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Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales
Advances in Difference Equations volume 2013, Article number: 248 (2013)
Abstract
In this paper, we investigate the oscillation of the following higher-order dynamic equation:
on an arbitrary time scale T, where , () are positive rd-continuous functions on T, and γ is the quotient of two odd positive integers, with and . We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.
MSC:34K11, 39A10, 39A99.
1 Introduction
Let R be the set of all real numbers, and let T be a time scale (i.e., a closed nonempty subset of R) with . In this paper, we study Kamenev-type oscillation criteria of solutions of the following higher-order dynamic equation:
where is a constant and for any . Throughout this paper, we assume that the following conditions are satisfied:
() ().
() γ is the quotient of two odd positive integers.
() ().
() is a nondecreasing function with for any .
() and there exists a positive rd-continuous function defined on T such that for any ,
Write
Then Eq. (1.1) reduces to the equation
By a solution of Eq. (1.3) we mean a nontrivial real-valued function with , which has the property that for and satisfies Eq. (1.3) on , where is the space of differentiable functions whose derivative is rd-continuous. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of Eq. (1.3) 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 his PhD thesis [1] in order to unify continuous and discrete analysis. The cases when a time scale T is equal to R or 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 helps avoid proving results twice - once for differential equations and once again for difference equations. The general idea 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 the 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 the last years there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales, and we refer the reader to the papers [3–17] and the references cited therein.
Recently, Wang [18] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation
Erbe et al. in [19–21] considered the third-order dynamic equations
and
respectively, and established some sufficient conditions for oscillation.
Hassan [22] studied the third-order dynamic equation
and obtained some oscillation criteria, which improved and extended the results that were established in [19–21].
Hassan [23] studied the Kamenev-type oscillation criteria of the second-order dynamic equation
and established some new sufficient conditions, which improved some oscillation results for second-order differential and difference equations.
2 Some auxiliary lemmas
We shall employ the following lemmas.
Lemma 2.1 [24]
Let . Then:
-
(1)
implies for .
-
(2)
implies for .
Lemma 2.2 Suppose that is an eventually positive solution of Eq. (1.3), then there exist an integer and such that:
-
(1)
is even.
-
(2)
implies that for any and .
-
(3)
implies that for any and .
Proof Since is an eventually positive solution of Eq. (1.3), there exists a such that and on . It follows from (1.3) that
Hence is decreasing on .
We claim that for all . If not, there exists a such that
Then we obtain
By Lemma 2.1, we get , which contradicts with eventually. Then for all . This implies that exactly one of the following is true:
() for ;
() There exists a such that for .
If () holds, then we obtain by Lemma 2.1
Thus the conclusions of Lemma 2.2 hold.
If () holds, then is strictly decreasing on and exactly one of the following is true:
() for ;
() There exists a such that for .
If () holds, then we obtain by Lemma 2.1
which contradicts with eventually. Hence () is impossible.
From (), we see that is strictly increasing on and exactly one of the following is true:
() for ;
() There exists a such that for .
Therefore we can repeat the above argument and show that the conclusions of Lemma 2.2 hold. The proof is completed. □
Remark 2.3 Let and T be the set of integers. Then Lemma 2.1 and Lemma 2.2 are Lemma 1.8.10 and Theorem 1.8.11 of [3] respectively.
Lemma 2.4 Assume that
holds and is an eventually positive solution of Eq. (1.3). Then there exists sufficiently large such that either of the following cases holds:
-
(1)
for any and .
-
(2)
.
Proof Since is an eventually positive solution of Eq. (1.3), there exists a such that and on . It follows from (1.3) that
By Lemma 2.2, we see that there exists an integer , with being even, such that for and , and is eventually monotone.
We claim that implies . If not, then () and (). It is easy to see that there exist a and a constant such that on . Integrating Eq. (1.3) from t to ∞, we get that for ,
Thus
Again, integrating the above inequality from to t, we obtain that for ,
It follows from (2.1) that , which is a contradiction to (). Thus . The proof is completed. □
Lemma 2.5 Let be a solution of Eq. (1.3) such that case (1) of Lemma 2.4 holds for with some . Then we have that for ,
and
where
Proof Because is an eventually positive solution of Eq. (1.3), there exists a sufficiently large such that and for . Note , we know that is strictly decreasing on . Then for ,
Repeating the above process, we have
Thus it follows
That is,
The proof is completed. □
Lemma 2.6 [25]
Let be continuously differentiable and suppose that is delta differentiable. Then is delta differentiable and the formula
holds.
Lemma 2.7 [23]
Suppose that a and b are nonnegative real numbers and . Then
where the equality holds if and only if .
3 Main results
For convenience, we write . Now we state and prove our main results.
Theorem 3.1 Assume that (2.1) holds. Furthermore, suppose that there exist with such that
where is the △-partial derivative with respect to the second variable, and there exist , such that is a delta differentiable function, and a delta differentiable function such that
and
for all sufficiently large T, where
and
Then every solution of Eq. (1.3) is either oscillatory or tends to zero.
