FUNCTIONAL
DIFFERENTIAL
EQUATIONS
VOLUME 16
2009, NO 1
PP. 11- 17
AN OSCILLATION CRITERION FOR FIRST ORDER DELAY
DYNAMIC EQUATIONS •
RAVI AGARWAL t
AND
MARTIN BOHNER;
Abstract. In this paper we improve some known oscillation criteria for dynamic
equations on time scales. The improvement is new even in the case of difference equations.
Dynamic equations on time scales contain differential equations and difference equations
as well as other kinds of equations such as for example q-di:fference equations.
Key Words. Dynamic equations, time scales, oscillation, difference equations.
AMS{MOS) subject classification. 39All, 34Cl0, 34Kll, 39A10, 39A12, 39Al3.
1. Introduction.
In this paper we consider a first order delay
dynamic equation (see [2]) of the form
(1)
xe:..(t) + p(t)x(r(t))
= 0,
where t E 'll.', '1I.' is a time scale (i.e., any nonempty closed subset of the
reals) that is unbounded above, p is rd-continuous and nonnegative, the
delay function r : 'Jl' ~ 'Jl' is nondecreasing and satisfies r(t) < t for all t E 1'
and limt->oo r(t) = oo, and xe:..(t) is the delta derivative of x : '1I.' ~ JR at
t E 1'. If 1' = R, then xe:.. = x' (the usual derivative), while if 1' = Z, then
xA = l:!..x (the usual forward difference). For further details concerning the
time scales calculus we refer to [3, 4]. The equation (1) is called oscillatory if
it does not possess any eventually positive solution. The following oscillation
criteria are known in the dynamic equations case.
* In Memory of Professor Michael Drakhlin
t Florida Institute of Technology, Department of Mathematical Sciences, Melbourne,
FL 32901-6975, USA, E-mail: agarwal<Qfit.edu
t Missouri University of Science and Technology, Department of Mathematics and
Statistics, Rolla, MO 65409-0020, USA, E-mail: bohner(Qmst. edu
11
12
RAVI AGARWAL AND MARTIN BOHNER
Theorem 1 (see [2, Theorem 1 and Remark 3]). The equation (1) is
oscillatory provided
liminflt p(s)As >
t-+oo
r(t)
~e -
Theorem 2 (see [6, Theorem 2.5]). The equation (1) is oscillatory provided there exists ME (0, 1) with
(2)
liminf
t-+oo
l
t
p(s)As > M
lim sup
and
r(t)
t-+oo
lt
M2
p(s)As > 1- -
r(t)
4
.
In this paper we will prove the following result.
Theorem 3. The equation (1) is oscillatory provided there exists ME (0, 1)
with
(3)
liminflt p(s)As
t-+oo
>M
and
r(t)
Hmsuplt p(s)As > 1- (1- v'l- M )2 •
t->oo
r(t)
Remark 1. Theorem 3 improves Theorem 2 because the right-hand side of
the second inequality in (3) is for all M E (0, 1) smaller than the right-hand
side of the second inequality in (2). To show this, consider f : [0, 1) ~ R
defined by
f(x) = [1-
~] - [1- (1- Vf=x)2]
for x E [0, 1).
Then
f'(x) =
1
Vf=X
1- x
X
1-2
and
1[ 1
J"(x) = -
2
y'(1 - x) 3
l
- 1
for x E [0, 1).
Clearly, f"(x) > 0 for all x E (0, 1), so that f' is strictly increasing on [0, 1).
Since f'(O) 0, this implies f'(x) > 0 for all x E (0, 1), so that f is strictly
increasing on [0, 1). Since f(O) = 0, this implies f(x) > .0 for all x E (0, 1).
=
2. The Oscillation Criterion. We first prove the following auxiliary
result.
Lemma 1. If x is a positive solution of (1), then x is nonincreasing,
(4)
x(s) ;::: x(t) + x(-r(t))
1t
p(r)Ar
if s
~ t,
and
o-(t)
(5)
x(s);::: x(u(t)) + x(r(t))
1
s
p(r)~r
if s ~ cr(t).
OSCILLATION FOR FIRST ORDER DYNAMIC EQUATIONS
13
Proof. Assume that x is a positive solution of (1). Then
xA(t) = -p(t)x('r(t)) ~ 0
so that x is indeed nondecreasing. To show (4), we integrate both sides of
(1) from s tot~ s to arrive at
it
0 -
-
[xA(r) + p(r)x(r(r))] D..r
x(t)- x(s)
+it
p(r)x(r(r))D..r
> x(t)- x(s) + x(7(t))
lt
p(r)D..r.
where the last inequality is true as xis nonincreasing and 7 is nondecreasing.
Finally, to show (5), we integrate both sides of (1) from s to cr(t) ~ s to arrive
at
t'(t)
0
= Js
[xA(r)
+ p(r)x(7(r))] !:l.r
= . x(cr(t))- x(s) +
l
s
q(t)
p(r)x(7(r) )D..r
;:::: x(cr(t))- x(s) + x(7(t))
l
s
u(t)
p(r)!:l.r.
where the last inequality is true due to [6, Lemma 2.1] asp is rd-continuous,
x is nonincreasing, and 7 is nondecreasing.
0
Now we are ready to prove the main result of this paper.
Proof of Theorem 3. We assume that (3) holds for some ME (0,1). Suppose
now that xis an eventually positive solution of (1). Define
1- Vl- M
a=---M
(6)
and note that 1-Vl - M > 0 so that a > 0. Moreover, Vl - M [ Vl - M- 1]
0 implies 1-M -Vl- M < 0 and therefore 1-Vl- M < M so that a< 1.
