Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations
By introducing auxiliary functions, we investigate the oscillation of a class of second-order sub-half-linear neutral impulsive differential equations of the form where Several oscillation criteria for the above equation are established in both the case and the case which generalize and complement some existing results in the literature. Two examples are also given to illustrate the effect of impulses on the oscillatory behavior of solutions to the equation.
Impulsive differential equations appear as a natural description of observed evolution phenomena of several real-world problems involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems [1–5]. In recent years, impulsive differential equations have received a lot of attention.
We are here concerned with the following second-order sub-half-linear neutral impulsive differential equation: where , and for some , is a strictly increasing unbounded sequence of real numbers, for , and
Let denote the set of all real-valued functions defined on such that is continuous for all except possibly at where exists and .
We assume throughout this paper that(a), and ;(b), or ;(c), ;(d) is a sequence of nonnegative real numbers;(e), and .
By a solution of (1.1) we mean a function defined on with such that and satisfies (1.1). It is tacitly assumed that such solutions exist. Note the assumption ; we have that each solution of (1.1) is continuous on . As usual, a nontrivial solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if its every nontrivial solution is oscillatory.
When , , and , (1.1) reduces to the following sublinear impulsive delay equation: which has received a lot of attention in the literature. However, for the general sub-half-linear neutral equation (1.1) under the impulse condition given in this paper, little has been known about the oscillation of (1.1) to the best of our knowledge, especially for the case when .
The main objective of this paper is to establish oscillation criteria for the sub-half-linear impulsive differential equation (1.1) in both the case and the case . By introducing an auxiliary function and a function defined below, we establish some new oscillation criteria for (1.1) which complement the oscillation theory of impulsive differential equations. Examples are also given to show the effect of impulses on oscillation of solutions of (1.1).
2. Main Results
Theorem 2.1. Let . If there exists a positive function such that where , then (1.1) is oscillatory.
Proof. Suppose to the contrary that (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that and for . The case being eventually negative can be similarly discussed. From (1.1), we have that Based on the impulsive condition , we can deduce that is nonincreasing on . We may claim that holds eventually. Otherwise, there exists such that . Noting that is continuous on , we have that which implies that is eventually negative since . This is a contradiction. Without loss of generality, say for . Choose sufficiently large such that for , and which is always possible because . Thus, we have By choosing sufficiently large such that for and using (2.6) and the nonincreasing character of , we have Since for and is continuous, we have By (1.1), (2.7), and (2.8), we get which implies From (2.1), we get Multiplying (2.10) by , we get Integrating (2.12) from to , we have that which implies that On the other hand, by the given impulsive condition, we get where Note that , for . Consequently, we see from (2.7), (2.8), and (2.15) that Substituting (2.17) into (2.14) yields which contradicts (2.2). This completes the proof.
Proof. Suppose to the contrary that there is a solution of (1.1) which is neither oscillatory nor tends to zero. Without loss of generality, we may let and for . Thus, is nonincreasing for . As a result, and are eventually of constant sign. Now, we consider the following two cases: (i) eventually; (ii) eventually. For the case (i), similar to the analysis as in the proof of Theorem 2.1, we have eventually and (2.6) holds. Notice that because ; from (1.1) and (2.6), we get
Following the similar arguments as in the proof of Theorem 2.1, we can get a contradiction with (2.19).
For the case (ii), assume that for . It must now hold that for . Let us consider two cases: (a) is unbounded; (b) is bounded. If is unbounded, then we have On the other hand, there exists a sequence satisfying , , and . Let be sufficiently large such that and . Then, we have which contradicts (2.21). If is bounded, then we can prove that . In fact, which implies that since . This is a contradiction. The proof of Theorem 2.2 is complete.
Proof. Suppose to the contrary that (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that and for . Similar to the proof of Theorem 2.1, we have that (2.12) holds. Multiplying on both sides of (2.12) and integrating it from to , we get which implies that Therefore, Proceeding as in the proof of Theorem 2.1, we get a contradiction with (2.26). This completes the proof.
For the case , we have the following oscillation result. Since the proof is similar to that of Theorem 2.2, we omit it here.
The following two corollaries for (2.23) are immediate.
We now present two examples to illustrate the effect of impulses on oscillation of solutions of (1.1).
Example 3.1. Consider the following impulsive delay differential equation: where , is a constant, , and . We see that , , , , , . Let . A straightforward computation yields . Therefore, when , it is not difficult to verify that (2.1) and (2.2) hold. Thus, (3.1) is oscillatory by Theorem 2.1. However, when there is no impulse in (3.1), Corollary 2.3 cannot guarantee the oscillation of (3.1) since condition (2.24) is invalid for this case. Therefore, the impulsive perturbations may greatly affect the oscillation of (3.1). If , then we have that every solution of (3.1) is either oscillatory or tends to zero by Theorem 2.2. Such behavior of solutions of (3.1) is determined by the impulsive perturbations to a great extent, since Corollary 2.4 fails to apply for this case.
Example 3.2. Consider the following impulsive delay differential equation: where , is a constant, , and . We see that , , , , , . Let . It is not difficult to verify that (2.1) and (2.2) hold if , which implies that (3.2) is oscillatory by Theorem 2.1. We also can verify that and (2.1) and (2.19) hold if . Thus, by Theorem 2.2, every solution of (3.2) is either oscillatory or tends to zero. However, Corollaries (1.1) and (2.1) do not apply for this case. Therefore, the impulsive perturbations play a key role in the oscillation problem of (3.2).
The author thanks the reviewers for their valuable comments and suggestions on this paper. This paper was supported by the Natural Science Foundation of Shandong Province under the Grant ZR2010AL002.
L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation theory for functional-differential equations, vol. 190, Marcel Dekker, New York, NY, USA, 1995.