Research Article
Received 24 November 2012 Published online 20 June 2013 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2884
MOS subject classification: 34K11; 39A10
Necessary and sufficient conditions on the
asymptotic behavior of second-order neutral
delay dynamic equations with positive and
negative coefficients
B. Karpuz
*
†
Communicated by M. Brokate
In this paper, we establish necessary and sufficient conditions for the solutions of a second-order nonlinear neutral delay
dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider
three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory.
Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on
nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.
Keywords: asymptotic behavior; nonoscillation; oscillation; positive and negative coefficients; second-order nonlinear dynamic
equations; time scales
1. Introduction
In this paper, we are concerned with the oscillation and asymptotic behavior of bounded and unbounded solutions (depending on the
range of the coefficient A) of second-order neutral dynamic equations of the form
Œx.t/ C A.t/x.˛.t//
2
C B.t/F.x.ˇ.t/// C.t/F.x..t/// D '.t/ for t 2 Œt
0
, 1/
T
, (1)
where t
0
2 T , T is a time scale unbounded above, A 2 C
rd
.Œt
0
, 1/
T
, R/, B, C 2 C
rd
Œt
0
, 1/
T
, R
C
0
, ˛, ˇ, 2 C.Œt
0
, 1/
T
, T / are strictly
increasing unbounded functions satisfying ˛.t/, .t/, ˇ.t/ t for all t 2 Œt
0
, 1/
T
and ' 2 C
rd
.Œt
0
, 1/
T
, R/.
We find useful to recall some basic concepts of time scale calculus. A time scale, which inherits the standard topology on R, is a
nonempty closed subset of reals. Here, and later throughout this paper, a time scale will be denoted by the symbol T , and the intervals
with a subscript T are used to denote the intersection of the usual interval with T . For t 2 T , the forward jump operator : T ! T is
defined by .t/ :D inf.t, 1/
T
, whereas the backward jump operator : T ! T is defined by .t/ :D sup.1, t/
T
, and the graininess
function : T ! R
C
0
is defined to be .t/ :D .t/ t. A point t 2 T is called right-dense if .t/ D t and/or equivalently .t/ D 0 holds;
otherwise, it is called right-scattered, and similarly left-dense and left-scattered points are defined with respect to the backward jump
operator. The Hilger derivative (in short: derivative) of a function f : T ! R is defined by
f
.t/ :D lim
t¤.t/
s!t
f
.t/ f .s/
.t/ s
for t 2 T
(provided that limit exists), and T
:D T nfsup T g if sup T D max T and satisfies .max T / ¤ max T ; otherwise, T
:D T .A
function f is called rd-continuous provided that it is continuous at right-dense points in T and has finite limit at left-dense points, and
the set of rd-continuous functions are denoted by C
rd
.T , R/. The set of functions C
1
rd
.T , R/ consists of functions whose derivative is in
Department of Mathematics, Faculty of Science and Literature, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey
*Correspondence to: B. Karpuz, Department of Mathematics, Faculty of Science and Literature, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey.
†
E-mail: bkarpuz@gmail.com
Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 1219–1231
1219