Journal of Difference Equations and Applications,
Vol. 12, No. 3–4, March –April 2006, 343–355
Monotone solutions of dynamic systems
on time scales
L. ERBE†*, A. PETERSON†{ and C.C. TISDELL‡§
†Department of Mathematics, University of Nebraska—Lincoln, Lincoln, NE 68588-0130, USA
‡School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia
(in final form 14 September 2005)
This paper is dedicated to the memory of Bernd Aulbach
We are concerned with proving that solutions of certain dynamical systems on time scales satisfy some
monotoneity conditions. These results then give important results for n-th order linear scalar equations.
We then give a related result for a third order nonlinear (Emden– Fowler type) dynamic equation.
Keywords: Time scales; Dynamic equations; Monotone solutions; Nonlinear equation
AMS Subject Classification: 39A10
1. Introduction
First we give some introductory definitions and results concerning the time scale calculus that
will be used in this paper. For more detailed information see the books [2,3,8] and the papers
[1,7]. The set T is a time scale provided it is a nonempty closed subset of the real numbers R.
The forward jump operator s and the backward jump operator r are defined by
sðtÞ :¼ inf{t . t : t [ T};
and
rðtÞ :¼ sup{t , t : t [ T};
for all t [ T, where inf Y: ¼ sup T and sup Y: ¼ inf T, where Y denotes the empty set.
We assume throughout that T has the topology that it inherits from the standard topology on
the real numbers R. If s(t) . t, we say t is right-scattered, while if r(t) , t we say t is leftscattered. If s(t) ¼ t and t , sup T we say t is right-dense, while if r(t) ¼ t and t . inf T
we say t is left-dense. The function x:T ! R is said to be right-dense continuous
(rd-continuous) and we write x [ Crd provided x is continuous at each right-dense point in T
and at each left-dense point in T left-hand limits exist (finite). The function x:T ! R is said to
be regressive provided the regressivity condition
1 þ mðtÞxðtÞ – 0;
t[T
*Corresponding author. Email: lerbe@math.unl.edu
{Email: apeterso@math.unl.edu
§Email: cct@maths.unsw.edu.au
Journal of Difference Equations and Applications
ISSN 1023-6198 print/ISSN 1563-5120 online q 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/10236190500489582
344
L. Erbe et al.
holds. Let R denote the set of all functions x:T ! R such that x is rd-continuous and
regressive and let
R þ :¼ {x [ R : 1 þ mðtÞxðtÞ . 0;
t [ T}:
The set R is called the set of regressive functions and the set R þ is called the set of
positively regressive functions.
Throughout this paper we make the blanket assumption that a , b are points in T.
We define the time scale interval
½a; bT :¼ {t [ T such that a # t # b}
and other types of time scale intervals are defined similarly.
Time scale calculus unifies continuous and discrete calculus and is much more general as
T can be any nonempty closed subset of the reals R. For example, it includes quantum
calculus [5], which is time scale calculus on the time scales
q Z < {0} :¼ {0; 1; q ^1 ; q ^2 ; q ^3 ; . . .};
q . 0;
q – 1;
and hZ: ¼ {0, ^ h, ^ 2 h, ^ 3 h,. . .}.
Definition 1 Assume x:T ! R and fix t [ Tk then we define x D(t) to be the number
(provided it exists) with the property that given any e . 0, there is a neighborhood U of t
such that
j½xðsðtÞÞ 2 xðsÞ 2 x D ðtÞ½sðtÞ 2 sj # e jsðtÞ 2 sj;
for all s [ U. We call x D(t) the (delta) derivative of x at t.
n
We write x [ Cnrd if the n-th delta derivative function, denoted by x D , is rd-continuous.
The following theorem concerning (delta) differentiation is due to Hilger [7]. See also
[2, Theorem 1.16].
Theorem 1
Assume that g:T ! Rn and let t [ Tk.
(i) If g is differentiable at t, then g is continuous at t.
