On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations
We investigate the oscillation of the following higher-order functional differential equation: where and are continuous functions on , and are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case and the case under the assumption that the potentials and change signs on .
Consider the following th-order forced functional differential equation of the form: where is an integer, , , , and are constants.
We are here only concerned with the nonconstant solutions of (1.1) that are defined for all large . The oscillatory behavior is considered in the usual sense, that is, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory.
The oscillatory behavior of (1.1) with has been studied by many authors. In early papers [1, 2], by assuming that , where is an oscillatory function satisfying , the author proved that the forced equation would remain oscillatory if the unforced equation is oscillatory. However, the potential is usually assumed to be nonnegative in [1, 2].
When , , and , Agarwal and Grace  studied the oscillation of (1.1) by using a method of general means without imposing the Kartsatos condition. Following this method, the oscillation of (1.1) with was studied in  for both the case and on . When changes its sign on , , and , oscillation criteria for (1.1) were given in . Sun and Saker , Sun and Mingarelli , and Yang  studied the oscillation for a generalized form of (1.1) with . When , there have been many oscillation criteria for equations of the type (1.1). For example, see [9–14] and references cited therein. We see that all these oscillation criteria depend on time delay.
To the best of our knowledge, little has been known about the oscillatory behavior of (1.1) in the case of oscillatory potentials when . Particularly, little has been known about the delay-independent criteria for oscillation of (1.1). Unlike most of papers devoted on delay-dependent oscillation criteria for functional differential equations, the main purpose of this paper is to establish two delay-independent oscillation criteria for (1.1) in both the case and the case , where the potential is not imposed on the Kartsatos condition, and the potential may change its sign. Finally, two interesting examples are worked out to illustrate the main results.
2. Main Results
Theorem 2.1. Assume that and . If where , then all solutions of (1.1) are oscillatory for any .
Proof. Let be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that for . When is eventually negative, the proof follows the same argument. Multiplying (1.1) by and integrating it from to yields where , , and . Since , we have This together with (2.4) and (2.5) yield For given and , set It is not difficult to see that obtains its minimum at and It implies that Therefore, for any , multiplying (2.7) by , using (2.10), and taking on both sides of (2.7), we get a contradiction with (2.1). This completes the proof of Theorem 2.1.
Proof. Let be a nonoscillatory solution of (1.1) satisfying . Without loss of generality, we may assume that for , and there exists a positive constant such that . Similar to the corresponding computation in Theorem 2.1 and noting that , we have Since , we get Then, for any , multiplying (2.11) by , using (2.10) and (2.12), and taking on both sides of (2.11), we get a contradiction with (2.1). This completes the proof of Theorem 2.2.
The main results in this paper can also be extended to the case of time-varying delay. That is, we can consider the following equation: where is continuously differentiable on , , and for sufficiently large. Without loss of generality, say for . Similar to the analysis as before, we have the following delay-independent and derivative-dependent oscillation criteria for (2.13).
Theorem 2.3. Assume that and . If where , then all solutions of (2.13) are oscillatory.
In this section, we work out two examples to illustrate the main results.
Example 3.1. Consider the following equation: where , , , and are constants. Note that where . We have where the Beta function is a positive constant. On the other hand, where has the asymptotic formula as [15, pages 49 and 50]. By Theorems 2.1 and 2.2, we have that if then all solutions of (3.1) are oscillatory for any , and all solutions of (3.1) satisfying are oscillatory for any .
Example 3.2. Consider the following equation: where , , and are defined as in Example 3.1. Similar to the computation in Example 3.1, we have where . Following the same argument in Example 3.1, we have that all solutions of (3.7) are oscillatory if .
This paper was supported by the Natural Science Foundation of Shandong Province under Grant no. ZR2010AL002.
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