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Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities
Advances in Difference Equations volume 2011, Article number: 513757 (2011)
Abstract
This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form where
and
with
. Further the results obtained here generalize and complement to the results obtained by Han et al. (2010). Examples are provided to illustrate the results.
1. Introduction
Since the introduction of time scale calculus by Stefan Hilger in 1988, there has been great interest in studying the qualitative behavior of dynamic equations on time scales, see, for example, [1–3] and the references cited therein. In the last few years, the research activity concerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamic equations on time scales has been received considerable attention, see, for example, [4–8] and the references cited therein. Moreover the oscillatory behavior of solutions of second order differential and dynamic equations with mixed nonlinearities is discussed in [9–16].
In 2004, Agarwal et al. [5] have obtained some sufficient conditions for the oscillation of all solutions of the second order nonlinear neutral delay dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ1_HTML.gif)
on time scale , where
,
is a quotient of odd positive integers such that
,
,
are real valued rd-continuous functions defined on
such that
,
, and
.
In 2009, Tripathy [17] has considered the nonlinear neutral dynamic equation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ2_HTML.gif)
where is a quotient of odd positive integers,
,
are positive real valued rd-continuous functions on
,
is a nonnegative real valued rd-continuous function on
and established sufficient conditions for the oscillation of all solutions of (1.2) using Ricatti transformation.
Saker et al. [18], Şahíner [19], and Wu et al. [20] established various oscillation results for the second order neutral delay dynamic equations of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ3_HTML.gif)
where ,
is a quotient of odd positive integers,
,
are real valued nonnegative rd-continuous functions on
such that
, and
.
In 2010, Sun et al. [21] are concerned with oscillation behavior of the second order quasilinear neutral delay dynamic equations of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ4_HTML.gif)
where ,
,
,
are quotients of odd positive integers such that
and
,
,
,
, and
are real valued rd-continuous functions on
.
Very recently, Han et al. [22] have established some oscillation criteria for quasilinear neutral delay dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ5_HTML.gif)
where ,
are quotients of odd positive integers such that
,
,
,
, and
are real valued rd-continuous functions on
.
Motivated by the above observation, in this paper we consider the following second order neutral delay dynamic equation with mixed nonlinearities of the form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ6_HTML.gif)
where is a time scale,
and
, and this includes all the equations (1.1)–(1.5) as special cases.
By a proper solution of (1.6) on we mean a function
which has a property that
and satisfies (1.6) on
. For the existence and uniqueness of solutions of the equations of the form (1.6), refer to the monograph [2]. As usual, we define a proper solution of (1.6) which is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is known as nonoscillatory.
Throughout the paper, we assume the following conditions:
(C1)the functions are nondecreasing right-dense continuous and satisfy
,
,
with
,
, and
for
;
(C2) is a nonnegative real valued rd-continuous function on
such that
;
(C3) and
,
are positive real valued rd-continuous functions on
with
;
(C4) ,
,
are positive constants such that
.
We consider the two possibilities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ8_HTML.gif)
Since we are interested in the oscillatory behavior of the solutions of (1.6), we may assume that the time scale is not bounded above, that is, we take it as
.
The paper is organized as follows. In Section 2, we present some oscillation criteria for (1.6) using the averaging technique and the generalized Riccati transformation, and in Section 3, we provide some examples to illustrate the results.
2. Oscillation Results
We use the following notations throughout this paper without further mention:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ9_HTML.gif)
In this section, we obtain some oscillation criteria for (1.6) using the following lemmas. Lemma 2.1 is an extension of Lemma 1 of [13].
Lemma 2.1.
Let ,
be positive constants satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ10_HTML.gif)
Then there is an -tuple
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ11_HTML.gif)
which also satisfies either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ12_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ13_HTML.gif)
In the following results we use the Keller's Chain rule [1] given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ14_HTML.gif)
where is a positive and delta differentiable function on
.
Lemma 2.2 (see [23]).
Let , where
and
are constants,
is a positive integer. Then
attains its maximum value on
at
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ15_HTML.gif)
Lemma 2.3.
Assume that (1.7) holds. If is an eventually positive solution of (1.6), then there exists a
such that
,
, and
for
. Moreover one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ16_HTML.gif)
Since the proof of Lemma 2.3 is similar to that of Lemma 2.1 in [6], we omit the details.
Lemma 2.4.
Assume that (1.7) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ17_HTML.gif)
hold. If is an eventually positive solution of (1.6), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ18_HTML.gif)
and is strictly decreasing.
Proof.