Proof Assume that Eq. (1.3) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . By Lemma 2.4, there are two possible cases:
-
(1)
for any and .
-
(2)
.
If case (1) holds, then set
we have
It follows from (1.3) and the definition of that for all ,
Using the fact that and is increasing on , we get
Now we consider the following two cases.
Case 1: If , then it follows from and Lemma 2.6 that and
By (3.9) and (3.10), we have
It follows from Lemma 2.5 that
Then
Combining (3.11) with (3.13), we get
Case 2: If , then it follows from and Lemma 2.6 that and
By (3.9) and (3.15), we have
It follows from (3.12) that
Combining (3.16) with (3.17) gives
Noting that the definitions of , and . It follows from (3.14), (3.18) and the fact
that for ,
Multiplying both sides of (3.19), with t replaced by s, by and integrating with respect to s from T to t (), one gets
Integrating by parts and using (3.1) and (3.2), we have
It is easy to check that
which implies
Combining (3.21) with (3.20), it follows
which contradicts assumption (3.3). Thus every solution of Eq. (1.3) is either oscillatory or tends to zero. The proof is completed. □
Theorem 3.2 Assume that (2.1) holds. Furthermore, suppose that there exist with such that
where is the △-partial derivative with respect to the second variable, and there exists a delta differentiable function such that
and
for all sufficiently large T, where
Then every solution of Eq. (1.3) is either oscillatory or tends to zero.
Proof Assume that Eq. (1.3) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . By Lemma 2.4, there are two possible cases:
-
(1)
for any and .
-
(2)
.
If case (1) holds, then set
By (3.9), we have
It follows from Lemma 2.6 that
Case 1. If , then
Case 2. If , then
Noting that , we have
By (3.12), we obtain
where . Multiplying both sides of (3.32), with t replaced by s, by and integrating with respect to s from T to t (), one gets
Integrating by parts and using (3.22) and (3.23), we have
Write
It follows from Lemma 2.7 that
Combining the above inequality with (3.33) gives
which implies
which contradicts assumption (3.24). Thus every solution of Eq. (1.3) is either oscillatory or tends to zero. The proof is completed. □
Theorem 3.3 Assume that (2.1) holds. Furthermore, suppose that for all sufficiently large T,
holds. Then every solution of Eq. (1.3) is either oscillatory or tends to zero.
Proof Assume that Eq. (1.3) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, such that for . By Lemma 2.4, there are two possible cases:
-
(1)
for any and .
-
(2)
.
If case (1) holds, then using the fact that , we obtain
which implies
Combining (3.35) with (3.12) gives
Therefore
which contradicts assumption (3.34). Thus every solution of Eq. (1.3) is either oscillatory or tends to zero. The proof is completed. □
4 Examples
In this section, we give some examples to illustrate our main results.
Example 4.1 Consider the following higher-order dynamic equation:
where , , ρ is a positive constant, () are as in Eq. (1.3) with , and τ is defined as in (). If , then every solution of Eq. (4.1) is either oscillatory or tends to zero.
Proof Note that
and
Take , and if and if , then
and
Note that . It is easy to see that
From (3.4) and (4.2), we can find such that for all . Therefore we have that if , then
Thus conditions (), (2.1) and (3.3) are satisfied. By Theorem 3.1, every solution of Eq. (4.1) is either oscillatory or tends to zero if . The proof is completed. □
Example 4.2 Consider the following higher-order dynamic equation:
where , , () are as in Eq. (1.3) with , , , τ is defined as in () and ρ is a positive constant. If , then every solution of Eq. (4.3) is either oscillatory or tends to zero.
Proof Note that
and
Note that
Pick such that
Take and for and for , then
Therefore we have that if , then
Thus conditions (), (2.1) and (3.24) are satisfied. By Theorem 3.2 every solution of Eq. (4.3) is either oscillatory or tends to zero if . The proof is completed. □
Example 4.3 Consider the following higher order dynamic equation:
on an arbitrary time scale T, where , is the quotient of two odd positive integers, ρ is a positive constant, () are as in Eq. (1.3) with , , and τ is defined as in (). If , then every solution of Eq. (4.5) is either oscillatory or tends to zero.
Proof Note that
and
Pick that such that , then
Using arguments similar to that of (4.4), it is easy to see that . Therefore we have that if , then
Thus conditions (), (2.1) and (3.34) are satisfied. By Theorem 3.3 every solution of Eq. (4.5) is either oscillatory or tends to zero if . The proof is completed. □
Example 4.4 Consider the following third-order dynamic equation:
on , where , , , and for any . It is easy to see that conditions ()-() are satisfied and is an oscillatory solution of Eq. (4.6), which tends to zero as .
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Acknowledgements
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is project is supported by NNSF of China (11261005) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFDA276040) and SF of ED of Guangxi (2013ZD061).
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Wu, X., Sun, T., Xi, H. et al. Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales. Adv Differ Equ 2013, 248 (2013). https://doi.org/10.1186/1687-1847-2013-248
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DOI: https://doi.org/10.1186/1687-1847-2013-248