(ii) If g is continuous at t and t is right-scattered, then g is differentiable at t with
g D ðtÞ ¼
gðsðtÞÞ 2 gðtÞ
:
sðtÞ 2 t
(iii) If g is differentiable and t is right-dense, then
g D ðtÞ ¼ lim
s!t
gðtÞ 2 gðsÞ
:
t2s
(iv) If g is differentiable at t, then
gðsðtÞÞ ¼ gðtÞ þ mðtÞg D ðtÞ:
ð1Þ
Note that
g D ðtÞ ¼ g0 ðtÞ;
if
T ¼ R;
and g D ðtÞ ¼ DgðtÞ :¼ gðt þ 1Þ 2 gðtÞ if
T ¼ Z;
Monotone solutions
345
where D is the forward difference operator. If T ¼ q N 0 :¼ {1; q; q 2 ; q 3 ; . . .}, q . 1, then one
gets the so-called q derivative (quantum derivative) [5]
g D ðtÞ ¼ Dq gðtÞ :¼
gðqtÞ 2 gðtÞ
:
ðq 2 1Þt
See [5] for some important applications of this quantum derivative. This q derivative is
called the Hahn derivative in orthogonal polynomial theory (where it is usually assumed that
0 , q , 1 with a related time scale).
2. Main results
Our first main result concerns the first-order linear vector dynamic equation:
x D ¼ AðtÞx s ;
ð2Þ
where x s denotes the composite function x + s.
Let us recall some notation. We write A(t) # 0 provided each element aij(t) of A(t) satisfies
aij(t) # 0. We say A is rd-continuous on T provided each element of A is rd-continuous on T.
Finally we say A is a regressive matrix function on T provided I þ m(t)A(t) is invertible for
t [ T. In the proof of the next theorem, we will use the fact [2, chapter 5] that if the n £ n
matrix function is regressive and rd-continuous on T, t0 [ Tk, and x0 [ Rn, then the IVP
x D ¼ AðtÞx s ;
xðt0 Þ ¼ x0
has a unique solution defined on all of T. Throughout the remainder of the paper, we assume
v: ¼ sup T ¼ 1 or that v [ T is left-dense and we will be concerned with the behavior
of solutions on [a,v)T. If v , 1, then we do not assume that the matrix function A is defined
at v. For monotonicity results in the continuous case, see Chapter 14 in Hartman [6].
Theorem 2 Assume that the n £ n matrix function A is regressive and rd-continuous on
[a,v)T, with A(t) # 0 on T. Then the linear dynamic system (2) has a nontrivial solution x
satisfying
xðtÞ $ 0;
x D ðtÞ # 0;
t [ ½a; vÞT :
Proof. Assume t [ (a,v)T, y0 [ Rn with y0 . 0, and let y(t,t) be the solution of the IVP
y D ¼ AðtÞy s ;
yðtÞ ¼ y0 :
We claim that y(t,t) . 0 on [a,t]T. Assume not, then there is a t1 [ [a,t)T such that either
s(t1) ¼ t1, y(t,t) . 0 on (t1,t]T and at least one component of y(t,t) is zero at t1 or s(t) . t1,
y(t,t) . 0 on [s(t1),t]T, and at least one component of y(t,t) is nonpositive at t1. In either case
y D ðt; tÞ ¼ AðtÞy s ðt; tÞ # 0
for t [ [t1,t)T. It follows from this that
yðt1 ; tÞ $ yðt; tÞ ¼ y0 . 0;
346
L. Erbe et al.
which is a contradiction. Hence y(t,t) . 0 on [a,t]T for each t [ (a,v)T. Let {tn }1
n¼1 ,
ða; vÞT with limn!1 tn ¼ v and let
xn ðtÞ :¼
yðt; tn Þ
;
kyða; tn Þk
t [ ½a; vÞT ;
n $ 1:
Then for each n $ 1, xn is a solution of equation (2) with kxn ðaÞk ¼ 1: It follows that there
is a subsequence {xnk ðaÞ}1
k¼1 such that
lim xnk ðaÞ ¼ x0 ;
k!1
where
kx0 k ¼ 1:
Let x be the solution of the limit IVP
x D ¼ AðtÞx s ;
xðaÞ ¼ x0 :
Then
xðtÞ ¼ lim xnk ðtÞ $ 0;
t [ ½a; vÞT
x D ðtÞ ¼ AðtÞx s ðtÞ # 0;
t [ ½a; vÞT :
k!1
and so it follows that
A
We next give the corresponding result for an alternative form of a first order linear
system,
x D ¼ BðtÞx:
ð3Þ
Corollary 3 Assume that B is a regressive and rd-continuous matrix function on [a,v)T.
