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Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 562329 (2009)
Abstract
We employ Kranoselskii's fixed point theorem to establish the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation on a time scale T. To dwell upon the importance of our results, one interesting example is also included.
1. Introduction
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [1]). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale calculus; we refer also to the last book by Bohner and Peterson [4] for advances in dynamic equations on time scales. For the notation used below we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson [3].
In recent years, there has been much research activity concerning the oscillation of solutions of various equations on time scales, and we refer the reader to Erbe [5], Saker [6], and Hassan [7]. And there are some results dealing with the oscillation of the solutions of second-order delay dynamic equations on time scales [8–22].
In this work, we will consider the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form

on a time scale (an arbitrary closed subset of the reals).
The motivation originates from Kulenović and Hadžiomerpahić [23] and Zhu and Wang [24]. In [23], the authors established some sufficient conditions for the existence of positive solutions of the delay equation

Recently, [24] established the existence of nonoscillatory solutions to the neutral equation

on a time scale
Neutral equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines. So, we try to establish some sufficient conditions for the existence of equations of (1.1). However, there are few papers to discuss the existence of nonoscillatory solutions for neutral delay dynamic equations on time scales.
Since we are interested in the nonoscillatory behavior of (1.1), we assume throughout that the time scale under consideration satisfies
and
As usual, by a solution of (1.1) we mean a continuous function which is defined on
and satisfies (1.1) for
A solution of (1.1) is said to be eventually positive (or eventually negative) if there exists
such that
(or
) for all
in
A solution of (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative; otherwise, it is oscillatory.
2. Main Results
In this section, we establish the existence of nonoscillatory solutions to (1.1). For let
and
Further, let
denote all continuous functions mapping
into
and

Endowed on with the norm
(
) is a Banach space (see [24]). Let
we say that
is uniformly Cauchy if for any given
there exists
such that for any
for all
.
is said to be equicontinuous on
if for any given
there exists
such that for any
and
with
Also, we need the following auxiliary results.
Lemma 2.1 (see [24, Lemma ]).
Suppose that is bounded and uniformly Cauchy. Further, suppose that
is equicontinuous on
for any
Then
is relatively compact.
Lemma 2.2 (see [25, Kranoselskii's fixed point theorem]).
Suppose that is a Banach space and
is a bounded, convex, and closed subset of
Suppose further that there exist two operators
such that
-
(i)
for all
-
(ii)
is a contraction mapping;
-
(iii)
is completely continuous.
Then has a fixed point in
Throughout this section, we will assume in (1.1) that
,
,
=
,
=
,  
,
,
,
,
=
, and there exists a function
such that
=
,
=
Theorem 2.3.
Assume that holds and
Then (1.1) has an eventually positive solution.
Proof.
From the assumption we can choose
large enough and positive constants
and
which satisfy the condition

such that





Furthermore, from we see that there exists
with
such that
for
Define the Banach space as in (2.1) and let

It is easy to verify that is a bounded, convex, and closed subset of
Now we define two operators and
as follows:

Next, we will show that and
satisfy the conditions in Lemma 2.2.
We first prove that
for any
Note that for any
For any
and
in view of (2.3), (2.4) and (2.6), we have

Similarly, we can prove that for any
and
Hence,
for any
-
(ii)
We prove that
is a contraction mapping. Indeed, for
we have
(2.10)
for and

for Therefore, we have

for any Hence,
is a contraction mapping.
We will prove that
is a completely continuous mapping. First, by
we know that
maps
into
Second, we consider the continuity of Let
and
as
then
and
as
for any
Consequently, by (2.5) we have

for So, we obtain

which proves that is continuous on
Finally, we prove that is relatively compact. It is sufficient to verify that
satisfies all conditions in Lemma 2.1. By the definition of
we see that
is bounded. For any
take
so that

For any and
we have

Thus, is uniformly Cauchy.
The remainder is to consider the equicontinuous on for any
Without loss of generality, we set
For any
we have
for
and

for
Now, we see that for any there exists
such that when
with

for all This means that
is equicontinuous on
for any
By means of Lemma 2.1, is relatively compact. From the above, we have proved that
is a completely continuous mapping.
By Lemma 2.2, there exists such that
Therefore, we have

which implies that is an eventually positive solution of (1.1). The proof is complete.
Theorem 2.4.
Assume that holds and
Then (1.1) has an eventually positive solution.
Proof.
From the assumption we can choose
large enough and positive constants
and
which satisfy the condition

such that

Furthermore, from we see that there exists
with
such that
for
Define the Banach space as in (2.1) and let

