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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 253107, 18 pages
http://dx.doi.org/10.1155/2012/253107
Research Article

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

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 [431]).

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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