From Lemma 2.3, we have and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ19_HTML.gif)
Since , we have
. Now using the Keller's Chain rule, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ20_HTML.gif)
or . Let
. Clearly
. We claim that there is a
such that
on
. Assume the contrary, then
on
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ21_HTML.gif)
which implies that is strictly increasing on
. Pick
so that
and
for
. Then
, and
, so that
for
.
Using the inequality (2.8) in (1.6), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ22_HTML.gif)
Now by integrating from to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ23_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ24_HTML.gif)
which contradicts (2.4). Hence there is a such that
on
. Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ25_HTML.gif)
and we have that is strictly decreasing on
.
Theorem 2.5.
Assume that condition (1.7) holds. Let be
-tuple satisfying (2.3) of Lemma 2.1. Furthermore one assumes that there exist positive delta differentiable function
and a nonnegative delta differentiable function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ26_HTML.gif)
for all sufficiently large where
, and
. Then every solution of (1.6) is oscillatory.
Proof.
Suppose that there is a nonoscillatory solution of (1.6). We assume that
is an eventually positive for
(since the proof for the case
eventually is similar). From the definition of
and Lemma 2.3, there exists
such that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ27_HTML.gif)
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ28_HTML.gif)
Then from (2.19), we have and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ29_HTML.gif)
From Keller's chain rule, we have, from Lemma 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ30_HTML.gif)
Using (2.22) and the definition of in (2.21), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ31_HTML.gif)
From Lemma 2.4, we see that is strictly decreasing on
, and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ32_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ33_HTML.gif)
since for all
. Using (2.25) in (2.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ34_HTML.gif)
Now let . Then (2.26) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ35_HTML.gif)
By Lemma 2.1 and using the arithmetic-geometric inequality in (2.27), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ36_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ37_HTML.gif)
Set ,
,
, and
and applying Lemma 2.2 to (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ38_HTML.gif)
Now integrating (2.30) from to
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ39_HTML.gif)
which leads to a contradiction to condition (2.18). The proof is now complete.
By different choices of and
, we obtain some sufficient conditions for the solutions of (1.6) to be oscillatory. For instance,
,
and
,
in Theorem 2.5, we obtain the following corollaries:
Corollary 2.6.
Assume that (1.7) holds. Furthermore assume that, for all sufficiently large , for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ40_HTML.gif)
where is as in Theorem 2.5. Then every solution of (1.6) is oscillatory.
Corollary 2.7.
Assume that (1.7) holds. Furthermore assume that, for all sufficiently large , for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ41_HTML.gif)
where is as in Theorem 2.5. Then every solution of (1.6) is oscillatory.
Next we establish some Philos-type oscillation criteria for (1.6).
Theorem 2.8.
Assume that (1.7) holds. Suppose that there exists a function , where
,
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ42_HTML.gif)
and has a nonpositive continuous
-partial derivative
with respect to the second variable such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ43_HTML.gif)
and for all sufficiently large ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ44_HTML.gif)
where is same as in Theorem 2.5. Then every solution of (1.6) is oscillatory.
Proof.
We proceed as in the proof of Theorem 2.5 and define by (2.20). Then
and satisfies (2.28) for all
. Multiplying (2.28) by
and integrating, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ45_HTML.gif)
Using the integration by parts formula, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ46_HTML.gif)
Substituting (2.38) into (2.37), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ47_HTML.gif)
From (2.35) and (2.39), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ48_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ49_HTML.gif)
where .
By setting and
in Lemma 2.2, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ50_HTML.gif)
which contradicts condition (2.35). This completes the proof.
Finally in this section we establish some oscillation criteria for (1.6) when the condition (1.8) holds.
Theorem 2.9.
Assume that (1.8) holds and . Let
be
-tuple satisfying (2.3) of Lemma 2.1. Moreover assume that there exist positive delta differentiable functions
and
such that
and a nonnegative function
with condition (2.30) for all
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ51_HTML.gif)
where holds, then every solution of (1.6) either oscillates or converges to zero as
.
Proof.
Assume to the contrary that there is a nonoscillatory solution such that
,
,
, and
for
for some
. From Lemma 2.3 we can easily see that either
eventually or
eventually.
If eventually, then the proof is the same as in Theorem 2.5, and therefore we consider the case
.
If for sufficiently large t, it follows that the limit of
exists, say
. Clearly
. We claim that
. Otherwise, there exists
such that
and
. From (1.6) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ52_HTML.gif)
Define the supportive function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ53_HTML.gif)
and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ54_HTML.gif)
Now if we integrate the last inequality from to
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ55_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ56_HTML.gif)
Once again integrate from to
to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ57_HTML.gif)
which contradicts condition (2.43). Therefore , and there exists a positive constant
such that
and
. Since
is bounded,
and
. Clearly
. From the definition of
, we find that
; hence
and
. This completes proof of the theorem.
