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New oscillation criteria for higher order delay dynamic equations on time scales
Advances in Difference Equations volume 2014, Article number: 328 (2014)
Abstract
In this paper, we investigate the oscillation of the following higher order delay dynamic equation: on any time scale T with . Here , (), is an increasing differentiable function with and , with for some when , and , are two quotients of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.
MSC:34K11, 34N05, 39A10.
1 Introduction
In this paper, we investigate the oscillation of the following higher order delay dynamic equation:
on some time scale T. Here , (), is an increasing differentiable function with and , with for some when , and , are two quotients of odd positive integers. Write
then (E) reduces to the equation
Since we are interested in the oscillatory behavior of solutions near infinity, we assume that and is a constant. We define the time scale interval . A nontrivial real-valued function x is said to be a solution of (1.1) if , , which has the property that for , and satisfies (1.1) on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in [1] in order to unify continuous and discrete analysis. The cases when a time scale T is equal to R or the set of all integers Z represent the classical theories of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and it helps avoid proving results twice-once for differential equations and once again for difference equations. The general is to prove a result for a dynamic equation where the domain of the unknown function is a time scale T. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but it also extends these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations when , which has important applications in quantum theory (see [2]). In this work, knowledge and understanding of time scales and time scale notation are assumed, for an excellent introduction to the calculus on time scales; see Bohner and Peterson [3, 4]. In recent years, there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales.
In [5], Hassan studied the third-order dynamic equation
on a time scale T, where is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and the so-called delay function satisfies for and and and obtained some oscillation criteria, which improved and extended the results that have been established in [6–8].
Li et al. in [9] also discussed the oscillation of (1.2), where is the quotient of odd positive integers, is assumed to satisfy for and there exists a positive rd-continuous function p on T such that for . They established some new sufficient conditions for the oscillation of (1.2).
Wang and Xu in [10] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation
on a time scale T, where is a ratio of odd positive integers and the functions (), are positive real-valued rd-continuous functions defined on T.
Erbe et al. in [11] were concerned with the oscillation of the third-order nonlinear functional dynamic equation
on a time scale T, where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and satisfies and . The authors obtained some new oscillation criteria and extended many known results for oscillation of third-order dynamic equations.
Qi and Yu in [12] obtained some oscillation criteria for the fourth-order nonlinear delay dynamic equation
on a time scale T, where γ is the ratio of odd positive integers, p is a positive real-valued rd-continuous function defined on T, , , and .
Grace et al. in [13] were concerned with the oscillation of the fourth-order nonlinear dynamic equation
on a time scale T, where λ is the ratio of odd positive integers, q is a positive real-valued rd-continuous function defined on T. They reduced the problem of the oscillation of all solutions of (1.3) to the problem of oscillation of two second-order dynamic equations and gave some conditions to ensure that all bounded solutions of (1.3) are oscillatory.
Grace et al. in [14] established some new criteria for the oscillation of the fourth-order nonlinear dynamic equation
where a is a positive real-valued rd-continuous function satisfying , is continuous satisfying and for and . They also investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.
Agarwal et al. in [15] were concerned with oscillatory behavior of a fourth-order half-linear delay dynamic equation with damping
on a time scale T with , where λ is the ratio of odd positive integers, r, p, q are positive real-valued rd-continuous functions defined on T, , , and as . They established some new oscillation criteria of (1.4).
Zhang et al. in [16] concerned with the oscillation of a fourth-order nonlinear dynamic equation
on an arbitrary time scale T with , where with and there exists a positive constant L such that for all , they gave a new oscillation result of (1.5).
In [17], Sun et al. studied the following higher order dynamic equation:
and established some new oscillation criteria.
For much research concerning the oscillation and nonoscillation of solutions of higher order dynamic equations on time scales, please refer to the literature [18–27].
2 Some lemmas
In order to obtain the main results of this paper, we need the following lemmas.
Lemma 2.1 [28]
Assume that
and integer . Then:
-
(1)
implies for .
-
(2)
implies for .
Lemma 2.2 [28]
Assume that (2.1) holds. If and for , then there exists an integer such that:
-
(1)
is even.
-
(2)
for and .
-
(3)
If , then there exists such that for and .
Lemma 2.3 [28]
Assume that (2.1) holds. Furthermore, suppose that
If x is an eventually positive solution of (1.1), then there exists sufficiently large such that:
-
(1)
for .
-
(2)
Either or for and .
Lemma 2.4 [28]
Assume that x is an eventually positive solution of (1.1). If there exists such that:
-
(1)
for .
-
(2)
for and .
Then
and there exist and a constant such that
where
Lemma 2.5 [3]
Let be continuously differentiable and suppose that is delta differentiable. Then is delta differentiable and
Lemma 2.6 [17]
If A, B are nonnegative numbers and , then
Lemma 2.7 [29]
Assume that U, V are constants and is the quotient of odd positive integers. Then
3 Main results
Throughout this paper, we assume that:
-
(1)
, where the forward jump operator by .
-
(2)
and such that and are differentiable.