If (I þ m(t)B(t))21B(t) # 0 (or B(t)(I þ m(t)B(t))21 # 0) on [a,v)T. Then the linear dynamic
system (3) has a nontrivial solution x satisfying
xðtÞ $ 0;
x D ðtÞ # 0;
t [ ½a; vÞT :
Proof. Using x s ðtÞ ¼ xðtÞ þ mðtÞx D ðtÞ (see part (iv) of Theorem 1) is easy to see that the
vector dynamic equation (3) is equivalent to the vector dynamic equation
x D ¼ ðI þ mðtÞBðtÞÞ21 BðtÞx s :
Also, it is easy to verify that
ðI þ mðtÞBðtÞÞ21 BðtÞ ¼ BðtÞðI þ mðtÞBðtÞÞ21 :
This corollary then follows from Theorem 2.
A
Monotone solutions
347
We now can use Theorem 2 to prove the analogous result (Theorem 4) for the n-th order
scalar linear dynamic equation.
n
u D þ pn21 ðtÞu D
n21
s
þ pn22 ðtÞu D
n22
s
þ . . . þ p0 ðtÞu s ¼ 0:
ð4Þ
We say that equation (4) is regressive on [a,v)T in case pn21 [ R([a,v)T) and pi [
Crd([a,v)T), 0 # i # n 2 1. Under these conditions all initial value problems for equation
(4) have unique solutions that exist on [a,v)T (see [2, Section 5.5]).
Theorem 4 Assume equation (4) is regressive and that the coefficient functions pi in
equation (4) satisfy ( 2 1)nþipi21(t) $ 0 on [a,v)T, 1 # i # n. Then equation (4) has a
solution satisfying
uðtÞ . 0;
i
ð21Þi u D ðtÞ $ 0;
1 # i # n;
t [ ½a; vÞT :
ð5Þ
Proof. Let u be a solution of equation (4) on [a,v)T and set
2
uðtÞ
3
7
6 D
6 u ðtÞ 7
7
6
xðtÞ ¼ D6
7;
6 ... 7
4 n21 5
u D ðtÞ
t [ ½a; vÞT ;
where D is the diagonal matrix
D :¼ diag{1; 21; 1; . . .; ð21Þn21 }:
ð6Þ
Then
2
u D ðtÞ
3
6 D2 7
6 u ðtÞ 7
7
6
x D ðtÞ ¼ D6
7;
6 ... 7
4 n 5
u D ðtÞ
t [ ½a; vÞT ;
Using the formula (1), we get that
i
i
iþ1
u D s ðtÞ ¼ u D ðtÞ þ mðtÞu D ðtÞ;
ð7Þ
1#i#n21
and since u is a solution of equation (4) we have
n
u D ðtÞ ¼ 2p0 ðtÞu s ðtÞ 2 p1 ðtÞu Ds ðtÞ 2 · · · 2 pn21 ðtÞu D
n21
s
ðtÞ:
ð8Þ
348
L. Erbe et al.
The equations (7) and (8) can be written in the vector form
2
6
6
6
6
6
6
6
6
6
4
0
1
0
.
...
..
.
..
.
0
..
.
0
1
..
..
0
0
...
...
0
1
2p0
2p1
2p2
...
2pn21
.
0
..
.
3
2
7
76
76
76
76
76
76
76
74
5
us
3
2
m
1
6
6
uD 7
7 60 1
7
6
2
6
uD s 7
7 ¼ 6 ... . . .
7 6
6
... 7
5 60 ...
4
n21
uD s
0 0
0
...
m
..
.
..
.
..
.
...
1
...
0
3
0 2 D3
u
.. 7
6 D2 7
7
. 76
u 7
7
76
6
7 6 u D3 7
7:
7
7
0 76
6
76 . . . 7
7
m7
54 n 5
D
u
1
ð9Þ
It follows, after a simple calculation of the inverse of the matrix appearing on the right
hand side of equation (9), that
2
u
D
2
3
1
6
6 D2 7 6
6m 7 60
7 6
6
6 D3 7 6
6 u 7 ¼ 6 ..
7 6.
6
7 6
6
6 ... 7 6
5 60
4
n
4
uD
0
m2
. . . ð21Þn21 m n21
1
2m
..
.
.
..
..
.
...
0
1
0
...
0
76
76
0
ð21Þn22 m n22 7
76
76
6
..
76 ...
76
.
76
76 0
74
2m
5
2p0
1
uD
3
..
2
32
2m
.
0
1
0
.
...
..
.
..
.
0
1
0
...
...
0
1
2p1
2p2
...
2pn21
..
..
.
0
..
.
