Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales
We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.
In reality, it is known that the movement in the vacuum or ideal state is rare, while the movement with damping and disturbance is extensive. In recent years, the study of the oscillation of the second-order dynamic equations with damping on time scales is emerging; see [1–7], for example. Besides, the study of the oscillation for the second-order linear and nonlinear or semilinear dynamic equations can be found in [8–23] and of the oscillation for the high-order dynamic equations can be found in [24–33]. Then, inspired by the above work, this paper will study the oscillatory behavior of all solutions of a more extensive second-order half-linear neutral delay dynamic equation with damping, which is given as follows: where , , .
Here, we give the following hypotheses at first.(H1) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above and when with , we define the time scale interval of the form by .(H2) are positive rd-continuous functions such that , where is defined as the set of all regressive and rd-continuous functions and is all positively regressive elements of .(H3) is a strictly increasing and differentiable function such that (H4) is a continuous function such that, for some positive constant ,
The solution of (1) defines a nontrivial real-valued function satisfying (1) for . A solution of (1) is called oscillatory if it is neither eventually positive nor negative; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory. Here, we pay attention to those solutions of (1) which are not the eventually identical zero.
The purpose of this paper is to establish the oscillation criteria of Philos  for (1). The two famous results of Philos  about oscillation of second-order linear differential equations are extended to (1), while it satisfies Besides, two criteria of (1) about the fact that each solution is either oscillatory or converges to zero are obtained when
The paper is organized as follows. In Section 2, we present some basic definitions and results about the theory of calculus on time scales. In Section 3, we give some lemmas. Section 4 introduces the main results of this paper. We established four new oscillatory criteria when conditions (4) and (5) hold, respectively, for the solutions of (1) in this paper.
2. Some Preliminaries
On the time scale we define the forward and backward jump operators by A point is said to be left-dense if it satisfies , right-dense if it satisfies , left-scattered if it satisfies , and right-scattered if it satisfies . The graininess function of the time scale is defined by . For a function , the (delta) derivative is defined by if is continuous at and is right-scattered. If is right-dense, then the derivative is defined by provided this limit exists. A function is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit at every left-dense point. Denote by the set of rd-continuous functions , and denote by the set of functions which is -differentiable and the derivative is rd-continuous. The derivative of , the shift of , and the graininess function are related by the following formula:
We will make use of the following product and quotient rules for the derivative of the product and the quotient of two differentiable functions and : For , the Cauchy integral of is defined by The integration by parts formula reads and the infinite integral is defined by For more details, see [8, 9].
3. Several Lemmas
In this section, we present six lemmas that are needed in Section 4. The first lemma is well known, and it can be found in Chapter 2 of . Lemma 2 is Theorem 1.93 of ; Lemma 3 is the simple corollary of Theorem 1.90 in ; Lemma 4 is Theorem 41 in ; and Lemma 5 is Theorem 3 in .
Lemma 1. If , that is, is rd-continuous, such that for all , then the initial value problem has a unique and positive solution on , denoted by . This “exponential function” satisfies the semigroup property .
Lemma 2. Assume that is strictly increasing and is a time scale. Let . If and exist on , where then
Lemma 3. If is differentiable, then
Lemma 4. Assume that and are nonnegative real numbers; then where the equality holds if and only if .
Lemma 5. Let and . Then for positive rd-continuous functions we have where and .
Proof. Suppose that is an eventually positive solution of (1). There exists such that and for . From the definition of , we get for , and at the same time for , from (1), we get
Hence, from Lemma 1 and (11) we obtain
for . So
is decreasing. By Lemma 1, is either eventually positive or eventually negative. Therefore, for arbitrary , we have
Otherwise, we assume that (24) is not satisfied; then there exits such that for all . Because (23) is decreasing, from Lemma 1 we have
for , where . By (25) and Lemma 1, we get
After integrating the two sides of inequality (27) from to , we have
Next, we find the limits of the two sides of (28) when . From (4), we get . Therefore, is eventually negative, which is contradictory to . So the inequality (24) holds.
From (24) and (21), it is obvious that the second inequality of (20) holds. This completes the proof.
4. Main Results
Theorem 7. Assume that ()–() and (4) hold. Let , be rd-continuous function, such that and has a nonpositive continuous -partial derivative with respect to the second variable and satisfies (31). Let be a rd-continuous function and satisfies If there exist a positive and differentiable function such that for and a real rd-continuous function such that where , , and , then (1) is oscillatory on .
Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . By the definition of , it follows Since it satisfies , there exists such that for all . Then if it satisfies , we have By Lemma 6 and (), we obtain that on (where is short hand for ), and holds. Moreover, using Lemmas 3 and 6, it follows that In Lemma 2, let , and is unbounded above by , so , and by ; using Lemma 2, we get Thus By the above inequality and the first inequality in (37), we obtain that holds on . Now we define the function by Then we have on , and then we obtain on , where . Thus, for every with , by (13), we get where . So using Lemma 4, let Using the inequality (18), we have where . Thus From (46) and (50), we obtain that is, From condition (34), we have By (46), we have and from the above inequality, let , and denote that meanwhile noting (54), we obtain Now we assert that holds. Suppose to the contrary that By (31), there exists a constant such that From (59), there exists a for arbitrary real number such that for . By (13), we have From (60), there exists such that for , so . Since is arbitrary, we have Selecting a sequence with satisfying then there exists a constant such that for sufficiently large positive integer . From (63), we can easily see and (65) implies that From (65) and (66), we have that is, for sufficiently large positive integer , which together with (67) implies On the other hand, by Lemma 5, we obtain The above inequality shows that Hence, (70) implies which contradicts (32). Therefore (58) holds. Noting for , by using (58), we obtain which is contradicting with (33). This completes the proof.
Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . So and there exists such that for . Define the function by We proceed as in the proof of Theorem 7 to obtain (46) and (50), so that Hence, (76) implies then we have From the above inequality and (75), we have Therefore, there exists a sequence with such that Definitions of and are as in Theorem 7; from (46), and noting (81), we have For the above sequence , We obtain (58) by using reductio ad absurdum. The rest of the proof is similar to that of Theorem 7 and hence is omitted. This completes the proof.
Proof. As the proof of Theorem 7, assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . So and there exists such that
for . In the proof of Lemma 6, we find that is either eventually positive or eventually negative. Thus, we will distinguish the following two cases:(I) for ;(II) for .
Case (I). When is an eventually positive and the proof is similar to that of the proof of Theorem 7, we can obtain that (1) is oscillatory.
Case (II). When is an eventually negative, is decreasing and exists. Therefore, there exists such that for . Define the function . Equations (1) and (89) yield The inequality (90) is the assumed inequality of [8, Theorem 6.1] (see also [37, Lemma 1]). All assumptions of [8, Theorem 6.1], for example, , are satisfied as well. Hence the conclusion of [8, Theorem 6.1] holds; that is, for all , and thus for all . Assuming and using (87) in (92), we can get , and this is a contradiction to the fact that for . Thus ; that is, . Then, it follows from that holds. This completes the proof.
Remark 12. The theorems in this paper are new even for the cases of and .
Example 13. Consider a second-order half-linear delay 2-difference equation with damping where . Here, we have