Altogether,
O<a<l.
Now observe that
i
t
t
p(s)!:l.s = 0 < aM and
17'-l(t)
t
p(s)!:l.s
> M > aM
<
14
RAVI AGARWAL AND MARTIN BOHNER
(where the last inequality holds eventually due to the first condition in (3))
implies by the intermediate value theorem on time scales [3, Theorem 1.115]
that for each sufficiently large t E '[' there exists t* E [t, r- 1 (t)) such that
i
(7)
t
t*
p( s)~s :::; aM and
lu(t.) p(s
t
)~s
> aM.
Thus, eventually, we have the following estimate:
(5)
x(t) > x(cr(t.)) + x(r(t*))
lu(t.)
t
p(s)~s
(7)
> x(cr(t*)) + x(r(t.))aM > x(r(t*))aM
(4)
> { x(t) +x(r(t))
lt
r(t.)
-
{ x(t) + x(r(t))
p(s)~s} aM
[1::.) p(s)~s- it. p(s)~s]} aM
> { x(t) + x(r(t)) [M - i t. p(s)~s]} aM
(3)
(7)
> {x(t) + x(r(t)) [M- aM]} aM
- {x(t) + x(r(t))(1- a)M} aM
so that
(1 - aM)x(t) > x(r(t))a(1 - a)M 2
and thus
x(t)
a(1- a)M2 _
x(r(t)) >
1 - aM
=
=
-
1- yr::¥ ( 1 _ 1- yr::¥)
M
M
1-
1-.Jf=M M
M
(1- Jt- M)(M -1 + J1- M)
J1-M
(1- v~
i -M
~')Jt- M(l- Jt- M)
Jt-M
(1 - y ,--1--M.,.-,) 2 •
M2
OSCILLATION FOR FIRST ORDER DYNAMIC EQUATIONS
15
Therefore
(4)
1t p(s)~s
x(r(t)) [x(;~;)) - + 1:t) p(s)~s]
x(r(t)) [(1- \11-M?- 1 + 1t p(s)As]
x(r(t)) [it p(s)~s- { 1- (1- v'l- M? }] .
0 > x(t)- x(r(t)) + x(r(t))
r(t)
=
>
1
r(t)
=
1'(t)
Since the second condition in (3) implies that for any T E '.II.' there exists
t;::: T with J:(t)p(s)~s > 1- (1- v'1- M) 2 , we arrive at the contradiction
0 > 0. Therefore (1) cannot have an eventually positive solution and hence
is oscillatory.
0
Here we first consider '.II.'
3. Applications.
equation
x(t + 1) = x(t) + p(t)x(t- k),
(8)
= Z, i.e., the difference
t E Z,
where k E N is fixed. We recall the following recent result for (8).
Theorem 4 (see [5, Theorem 2.10]). Let k E N. The equation (8) is
oscillatory provided there exists ME (0, 1) with
k
(9)
lie!~Lp(t- i) > M
k
and
lim sup LP(t - i) > 1 t-• oo
i=l
i=1
4
M2
_ M.
2
Our Theorem 3 applied to (8) reads as follows.
Theorem 5. Let k E N. The equation (8) is oscillatory provided there exists
ME (0, 1) with
(10)
k
k
liminfLp(t - i) > M
t->oo
and
i=l
lim sup
t -->oo
.L>(t- i) > 1- (1- v'1 - M)
2
•
i= l
Remark 2. Theorem 5 improves Theorem 4 because the right-hand side of
the second inequality in (10) is for all M E (0, 1) smaller than the right-hand
side of the second inequality in (9). To show this, consider f : [0, 1) -7 1R
defined by
2
f(x) = [1-
]
x
4 - 2x
-
[1 - (1 -
vr=xY]
for
X
E [0, 1).
16
RAVI AGARWAL AND MARTIN BOHNER
Then, by defining a as in (6) and using a calculation from the proof of
Theorem 9, we find for fixed x E (0, 1) that
(11) f(x) = g(a)- g(l/2),
a(1-a)x
1- ax
where g ()
a =
2
(
)
1
Jor
a E 0,1 .
We can easily calculate
2
and
"( a) = 2x
(X - )1) < 0
(
1- ax 3
g
I
J or
a E (0, 1).
Thus g has in (0, 1) a maximum at a, and f(x) ~ 0 for all x E (0, 1) follows
from (11). If f(x) = 0, then a = 1/2 so that 1 - M/2 = vl- M, hence
M 2 /4 = 0, and thus M = 0. Hence f(x) > 0 for all x E (0, 1).
Next we consider 1' = qz with q > 1, i.e., the q-difference equation
(12)
x(qt)
= x(t) + p(t)x(tjqk),
t E qz,
where kEN is fixed. Our Theorem 3 applied to (12) reads as follows.
Theorem 6. Let k E N. The equation {12) is oscillatory provided there
exists ME (0, 1) with
(13)
k
k
liminfLp(t/qi) > M
t__,oo
and
i=l
limsup LP(tjqi) > 1- (1- V1
t-+oo
-
M) 2•
i=l
Example 1. Define
p
(t) = {0.648
0.035
iflogq(t)/11 E Z
otherwise.
Then we have (see [5, Example 2.2})
10
lim in£" p(tjqi) = 0.35 > 0.349
t-+oo L..J
i=l
and
10
lim sup ~p(tjqi) = 0.963
t->oo
>1-
(1- vl- 0.349) 2 •
i=l
Hence, by Theorem 6, with k = 10, all solutions of (12) are oscillatory. Note
also that neither Theorem 1 nor Theorem 2 are applicable in this case.
OSCILLATION FOR FIRST ORDER DYNAMIC EQUATIONS
17
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