3
7
7
7
7
7
7
7
7
7
5
3
us
6 u Ds 7
7
6
7
6
6 u D2 s 7
6
7
7
6
6 ... 7
5
4
n21
uD s
Hence
2
2
us
3
6 D2 7
6 u Ds 7
6u 7
7
6
7
6
7
6
6
2
3 7
7
6
D
s
D
D
7 ¼ BCD6 u
u
x ¼ D6
7 ¼ BCx s ;
7
6
7
6
7
6
6 ... 7
6 ... 7
5
4
5
4
Dn21 s
Dn
u
u
where
2
1 2m
6
6
60
6
6
.
B ¼ D6
6 ..
6
6
60
4
0
m2
. . . ð21Þn21 m n21
..
.
.
2m
..
.
..
.
...
...
1
0
...
0
1
..
3
2
1
7 6
7 6
6
ð21Þn22 m n22 7
7 60
7 6.
..
7¼6.
7 6.
.
7 6
7 6
7 60
2m
5 4
0
1
2m
m2
21
..
.
.
m
..
.
..
.
...
...
0
0
...
0
..
...
ð21Þn21 m n21
3
7
7
ð21Þn21 m n22 7
7
7
..
7
7
.
7
7
n21
ð21Þ m 7
5
ð21Þn21
Monotone solutions
349
and
2
6
6
6
6
6
C¼6
6
6
6
4
0
1
0
.
...
..
.
..
.
0
..
.
0
1
..
0
0
...
...
0
1
2p0
2p1
...
.
..
0
..
.
2pn21
3
2
7
7
7
7
7
7
7
7
7
5
6
6
6
6
6
D¼6
6
6
6
4
0
21
0
0
..
.
0
1
..
0
2p0
.
...
..
.
..
.
0
..
.
..
.
...
...
0
ð21Þn21
p1
2p2
.
..
. . . ð21Þn pn21
3
7
7
7
7
7
7:
7
7
7
5
Since the sign of every element in the i-th column of B(t) is ( 2 1)i21 and from the sign
assumptions on the coefficient functions pi we see that the sign of every element in the i-th
row of C is (2 1)i it follows that
AðtÞ :¼ BðtÞCðtÞ # 0
on [a,v)T. Therefore, from Theorem 2 there is a nontrivial solution u of equation (4) on
[a,v)T satisfying
2
3
uðtÞ
7
6 D
6 u ðtÞ 7
7
6
xðtÞ ¼ D6
7 $ 0;
6 ... 7
4 n21 5
u D ðtÞ
t [ ½a; vÞT ;
and
2
u D ðtÞ
3
6 D2 7
6 u ðtÞ 7
7
6
x D ðtÞ ¼ D6
7 # 0;
6 ... 7
4 n 5
u D ðtÞ
t [ ½a; vÞT :
It follows that u satisfies equation (5).
A
A second important form of an n-th order linear scalar equation ([2, Section 5.5]) is
n
u D þ qn21 ðtÞu D
n21
þ · · · þ q0 ðtÞu ¼ 0:
ð10Þ
We say the dynamic equation (10) is regressive provided the coefficient functions qi(t),
0 # i # n 2 1, are rd-continuous on T and the regressivity condition
RðtÞ :¼ 1 þ
n21
X
ð2mðtÞÞn2j qj ðtÞ – 0;
t[T
j¼0
holds. It follows ([2, Corollary 5.90]) that if the dynamic equation (10) is regressive on
[a,v)T, then every initial value problem has a unique solution and all solutions exist on
[a,v)T.
350
L. Erbe et al.
Corollary 5
Assume that the dynamic equation (10) is regressive and
RðtÞ
i21
X
ð21Þn2j21 m i2j21 ðtÞqj ðtÞ $ 0;
ð11Þ
j¼0
for t [ [a,v)T, 1 # i # n. Then the dynamic equation (10) has a nontrivial solution u
satisfying equation (5).
Proof. This follows from the fact that (see [2, Theorem 5.99]) the dynamic equations (4) and
(10) are equivalent if
pi ðtÞ :¼
i
1 X
ð2mðtÞÞi2j qj ðtÞ;
RðtÞ j¼0
0 # i # n 2 1:
Hence, we have from equation (11) that
ð21Þnþi pi21 ðtÞ ¼
i21
1 X
ð21Þn2j21 m i2j21 ðtÞqj ðtÞ $ 0
RðtÞ j¼0
for t [ [a,v)T, 1 # i # n. The result then follows from Theorem 4.
