Abstract and Applied Analysis

Volume 2012 (2012), Article ID 253107, 18 pages

doi:10.1155/2012/253107

Research Article

Kamenev-Type Oscillation Criteria for the Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

M. Tamer Şenel

Department of Mathematics, Faculty of Sciences, Erciyes University, 38039 Kayseri, Turkey

Received 6 March 2012; Accepted 22 March 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 M. Tamer Şenel. 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.

The oscillation of solutions of the second-order nonlinear dynamic equation , with damping on an arbitrary time scale , is investigated. The generalized Riccati transformation is applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic equation. Several new sufficient conditions for oscillatory solutions of this equation are obtained.

1. Introduction

Much recent attention has been given to dynamic equations on time scales, or measure chains, and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatment of the subject. Since then, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus.

A time scale is an arbitrary nonempty closed subset of the real numbers . The forward and the backward jump operators on any time scale are defined by . A point , , is said to be left-dense if , right dense if and , left scattered if , and right scattered if . The graininess function for a time scale is defined by . For a function the (delta) derivative is defined by

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

provided this limit exists. A function is said to be right-dense continuous if it is right continuous at each right-dense point and there exists a finite left limit at all left-dense points, and is said to be differentiable if its derivative exists. A useful formula dealing with the time scale is that

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

The integration by parts formula is

The function is called -continuous if it is continuous at the right-dense points and if the left-sided limits exist in left-dense points. We denote the set of all which are -continuous and regressive by . If , then we can define the exponential function by

for , where is the cylinder transformation, which is defined by

Alternately, for one can define the exponential function , to be the unique solution of the IVP with .

The various-type oscillation and nonoscillation criteria for solutions of ordinary and partial differential equations have been studied extensively in a large cycle of works (see [4–31]).

In [27], the authors have considered second-order nonlinear neutral dynamic equation

on a time scale . They have assumed that is a quotient of odd positive integers, and positive constants such that the delay functions satisfy for all , and real-valued positive functions defined on and also they have supposed that

(H1),(H2) is continuous such that for all and there exists a nonnegative function defined on such that

and were concerned with oscillation properties of (1.8). In [28], Saker has considered second-order nonlinear neutral delay dynamic equation

when is an odd positive integer with and real-valued positive functions defined on . The author also has improved some well-known oscillation results for second-order neutral delay difference equations. Agarwal et al. [29] have considered the second-order perturbed dynamic equation

where is odd and they have interested in asymptotic behavior of solutions of (1.10). Saker et al. [30] have studied the second-order damped dynamic equation with damping

when , and are positive real-valued -continuous functions and they have proved that if and , then every solution of (1.11) is oscillatory.

In the present paper, we consider the second order nonlinear dynamic equation

where , are real-valued, nonnegative, and right-dense continuous function on a time scale , with and is a quotient of odd positive integers. We assume that is a nondecreasing function and such that , for and . The function is assumed to satisfy , for and there exists a positive -continuous function defined on such that . Throughout this paper we assume that

Since we are interested in the oscillatory of solutions near infinity, we assume that and define the time scale interval by . The oscillation of solutions of the second-order nonlinear dynamic equation (1.12) with damping on an arbitrary time scale is investigated. The generalized Riccati transformation is applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic differential equation. Several new sufficient conditions for oscillatory solutions of this equation are obtained.

A solution of (1.12) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.

2. Preliminary Results

Lemma 2.1. Assume that the condition () is satisfied and (1.12) has a positive solution on . Then there exists a sufficiently large such that

Proof. Let such that on . Since is positive nonoscillatory solution of (1.12) we can assume that for all large . Then without loss of generality we take for all . From (1.12) it follows that

and so

Define . Hence

and it implies that

Then

and therefore

Next an integration for and by () gives

which is a contradiction. Hence is not negative for all large and so for all . This completes the proof of Lemma 2.1.

We now define

Lemma 2.2. Assume that () holds and (1.12) has a positive solution on . Then there exists a sufficiently large such that if for one has

Whereas, if , one has

Proof. As in the proof of Lemma 2.1, there is a sufficiently large such that

From (1.12) and (2.12) it follows that

and so

Then

Next, when , we get

Finally, since is decreasing on for , we get

and we obtain

3. Main Results

Theorem 3.1. Assume that () holds and there exist a function such that is a -differentiable function and a positive real -functions -differentiable function such that

where

Then every solution of (1.12) is oscillatory.

Proof. Suppose to the contrary that is a nonoscillatory solution of (1.12). Without loss of generality, there is a , sufficiently large, so that satisfies the conclusions of Lemmas 2.1 and 2.2 on . Define the function by Riccati substitution

Then satisfies

From (1.12) and the definition of for it follows that

Using the fact that and is a increasing function, we obtain

Now we consider the following two cases: and .

