have for , hence

(3.5)

which implies by (1.8) that

(3.6)

and this contradicts the fact that for all . Hence is eventually positive. So is eventually positive. Then is eventually increasing.

By (2.5), we get

(3.7)

From (3.4), (3.7), and , we can easily verify that is eventually negative. Therefore, we see that there is some such that (3.3) holds. The proof is complete.

Lemma 3.3.

Let and be differentiable on time scale with for all . Then we have

(3.8)

The proof is similar to that of Akin-Bohner et al. [7, Lemma ].

Theorem 3.4.

Assume that (1.8) holds, , and , . Define

(3.9)

If

(3.10)

then (1.1) is oscillatory on .

Proof.

Suppose that (1.1) has a nonoscillatory solution . We may assume without loss of generality that and for all . So by Lemma 3.2, (3.3) holds. Define the function by

(3.11)

Then, and using (2.5) we get

(3.12)

By Lemma 3.1, for each , there exists a constant , such that

(3.13)

note that , (3.1) and (3.3) imply

(3.14)

So,

(3.15)

Upon integration, we arrive at

(3.16)

which contradicts (4.3). The proof is complete.

Remark 3.5.

Theorem 3.4 includes results of Akin-Bohner et al. [7, Theorem ].

Putting that is, in Theorem 3.4, we obtain the following corollary.

Corollary 3.6.

Assume , . If

(3.17)

then (1.1) is oscillatory on .

Theorem 3.7.

Assume that (1.8) holds, , and , . Furthermore, assume that there exists a function such that

(3.18)

holds for all constants . Then (1.1) is oscillatory on .

Proof.

Suppose that (1.1) has a nonoscillatory solution . We may assume without loss of generality that and for all . So by Lemma 3.2, (3.3) holds. Define the function by

(3.19)

Then, using (2.5) we get

(3.20)

So, from (1.1) and Lemma 3.3, we have

(3.21)

By Lemma 3.1, for each , there exists a constant , such that

(3.22)

note that (3.1) and (3.3) imply

(3.23)

then

(3.24)

where if . If , we chose .

Therefore,

(3.25)

which contradicts (3.18). The proof is complete.

Remark 3.8.

Theorem 3.7 not only includes results of Agarwal et al. [14, Theorem ], Akin-Bohner et al. [7, Theorem ], and Han et al. [17, Theorem ], but also improves conditions of Agarwal et al. [14, Theorem ] and Han et al. [17, Theorem ].

From Theorem 3.7, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of .

For example, let . Now Theorem 3.4 yields the following results.

Corollary 3.9.

Assume that (1.8) holds, , and , . If

(3.26)

holds for all constants , then (1.1) is oscillatory on .

Sometimes the following criterion is easier to check than the one given in Corollary 3.6, but it follows easily from Corollary 3.6 as we always have for all .

Corollary 3.10.

Assume that (1.8) holds, , and , . If

(3.27)

holds for all constants , then (1.1) is oscillatory on .

Now, using Lemma 3.2, we can give some sufficient conditions when (1.9) holds, which guarantee that every solution of (1.1) oscillates or converges to zero in .

Theorem 3.11.

Assume that (1.9) holds, , and , and . Furthermore, assume that there exists a positive function such that (3.18) holds. If

(3.28)

then every solution of (1.1) is oscillatory or converges to zero on .

Theorem 3.12.

Assume that (1.8) holds, , and , . Furthermore, assume that there exists a function such that

(3.29)

holds for all constants . Then (1.1) is oscillatory on .

Proof.

We assume that (1.1) has a nonoscillatory solution such that and , for all . By Lemma 3.2, we obtain (3.3). We calculate

(3.30)

where and . By putting and , we find that

(3.31)

Now note that , (3.1) and (3.31) imply

(3.32)

So,

(3.33)

where we put if . If , we chose . Note . Now define the function by (3.19). Therefore, using (3.21), Lemma 3.1, for each , there exists a constant , such that

(3.34)

noting that (3.3) and (3.33), we obtain

(3.35)

where if . If , we chose . Upon integration, we arrive at

(3.36)

which contradicts (3.29). The proof is complete.

Remark 3.13.

Theorem 3.12 includes results of Akin-Bohner et al. [7, Theorem ] and Han et al. [17, Theorem ] and improves conditions of Han [17, Theorem ].

From Theorem 3.12, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of .

For example, let . Now Theorem 3.12 yields the following results.

Corollary 3.14.

Assume that (1.8) holds, , and , . If

(3.37)

holds for all constants , then (1.1) is oscillatory on .

Sometimes the following criterion is easier to check than the one given in Corollary 3.14, but it follows easily from Corollary 3.14 as we always have for all .

Corollary 3.15.

Assume that (1.8) holds, , and , . If

(3.38)

holds for all constants , then (1.1) is oscillatory on .

4. Applications

In this section, we give one example to illustrate our main results. To obtain the conditions for oscillation, we will use the following fact:

(4.1)

For more details, we refer the reader to [4, Theorem ].

Akin-Bohner et al. [7] considered the second-order dynamic equations on time scales

(4.2)

where is a quotient of odd positive integer, and are positive, real-valued rd-continuous functions defined on , and established some new oscillation criteria of (4.2).

Theorem 4.1 (Akin-Bohner et al. [7, Theorem ]).

Assume that (1.8) holds. If

(4.3)

then (4.2) is oscillatory on .

We note that (1.1) becomes (4.2) when , and Theorem 3.4 becomes Theorem 4.1, so Theorem 3.4 essentially includes results of Akin-Bohner et al. [7, Theorem ].

Example 4.2.

Consider the second-order delay dynamic equations on time scales

(4.4)

where . By Corollary 3.10, we have

(4.5)

Let , then , pick , so for all . Therefore,

(4.6)

for all constants . Then (4.4) is oscillatory on .