Write
Theorem 3.1 Suppose that (2.1) and (2.2) hold. If there exist differentiable functions and with being differentiable such that for all sufficiently large and for any positive constants , , there is a with such that
then every solution of (1.1) is either oscillatory or tends to 0.
Proof Assume that (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a sufficiently large such that for . Therefore from Lemma 2.3, we know that there exists sufficiently large such that:
-
(1)
for .
-
(2)
Either or for and .
Let for and . Consider
Then and for .
By the product rule and the quotient rule
Since (), we get
Using the fact that x and Ï„ are differentiable functions and , we see that is a differentiable function and . Note . From Lemma 2.5, we get
which implies
We choose such that for . Then, from (2.3) and the fact that for , we get
From (2.3), we have
Thus
which combines with (3.8) to imply
Combining (3.9) with (3.6) and from , we obtain that
Now we consider the following three cases.
Case (i). If , then for and . Thus
Case (ii). If , then
Case (iii). If , then from (2.4) we get that there exist and a constant such that
Thus
where .
We obtain from the above that
Thus
Since
from (3.10) and (3.11) and the definitions of and , we get
It is easy to check that
Integrating both sides of the above inequality from to t, we get
which leads to a contradiction to (3.3). The proof is completed. □
Theorem 3.2 Suppose that (2.1) and (2.2) hold. If there exist differentiable functions and with being differentiable such that for all sufficiently large and for any positive constants , , there is a with such that
then every solution of (1.1) is either oscillatory or tends to 0.
Proof Assume that (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a sufficiently large such that for . Therefore from Lemma 2.3, we know that there exists sufficiently large such that:
-
(1)
for .
-
(2)
Either or for and .
Let for and . From (3.7), we get
Define as (3.4). Choosing such that for . Combining (3.14) with (3.6), we see that for ,
Note since , we obtain
Now we consider the following three cases.
Case (i). If , then
since for .
Case (ii). If , then
Case (iii). If , then we get from (2.4) that there exist and a constant such that
Thus
where .
We obtain from the above
Then
From Lemma 2.7, we have
Combining (3.16) with (3.15) and the definitions of and , we get
Let
We have from Lemma 2.6
Then
Integrating both sides of the above inequality from to t, we get
which leads to a contradiction to (3.13). The proof is completed. □
4 Further results
For convenience, let . For any function , denote by the partial derivative of with respect to s. Define
Theorem 4.1 Suppose that (2.1) and (2.2) hold. If there exist functions and differentiable functions and with being differentiable such that for all sufficiently large and for any positive constants , , there is a with such that
and
then every solution of (1.1) is either oscillatory or tends to 0.
Proof Assume that (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a sufficiently large such that for . Therefore from Lemma 2.3, we know that there exists sufficiently large such that:
-
(1)
for .
-
(2)
Either or for and .
Let for and . Define as (3.4). Choosing such that (3.12) holds for . Then for
In (4.3), replace t by s and multiply both sides by , integrate with respect to s from to , we have
Integrating by parts and using (4.1), we get
This implies
Thus
which leads to a contradiction to (4.2). The proof is completed. □
Theorem 4.2 Suppose that (2.1) and (2.2) hold. If there exist functions and differentiable functions and with being differentiable such that for all sufficiently large and for any positive constants , , there is a with such that
and
then every solution of (1.1) is either oscillatory or tends to 0.
Proof Assume that (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a sufficiently large such that for . Therefore from Lemma 2.3, we know that there exists sufficiently large such that:
-
(1)
for .
-
(2)
Either or for and .
Let for and . Define as (3.4). Choosing such that (3.17) holds for . Then for , we have
In (4.6), replace t by s and multiply both sides by and integrate with respect to s from to , it follows that
Integrating by parts and using (4.4), we get
Let
From Lemma 2.6, we have
which implies
Then
which leads to a contradiction to (4.2). The proof is completed. □
5 Example
In this section, we give an example to illustrate our main results.
Example 5.1 Consider the following higher order dynamic equation:
on time scale , where , () is as in (1.1) with , , , , , , and . Then and the forward jump operator satisfies . Thus
Therefore (2.1) and (2.2) hold. Note that
It is easy to check that
and
Then for any positive constants , , there is a sufficiently large such that for , , and for ,
Choosing and . Then and . Thus
The conditions of Theorem 3.1 are satisfied. Then every solution of (5.1) is either oscillatory or tends to 0.
Remark 5.1 If , then the conditions of Theorem 3.1 are also satisfied and every solution of (5.1) is also either oscillatory or tends to 0.
Remark 5.2 In Example 5.1, let for and for . Then the conditions of Theorem 4.1 are satisfied. It also follows from Theorem 4.1 that every solution of (5.1) is either oscillatory or tends to 0.
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This project is supported by NNSF of China (11461003) and NSF of Guangxi (2012GXNSFDA276040).
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Sun, T., Yu, W. & He, Q. New oscillation criteria for higher order delay dynamic equations on time scales. Adv Differ Equ 2014, 328 (2014). https://doi.org/10.1186/1687-1847-2014-328
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DOI: https://doi.org/10.1186/1687-1847-2014-328
Keywords
- oscillation
- dynamic equation
- time scale