Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations
© M. Ünal and A. Zafer. 2010
Received: 19 January 2010
Accepted: 20 March 2010
Published: 30 March 2010
New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative.
In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form
on an arbitrary time scale , where
the functions , , : are right-dense continuous with nondecreasing; the delay functions are nondecreasing right-dense continuous and satisfy for with as .
We assume that the time scale is unbounded above, that is, and define the time scale interval by . It is also assumed that the reader is already familiar with the time scale calculus. A comprehensive treatment of calculus on time scales can be found in [1–3].
By a solution of (1.1) we mean a nontrivial real valued function such that and for all with , and that satisfies (1.1). A function is called an oscillatory solution of (1.1) if is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if and only if every solution of (1.1) is oscillatory.
Notice that when , (1.1) is reduced to the second-order nonlinear delay differential equation
while when , it becomes a delay difference equation
Another useful time scale is and is a real number , which leads to the quantum calculus. In this case, (1.1) is the -difference equation
where , , and .
Interval oscillation criteria are more natural in view of the Sturm comparison theory since it is stated on an interval rather than on infinite rays and hence it is necessary to establish more interval oscillation criteria for equations on arbitrary time scales as in . As far as we know when , an interval oscillation criterion for forced second-order linear differential equations was first established by El-Sayed . In 2003, Sun  demonstrated nicely how the interval criteria method can be applied to delay differential equations of the form
where the potential and the forcing term may oscillate. Some of these interval oscillation criteria were recently extended to second-order dynamic equations in [6–10]. Further results on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations on time scales can be found in [11–23], and the references cited therein.
Therefore, motivated by Sun and Meng's paper , using similar techniques introduced in  by Kong and an arithmetic-geometric mean inequality, we give oscillation criteria for second-order nonlinear delay dynamic equations of the form (1.1). Examples are considered to illustrate the results.
2. Main Results
We need the following lemmas in proving our results. The first two lemmas can be found in [25, Lemma ].
The last important lemma that we need is a special case of the one given in . For completeness, we provide a proof.
which completes the proof.
In what follows we say that a function belongs to if and only if is right-dense continuous function on having continuous -partial derivatives on , with for all and for all . Note that in case , the -partial derivatives become the usual partial derivatives of . The partial derivatives for the cases and will be explicitly given later.
Denoting the -partial derivatives and of with respect to and by and , respectively, the theorems below extend the results obtained in  to nonlinear delay dynamic equation on arbitrary time scales and coincide with them when is replaced by . Indeed, if we set , then it follows that
When , they become
as in . However, we prefer using instead of for simplicity.
then (1.1) is oscillatory.
Suppose on the contrary that is a nonoscillatory solution of (1.1). First assume that and are positive for all for some . Choose sufficiently large so that . Let .
Similarly, by following the above calculation step by step, that is, multiplying both sides of (2.32) this time by after taking into account that
This contradiction completes the proof when is eventually positive. The proof when is eventually negative is analogous by repeating the above arguments on the interval instead of .
then (1.3) is oscillatory.
then (1.4) is oscillatory.
then (1.5) is oscillatory.
Notice that Theorem 2.6 does not apply if there is no forcing term, that is, . In this case we have the following theorem.
then (1.1) with is oscillatory.
where and , are as in Lemma 2.2.
then (1.3) with is oscillatory.
then (1.4) with is oscillatory.
then (1.5) with is oscillatory.
It is obvious that Theorem 2.6 is not applicable if the functions are nonpositive for . In this case the theorem below is valid.
then (1.1) is oscillatory.
Suppose that (1.1) has a nonoscillatory solution. Without losss of generality, we may assume that and are eventually positive on when is sufficiently large. If is eventually negative, one may repeat the same proof step by step on the interval
Rewriting (1.1) for as
The rest of the proof is the same as that of Theorem 2.6 and hence it is omitted.
then (1.3) is oscillatory.
then (1.4) is oscillatory.
then (1.5) is oscillatory.
In this section we give three examples when , and , in (1.1). That is, we consider
For simplicity we take , thus . Note that and by Lemma 2.2.
Let , , and , .
Since all conditions of Corollary 2.11 are satisfied, we conclude that (3.2) is oscillatory when (3.4) holds.
Let , , and . We derive
Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.5) is oscillatory if (3.7) holds.
Let , , and . We have
We see that (2.67) holds for all and . Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.8) is oscillatory if and are positive.
The paper is supported in part by the Scientific and Research Council of Turkey (TUBITAK) under Contract 108T688. The authors would like to thank the referees for their valuable comments and suggestions.
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