A
In the next theorem, we see that we can relax the sign condition on the coefficient function
pn21 in Theorem 4 and get a slightly different conclusion. In Theorem 6, we consider the
generalized exponential function eq(t,t0) for q [ R. See [2, Section 2.2] for an elementary
development of this generalized exponential function.
Theorem 6
Assume pn21 [ R þ and that the coefficient functions pi satisfy
ð21Þnþi pi21 ðtÞ $ 0
on
½a; vÞT ;
1 # i # n 2 1:
Then the dynamic equation (4) has a solution u satisfying
uðtÞ . 0;
i
ð21Þi u D ðtÞ $ 0;
1 # i # n 2 1;
ð21Þn ðpx n21 ÞD ðtÞ $ 0;
ð12Þ
for t [ [a,v)T, where pðtÞ :¼ epn21 ðt; aÞ:
Proof. Since pn21 [ R þ, we have by [2, Theorem 2.48] that
pðtÞ :¼ epn21 ðt; aÞ . 0
for all t [ T. Letting u be a solution of equation (4) and multiplying both sides of equation
(4) by the integrating factor p(t) we get that u is a solution of
n21 D
n22
pu D
þqn22 ðtÞu u s þ · · · þ q0 ðtÞu s ¼ 0;
ð13Þ
where qi(t): ¼ p(t)pi(t), 0 # i # n 2 2. Note that
ð21Þnþi qi ðtÞ ¼ pðtÞ ð21Þnþi pi ðtÞ $ 0;
t [ ½a; vÞT ;
1 # i # n 2 2:
ð14Þ
Monotone solutions
351
Let u be a solution of equation (4), then u is a solution of equation (13). Setting
3
2
uðtÞ
7
6
u D ðtÞ
7
6
7
6
7; t [ ½a; vÞT ;
6
xðtÞ ¼ D6
...
7
6
D 7
5
4
Dn21
pu
ðtÞ
where D is given by equation (6) we get using an argument very similar to that in the proof of
Theorem 4 that u(t) solves a system of the form
x D ¼ AðtÞx s ;
where A(t) ¼ B(t)C(t) where in this case (surpressing arguments)
3
2
n22
n21
1 2m m 2 . . . ð21Þn23 m n23 ð21Þn22ðm =pÞ ð21Þn21ðm =pÞ
7
6
6 0 21 m . . . ð21Þn23 m n24 ð21Þn22ðm n23 =pÞ ð21Þn21ðm n22 =pÞ 7
7
6
7
6
..
..
..
..
..
7
6 ..
7
6.
.
.
.
.
.
.
.
.
7
6
B¼6
7
2
n23
n22ð
m
=pÞ
n21ð
m
=pÞ
7
60
0
0
.
.
.
ð21Þ
ð21Þ
ð21Þ
7
6
7
6
60
0
0 ...
0
ð21Þn22ð1=pÞ
ð21Þn21ðm=pÞ 7
5
4
0
0
0 ...
0
0
ð21Þn21ð1=pÞ
and
2
6
6
6
6
C¼6
6
6
6
4
0
21
0
...
0
0
..
.
0
..
.
1
..
.
...
0
..
.
0
...
...
...
2q0
2q2
q1
...
0
. . . ð21Þ
7
7
7
7
7:
7
n21 7
7
ð21Þ
5
0
0
..
.
0
n22
3
qn22
Using the sign conditions (14) on the coefficient functions qi(t) the rest of the proof is
similar to the end of the proof of Theorem 4.
A
We can now slightly improve Corollary 5.
Corollary 7.
Assume that the dynamic equation (10) is regressive and q [ R þ, where
qðtÞ :¼
n21
1 X
ð2mðtÞÞn212j pj ðtÞ:
RðtÞ j¼0
Further assume that
RðtÞ
i21
X
j¼0
ð21Þn2j21 m i2j21 ðtÞqj ðtÞ $ 0;
ð15Þ
352
L. Erbe et al.
for t [ [a,v)T, 1 # i # n 2 1. Then the dynamic equation (10) has a nontrivial solution u
satisfying
uðtÞ . 0;
i
ð21Þi u D ðtÞ $ 0;
t [ ½a; vÞT ;
1 # i # n 2 1;
and
ð21Þn ðeq ðt; aÞu D
n21
ðtÞÞD $ 0;
t [ ½a; vÞT :
3. A third order nonlinear dynamic equation
In this section, we will be concerned with the third order nonlinear dynamic equation
x DDD þ pðtÞx Ds þ rðtÞx gs ¼ 0;
ð16Þ
where g is the quotient of odd integers and p, q are rd-continuous functions on [a,v)T. This
may be considered as an analogue of the third order Emden– Fowler equation. These results
are related to some results of Erbe [4] dealing with monotonicity properties of a third order
nonlinear differential equation. To help us prove our main result concerning the dynamic
equation (16), we first prove two important lemmas.
