Research Article | Open Access
Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales
We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale , , where , , and are positive right dense continuous (rd-continuous) functions on . Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.
The purpose of this paper is to give several oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale subject to the following hypotheses. is a time scale which is unbounded above and with . The time scale interval is defined by ., , and are positive right dense continuous functions on such that and or Consider , for all and there exists a positive constant such that .Consider , for all and for any fixed and there exist positive constants such that is a strictly increasing and differentiable function such that
By a solution of (1), we mean that a nontrivial real valued function satisfies (1) for . A solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called 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.
Many results have been obtained on the oscillation and nonoscillation of dynamic equations on time scales (see e.g., the papers [1–17], the books [18, 19] and the references cited therein). It is easy to see that (1) can be transformed into the equation where . If , , and , then (6) is simplified to the equation Also, if , (6) is simplified to the equation If , (8) takes the form If , (9) becomes In 2002, Guseinov and Kaymakçalan  studied (7) and established some sufficient conditions for nonoscillation. They proved that if then (7) is nonoscillatory. In 2005, Agarwal et al.  studied the linear delay dynamic equation (10) and Şahiner  considered the nonlinear delay dynamic equation (9) and gave some sufficient conditions for oscillation of (10) and (9). In 2007, Erbe et al.  considered the general nonlinear delay dynamic equations (8). They obtained some oscillation criteria which improve the results given by Şahiner . In 2011, Zhang and Gao  considered the oscillation of solutions of second-order nonlinear delay dynamic equation (6) with damping and establishing some new results. In this paper, we use the generalized Riccati transformation and the inequality technique to obtain some new oscillation criteria for (1). Our results generalize and improve the results in .
This paper is organized as follows. In Section 2, we present some preliminaries on time scales. In Section 3, we give some lemmas that we need through our work. In Section 4, we establish some new sufficient conditions for oscillation of (1). Finally, in Section 5, we present some examples to illustrate our results.
2. Some Preliminaries on Time Scales
A time scale is an arbitrary nonempty closed subset of the real numbers . On any time scale , we define the forward and backward jump operators by A point , inf is said to be left dense if , right dense if sup and , left-scattered if , and right-scattered if . The graininess function for a time scale is defined by . The set is derived from the time scale as if has a left-scattered maximum . Otherwise, .
A function is called rd-continuous provided that it is continuous at right dense points of and its left-sided limits exist at left dense points of . The set of rd-continuous functions is denoted by . By , we mean the set of functions whose delta derivative belong to .
A function is regressive provided that holds. The set of all regressive and rd-continuous functions is denoted by If , then we define the exponential function by where the cylinder function is defined by For a function (the range of may be actually replaced by any Banach space), the delta derivative is defined by provided is continuous at and is right-scattered. If is not right-scattered, then the delta derivative is defined by provided this limit exists.
A function is said to be differentiable if its derivative exists. The derivative and the shift of a function are related by the equation The delta derivative rules of the product and the quotient (where of two differentiable functions and are given by An integration by parts formula reads or and the infinite integral is defined by Throughout this paper, we use where , and positive constants where , , and are defined in , .
3. Several Lemmas
In this section, we present some lemmas that we need to prove our results in the next section.
Lemma 1. (Bohner and Peterson [18, Chapter 2]). If is rd-continuous such that for all , then the initial value problem , has a unique and positive solution on , denoted by .
Lemma 2. (Bohner [5, Lemma 2]). For nonnegative with , one has the inequality
Proof. Since is a positive solution of (1) on , we have
We claim that on . If not, then there is such that
By , we get
Integrating from to , we get
This implies that is eventually negative which is a contradiction. Hence, on . Therefore,
Using the fact that is strictly decreasing, we get
4. Main Results
Here, we establish some new sufficient conditions for oscillation of (1).
Proof. Assume that (1) has a nonoscillatory solution on . Also, assume that , for all , . Consider the generalized Riccati substitution
Using the delta derivative rules of product and quotient of two functions, we have
From the definition of , we have
Using the fact and , we get
Integrating the inequality from to , using the definition of and , we get
Now, substituting (41) in (40), we have
where and .
Therefore, Hence, Integrating the above inequality from to , we obtain and taking the limit supremum as , we obtain a contradiction to condition (36). Therefore, every solution of (1) is oscillatory on .
Theorem 5. Assume that and (2) hold. Let be an rd-continuous function defined as follows: such that and has a nonpositive continuous -partial derivative . If there exists a positive -differentiable function such that then every solution of (1) is oscillatory on .
Proof. Assume that (1) has a nonoscillatory solution on . Also, assume that , , for all ,. We proceed as in the proof of Theorem 4 to get (45) Multiplying the above inequality by , integrating from to and using (48), we get Thus, which is a contradiction to (49). This completes the proof.
Now, If is a function defined by then, we have the following result.
Theorem 7. Assume that and (2) hold. Let be an rd-continuous function defined as such that and has a nonpositive continuous -partial derivative . Let be an rd-continuous function satisfying If there exists a positive nondecreasing -differentiable function such that where , then every solution of (1) is oscillatory on .
Proof. Assume that (1) has a nonoscillatory solution on . Also, assume that , , for all , and there is such that satisfies the conclusion of Lemma 3 on . Proceeding as in the proof of Theorem 4, we get (43) which has the form
where and .
Multiplying the above inequality by , integrating from to , and using (56) and (57), we get Using and Lemma 3, we get Therefore, Thus which is a contradiction to (58). This completes the proof.
Proof. Assume that (1) has a nonoscillatory solution such that , for all . As in the proof of Lemma 3, we see that there exist two possible cases for the sign of . When is eventually positive, the proof is similar to the proof of Theorem 4. Next, suppose that for . Then, is decreasing and . Thus, Hence, by , we get Defining the function , using (1) and (66), we get The inequality (67) is the assumed inequality of [18, Theorem 6.1]. All other assumptions of [18, Theorem 6.1]; for example, , are satisfied. Hence, the conclusion of [18, Theorem 6.1] holds; that is, for all , and thus, for all , Assuming and using (64) in (69) yield , which is a contradiction to the fact that for . Thus, and then . This completes the proof.
Remark 9. Our results in this paper not only extend and improve some known results and show that some results of [2, 8–10, 12, 14] are special examples of our results but also unify the study of oscillation of second-order nonlinear delay differential equation with damping and second-order nonlinear delay difference equation with damping.
Example 1. Consider the second-order delay 2-difference equation with damping Here, The conditions , are clearly satisfied, and hold with , , and is satisfied as By Lemma 2, we get for , so that Hence, (2) is satisfied. Then, Therefore, and then we can find such that If , then Hence, according to Theorem 4, every solution of (70) is oscillatory on .
Example 2. Consider the second-order nonlinear delay dynamic equation with damping
where and .
Here, The conditions , are clearly satisfied, and hold with , and is satisfied as