It is easy to verify that is a bounded, convex, and closed subset of
Now we define two operators and
as in Theorem 2.3 with
replaced by
The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.
Theorem 2.5.
Assume that holds and
Then (1.1) has an eventually positive solution.
Proof.
From the assumption we can choose
large enough and positive constants
and
which satisfy the condition

such that

Furthermore, from we see that there exists
with
such that
for
Define the Banach space as in (2.1) and let

It is easy to verify that is a bounded, convex, and closed subset of
Now we define two operators and
as in Theorem 2.3 with
replaced by
The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.
We will give the following example to illustrate our main results.
Example 2.6.
Consider the second-order delay dynamic equations on time scales

where ,
,
,
,
,
,
,
Then
,
,
Let
It is easy to see that the assumption
holds. By Theorem 2.3, (2.26) has an eventually positive solution.
References
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
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
Erbe L: Oscillation results for second-order linear equations on a time scale. Journal of Difference Equations and Applications 2002,8(11):1061-1071. 10.1080/10236190290015317
Saker SH: Oscillation criteria of second-order half-linear dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,177(2):375-387. 10.1016/j.cam.2004.09.028
Hassan TS: Oscillation criteria for half-linear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2008,345(1):176-185. 10.1016/j.jmaa.2008.04.019
Agarwal RP, Bohner M, Saker SH: Oscillation of second order delay dynamic equations. The Canadian Applied Mathematics Quarterly 2005,13(1):1-17.
Zhang BG, Shanliang Z: Oscillation of second-order nonlinear delay dynamic equations on time scales. Computers & Mathematics with Applications 2005,49(4):599-609. 10.1016/j.camwa.2004.04.038
Şahiner Y: Oscillation of second-order delay differential equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):e1073-e1080.
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
Han Z, Sun S, Shi B: Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,334(2):847-858. 10.1016/j.jmaa.2007.01.004
Han Z, Shi B, Sun S: Oscillation criteria for second-order delay dynamic equations on time scales. Advances in Difference Equations 2007, 2007:-16.
Han Z, Shi B, Sun S-R: Oscillation of second-order delay dynamic equations on time scales. Acta Scientiarum Naturalium Universitatis Sunyatseni 2007,46(6):10-13.
Sun S-R, Han Z, Zhang C-H: Oscillation criteria of second-order Emden-Fowler neutral delay dynamic equations on time scales. Journal of Shanghai Jiaotong University 2008,42(12):2070-2075.
Zhang M, Sun S, Han Z:Existence of positive solutions for multipoint boundary value problem with
-Laplacian on time scales. Advances in Difference Equations 2009, 2009:-18.
Han Z, Li T, Sun S, Zhang C: Oscillation for second-order nonlinear delay dynamic equations on time scales. Advances in Difference Equations 2009, 2009:-13.
Sun S, Han Z, Zhang C: Oscillation of second-order delay dynamic equations on time scales. Journal of Applied Mathematics and Computing 2009,30(1-2):459-468. 10.1007/s12190-008-0185-6
Zhao Y, Sun S: Research on Sturm-Liouville eigenvalue problems. Journal of University of Jinan 2009,23(3):299-301.
Li T, Han Z: Oscillation of certain second-order neutral difference equation with oscillating coefficient. Journal of University of Jinan 2009,23(4):410-413.
Chen W, Han Z: Asymptotic behavior of several classes of differential equations. Journal of University of Jinan 2009,23(3):296-298.
Li T, Han Z, Sun S: Oscillation of one kind of second-order delay dynamic equations on time scales. Journal of Jishou University 2009,30(3):24-27.
Kulenović MRS, Hadžiomerspahić S: Existence of nonoscillatory solution of second order linear neutral delay equation. Journal of Mathematical Analysis and Applications 1998,228(2):436-448. 10.1006/jmaa.1997.6156
Zhu Z-Q, Wang Q-R: Existence of nonoscillatory solutions to neutral dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,335(2):751-762. 10.1016/j.jmaa.2007.02.008
Chen YS:Existence of nonoscillatory solutions of
th order neutral delay differential equations. Funkcialaj Ekvacioj 1992,35(3):557-570.
Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018), Shandong Research Funds (Y2008A28, Y2007A27), and also supported by the University of Jinan Research Funds for Doctors (B0621, XBS0843).
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Li, T., Han, Z., Sun, S. et al. Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales. Adv Differ Equ 2009, 562329 (2009). https://doi.org/10.1155/2009/562329
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DOI: https://doi.org/10.1155/2009/562329