In the first case . Using the Pötzsche chain rule (see, [3]), we obtain

Using (3.7) in (3.6) for , we get

By Lemmas 2.1 and 2.2, for , we have that

In the view of (3.8), and (3.9) we get

In the second case . Applying the Pötzsche chain rule (see, [3]), we obtain

In the view of (3.11), (3.6) yields

By Lemmas 2.1 and 2.2, we have that

By (3.13), (3.12), and then using the definition of , we get

Using (3.10), (3.14), and the definitions of , and for , we get

Then, we can write

and so, we get

Integrating (3.17) with respect to from to , we get

and this implies that

which contradicts to assumption (3.1). This completes the proof of Theorem 3.1.

Corollary 3.2. Assume that () holds. If

then every solution of (1.12) is oscillatory.

Example 3.3. Consider the nonlinear dynamic equation

where is the quotient of the odd positive integers. We have that and . If , then and . So we get . It is clear that () holds. Indeed,

and then

and so we can find such that for . Then we can see from Corollary 3.2 that it follows that

and therefore every solution of (3.21) is oscillatory.

Now, let us introduce the class of functions .

Let and . The function has the following properties:

and has a continuous -partial derivative on with respect to the second variable. ( is -continuous function if is -continuous function in and .)

Theorem 3.4. Assume that the conditions of Lemma 2.1 are satisfied. Furthermore, suppose that there exist functions such that (3.25) holds and there exist a function with a -differentiable function and a positive -differentiable function such that

where . Then every solution of (1.12) is oscillatory on .

Proof. Assume that (1.12) has a nonoscillatory solution on . Then without loss of generality, there is a sufficiently large such that satisfies the conclusions of Lemmas 2.1 and 2.2 on . Consider the generalized Riccati substitution

We proceed as Theorem 3.1 and from (3.15) it follows that

Multiplying both sides of (3.28) by and integrating with respect to from to (), we obtain

Integrating by parts, we get

It is easy to see that

where

Then we can write

Hence

which contradicts with assumption (3.26). This completes the proof of Theorem 3.4.

Corollary 3.5. Assume that () holds. Furthermore, suppose that there exist functions , and such that (3.25) holds and there exist a function such that is a -differentiable function and a positive -differentiable function such that

where is as defined in Theorem 3.1 and . Then every solution of (1.12) is oscillatory on .

Theorem 3.6. Assume that () holds and there exists a -differentiable positive function such that

where

Then every solution of (1.12) is oscillatory.

Proof. Suppose that (1.12) has a nonoscillatory solution on . Then without loss of generality, there is a sufficiently large such that satisfies the conclusions of Lemmas 2.1 and 2.2 on . Consider the generalized Riccati substitution

From (3.6) it follows that

In the same manner as in the proof of Theorem 3.1, we get

If , then we have that

whereas, if , we have that

Using the fact that is increasing and is decreasing on , we get

Using (3.41), (3.42), and (3.43), we obtain

where . Define and by

Then using the inequality (see [32])

we obtain

From this last inequality and (3.44) it follows that

which contradicts with the assumption (3.36). Theorem 3.6 is proved.

Example 3.7. Consider the second-order equation

where . Then it follows that

for , and so

Hence () is satisfied. Now let for . Then

and so (3.36) is satisfied as well. Hence by Theorem 3.6, we have that (3.49) is oscillatory.

Theorem 3.8. Assume that the conditions of Lemma 2.1 hold. Furthermore, suppose that there exist functions such that (3.25) holds and there exists a positive real -functions -differentiable function such that

where and . Then every solution of (1.12) is oscillatory on .

Proof . Assume that (1.12) has a nonoscillatory solution on . Then without loss of generality, there is a sufficiently large such that satisfies the conclusions of Lemmas 2.1 and 2.2 on . Consider the generalized Riccati substitution

By Theorem 3.6 and inequality (3.44)

where . Multiplying both sides of (3.55) with and integrating with respect to from to (), we get

Integrating by parts and using (3.25), we obtain

Define and by

Using the inequality (see [32])

we get

From this last inequality and (3.55) it follows that

which contradicts with the assumption (3.53). This completes the proof of Theorem 3.8.

Corollary 3.9. Assume that all conditions of Lemma 2.1 hold. Furthermore, suppose that there exist functions , and such that (3.25) holds and there exists a positive -differentiable function such that

where . Then every solution of (1.12) is oscillatory on .

Example 3.10. Consider the second-order dynamic equation

where , , . It is easy to check that () holds. For and , it immediately follows that

and so . Hence,

Therefore by Corollary 3.9, every solution of (3.63) is oscillatory.

Acknowledgments

The author would like to thank Professor A. Ashyralyev and anonymous referee for their helpful suggestions to the improvement of this paper. This work was supported by Research Fund of the Erciyes University Project no. FBA-11-3391.

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