Abstract

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form , on a time scale . We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was originally introduced by Hilger in his Ph.D. thesis [1], in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Many authors have expounded on various aspects of this new theory; see [2–4]. A time scale is a nonempty closed subset of the real numbers. If the time scale equals the real numbers or integer numbers, the obtained results represent the classical theories of the differential and difference equations. Many other interesting time scales exist and give rise to many applications. The new theory of the so-called “dynamic equation” not only unifies the theories of differential equations and difference equations, but also extends these classical cases to the so-called -difference equations (when for or ) which have important applications in quantum theory (see [5]). Also it can be applied on different types of time scales like and the space of the harmonic numbers .

In recent years, there have been many research activities concerning the oscillation of solutions of various forced second-order dynamic equations on time scales; we refer the reader to the articles [6–13] and the references cited therein.

In this paper, we are concerned with the interval oscillation of the second-order nonlinear dynamic equation:on a time scale , subject to the following conditions:(H1) is an unbounded above time scale, and with . We define the time scale interval by .(H2) such that(H3), , and are rd-continuous functions.(H4) is assumed to satisfy , for and for some and .(H5), , and , for all , and there exist positive constants such that(1), ,(2) for all ,(3) for all (H6) with and

By a solution of (1), we mean that a nontrivial real valued function satisfies (1) for . A solution of (1) is said to be oscillatory if it has arbitrarily a large number of zeros; that is, there exists a sequence of zeros such that and . Otherwise, is said to be nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory. In this work, we study the solutions of (1) which are not identically vanishing eventually.

In order to prove our results, we use the following Hardy et al. inequality [14].

Lemma 1 (Hardy et al. inequality [14]). If and are nonnegative, then

2. Main Results

In the following theorems, we apply Riccati techniques to establish some sufficient conditions for oscillation of (1) on a sequence of subintervals of the interval . Also, we do not require that , , and be of definite sign.

Theorem 2. Assume that hold, and suppose that for any there exist points in for such that and for and Further assume that there exist a function such that for , , on , and a positive delta differentiable function such thatwhere for and Then every solution of (1) is oscillatory.

Proof. Assume that (1) is nonoscillatory on . Then there is a solution of (1) and a point such that and are of the same sign on . Consider the cases and that are positive on . We use Riccati substitution:Then, from (1) and using , we haveSince such that , for and for , then, from (1), we have for . Hence is nonincreasing for . We claim that on . If not, then there is such that using , we have integrating from to , we get which implies that is eventually negative. This is a contradiction. Hence on .
Therefore, for , we haveNow, we use the fact that is nonincreasing for and . Then, for , we have which implies thatOn the other hand, for , we have which implies thatusing (15) and (17), we get Therefore,To see that (19) holds for , we note that if , then and clearly (19) holds. If , then and thus (19) holds for . Hence (19) holds for .
From (8), (13), and (19), we get Using , we get OrSince for all and on the interval , then from (22) we have for Now, we consider the following two cases: Case : ; Case : .
Case  1 (). Set , , and By Lemma 1, we get thatfor . From (23) and (25), we haveCase  2 (). When , this implies that (25) and (26) hold for .
Multiplying (26) by and integrating from to , we have Using integration by parts, we get Using the fact that , we obtain This implies that which is a contradiction of (5). The case , on is similar (in this case, we use on to get a similar contradiction). Therefore, any solution of (1) is oscillatory. This completes the proof.

If and , then (1) reduces to equationUsing Theorem 2 and choosing , we have the following corollary.

Corollary 3. Assume that and hold, and suppose that for any there exist points in for such that for and Further assume that there exist a function such that for , , on , and a positive delta differentiable function such thatwhere for and , Then every solution of (32) is oscillatory.

Proof. Assume that (32) is nonoscillatory on . Then there is a solution of (32) and a such that and have the same sign on . Consider that the cases and are positive on . As in the proof of Theorem 2, for and , inequality (26) holds for all eventually positive solutions of (32), where , , and . Thus, we get Multiplying (36) by and integrating from to , we have Using integration by parts, we get Using the fact that , we obtain This implies that which is a contradiction of (34). The case ,   on is similar (in this case, we use on to reach a similar contradiction). Therefore, any solution of (32) is oscillatory. This completes the proof.

Now, we assume that Then, (1) reduces toTherefore, we have the following theorem.

Theorem 4. Assume that hold, and suppose that for any there exist points in for such that and for and Further assume that there exist a function such that for , , on , and a positive delta differentiable function such thatwhere for and : Then every solution of (42) is oscillatory.

Proof. Assume that (42) is nonoscillatory on . Then there is a solution of (42) and a such that and have the same sign on . Consider that the cases and are positive on . Let be defined by (7). ThenUsing (13) and (19) and applying and , we get OrSince for all and on the interval , then from (48) we have for From (25) and (49), we have for Multiplying (50) by and integrating from to , we have Using integration by parts, we get From the fact that , we obtain This implies that which is a contradiction with (44). The case , on is similar (in this case, we use on and ).

In the following, we assume that is an rd-continuous function (i.e., ) and employ the generalized Riccati technique to establish new oscillation criteria for (1).

Theorem 5. Assume that and hold, and suppose that for any there exist points in for such that and for and Further assume that there is a function such that is a delta differentiable, a positive delta differentiable, function and there exist a function such that for , , on , ,where for and Then every solution of (1) is oscillatory.

Proof. Assume that (1) is nonoscillatory on . Then there is a solution of (1) and a such that and have the same sign on . Consider the case where and are positive on . Define the function by the generalized Riccati substitution: Then, from (1) and using , we have Using (19), we getTherefore,Since for all and on the interval , then from (61) we have for From (25) and (62), we haveFrom the definition of , we see thatAs in the proof of (19) in Theorem 2, we haveUsing (64) and (65) in (63), we get Hence, Multiplying (67) by and integrating from to , we have Using integration by parts, we get From the fact that , we obtainThis implies that which is a contradiction with (56). The case , on is similar (in this case, we use on and ).

Remark 6. From Theorem 5, we can establish different sufficient conditions for the oscillation of (1) by using different choices of and .

3. Examples

In this section, we give some examples to illustrate our results.

Example 1. Let and consider the following nonlinear forced delay differential equation:whereEquation (72) is of the form (1) where Applying Theorem 2 and for any , we choose sufficiently large so that . Let Note that , on If we take and , then , on , .
It is easy to verify thatwhere . Hence ; that is, (5) holds for . Similarly, for and , we can show that (5) holds for . Therefore, by Theorem 2, we get that (72) is oscillatory.

Remark 7. The results of [15] cannot be applied to (72) for and . But, according to Theorem 2, when , this equation is oscillatory.

Example 2. Let and consider the following nonlinear forced delay differential equation:where Equation (78) is of the form (1) where Applying Theorem 5 and for any , we choose sufficiently large so that . Let Note that , on If we take , , and , then ,   on ,  .
It is easy to verify that where . Hence (56) holds for . Similarly, for and , we can show that (56) holds for . Therefore, by Theorem 5, we get that (78) is oscillatory.

Remark 8. The results obtained in the above examples cannot be obtained by the results in either [9] or [10].

Competing Interests

The authors declare that they have no competing interests.