Remark 2.10.
If ,
, or
,
, and
,
, then Theorem 2.5 reduces to a result obtained in [20] or [24], respecively. If
, or
, and
, or
, and
,
, then the results established here complement to the results of [5, 9, 15] respectively.
3. Examples
In this section, we illustrate the obtained results with the following examples.
Example 3.1.
Consider the second order delay dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ58_HTML.gif)
for all . Here
,
,
,
,
,
, and
. Then
. By taking
, and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ59_HTML.gif)
By Theorem 2.5, all solutions of (3.1) are oscillatory if .
Example 3.2.
Consider the second order neutral delay dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F513757/MediaObjects/13662_2010_Article_54_Equ60_HTML.gif)
for all . Here
,
,
,
,
,
,
,
. From Corollary 2.6, every solution of (3.3) is oscillatory.
References
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston Inc., Boston, Mass, USA; 2003:xii+348.
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1-2):18-56.
Agarwal R, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002,141(1-2):1-26. 10.1016/S0377-0427(01)00432-0
Agarwal RP, O'Regan D, Saker SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. Journal of Mathematical Analysis and Applications 2004,300(1):203-217. 10.1016/j.jmaa.2004.06.041
Erbe L, Peterson A, Saker SH: Oscillation criteria for second-order nonlinear delay dynamic equations. Journal of Mathematical Analysis and Applications 2007,333(1):505-522. 10.1016/j.jmaa.2006.10.055
Saker SH, O'Regan D, Agarwal RP: Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. Acta Mathematica Sinica 2008,24(9):1409-1432. 10.1007/s10114-008-7090-7
Shao J, Meng F: Oscillation theorems for second-order forced neutral nonlinear differential equations with delayed argument. International Journal of Differential Equations 2010, 2010:-15.
Agarwal RP, Anderson DR, Zafer A: Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities. Computers & Mathematics with Applications 2010,59(2):977-993. 10.1016/j.camwa.2009.09.010
Agarwal RP, Zafer A: Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities. Advances in Difference Equations 2009, 2009:-20.
Li C, Chen S: Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials. Applied Mathematics and Computation 2009,210(2):504-507. 10.1016/j.amc.2009.01.014
Murugadass S, Thandapani E, Pinelas S: Oscillation criteria for forced second-order mixed type quasilinear delay differential equations. Electronic Journal of Differential Equations 2010, No. 73, 9.
Sun YG, Meng FW: Oscillation of second-order delay differential equations with mixed nonlinearities. Applied Mathematics and Computation 2009,207(1):135-139. 10.1016/j.amc.2008.10.016
Sun YG, Wong JSW: Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. Journal of Mathematical Analysis and Applications 2007,334(1):549-560. 10.1016/j.jmaa.2006.07.109
Ünal M, Zafer A: Oscillation of second-order mixed-nonlinear delay dynamic equations. Advances in Difference Equations 2010, 2010:-21.
Zheng Z, Wang X, Han H: Oscillation criteria for forced second order differential equations with mixed nonlinearities. Applied Mathematics Letters 2009,22(7):1096-1101. 10.1016/j.aml.2009.01.018
Tripathy AK: Some oscillation results for second order nonlinear dynamic equations of neutral type. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1727-e1735. 10.1016/j.na.2009.02.046
Saker SH, Agarwal RP, O'Regan D: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Applicable Analysis 2007,86(1):1-17. 10.1081/00036810601091630
Şahíner Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Advances in Difference Equations 2006, 2006:-9.
Wu H-W, Zhuang R-K, Mathsen RM: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Applied Mathematics and Computation 2006,178(2):321-331.
Sun Y, Han Z, Li T, Zhang G: Oscillation criteria for second-order quasilinear neutral delay dynamic equations on time scales. Advances in Difference Equations 2010, 2010:-14.
Han Z, Sun S, Li T, Zhang C: Oscillatory behavior of quasilinear neutral delay dynamic equations on time scales. Advances in Difference Equations 2010, 2010:-24.
Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.
Saker SH: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics 2006,187(2):123-141. 10.1016/j.cam.2005.03.039
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Thandapani, E., Piramanantham, V. & Pinelas, S. Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities. Adv Differ Equ 2011, 513757 (2011). https://doi.org/10.1155/2011/513757
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DOI: https://doi.org/10.1155/2011/513757