Lemma 8
If x is a solution of equation (16) and b [ [a,v)T, then
ðb
g
D
s
rðtÞðx ðtÞx ðtÞÞ Dt ¼
a
ðb
a
½ðx DD ðtÞÞ2 2 pðtÞðx Ds ðtÞÞ2 Dt 2 x D ðtÞx DD ðtÞba :
ð17Þ
Proof. Assume x is a solution of equation (16), then multiplying both sides of equation (16)
by x Ds(t) and integrating from a to b we get
ðb
x
a
Ds
ðtÞx
DDD
ðtÞDt þ
ðb
pðtÞðx
a
Ds
2
ðtÞÞ Dt þ
ðb
rðtÞðx g ðtÞx D ðtÞÞs Dt ¼ 0:
a
After an integration by parts on the first term one easily gets the desired result (17).
A
In connection with the third order dynamic equation (16), we will be concerned with the
second order dynamic equation
y DD þ pðtÞy s ¼ 0:
ð18Þ
Definition We say that equation (18) is right-disfocal on [a,v)T provided if y is a solution
of equation (18), with y(s) ¼ 0, y D(s) . 0, then y D(t) . 0 on (s,v)T, for all s [ T.
Monotone solutions
Lemma 9
353
Assume v(t) . 0 with v [ C2rd and assume y [ C1rd : Then
2
b
ðb
v DD ðtÞ s 2
y ðtÞv D ðtÞ
2
y ðtÞÞ þ s ðy ðtÞÞ Dt ¼ F ðtÞDt þ
;
v ðtÞ
vðtÞ
a
a
a
qffiffiffiffiffiffiffiffi
D
ðtÞ ffi
ffiffiffiffiffiffiffiffiffiffiffiffi
pyðtÞv
.
where FðtÞ :¼ y D ðtÞ vvðtÞ
s ðtÞ 2
vðtÞv s ðtÞ
ðb
D
2
ð19Þ
Proof. Consider (here we suppress arguments)
ðb
v DD s 2
D 2
ðy Þ þ s ðy Þ Dt
v
a
¼
2 s
ðb
ðyÞ
ðy D Þ2 þ
v DD Dt
v
a
2 b
2 D #
ðb"
y
y D
D 2
v
ðy Þ 2
v D Dt þ
¼
v
v
a
a
ðintegrating by partsÞ
2D
2 b
ðb
vðy Þ 2 y 2 v D D
y D
D 2
¼
v
ðy Þ 2
v Dt þ
s
vv
v
a
a
¼
ðb
a
¼
ðquotient ruleÞ
2 b
2vyy D þ mvðy D Þ2 2 y 2 v D D
y D
v
ðy Þ 2
v Dt þ
s
vv
v
a
D 2
ða
b
ðproduct ruleÞ
2 b
2yy D v D y 2 ðv D Þ2 mðy D Þ2 v D
y D
v
ðy Þ 2
þ
2
Dt þ
s
s
s
v
vv
v
v
a
D 2
2 b
ðb D 2
ðy Þ
2yy D v D y 2 ðv D Þ2
y D
s
D
v
¼
ðv 2 mv Þ 2
þ
Dt þ
s
s
s
v
v
vv
v
a
a
¼
ðb
a
¼
ðb
a
¼
2 b
v D 2 2yy D v D y 2 ðv D Þ2
y D
v
ðy Þ 2
þ
Dt þ
s
s
s
v
v
vv
v
a
ðb
a
rffiffiffiffiffiffi
2 b
2
v
yv D
y D
v
y
2 pffiffiffiffiffiffiffiffis Dt þ
s
v
v
vv
a
D
F 2 Dt þ
b
y2 D
v
:
v
a
With the aid of Lemmas 8 and 9, we can now easily prove the following theorem.
Theorem 10 If equation (18) is right-disfocal on [a,1)T and r(t) # 0 on [a,1)T and not
identically zero on any nondegenerate time scale subinterval of [a,1)T, then equation (16)
354
L. Erbe et al.
has a solution x satisfying
x D ðtÞ . 0;
xðtÞ . 0;
x DD ðtÞ . 0
on (s(a),1)T.
Proof. Let x be a solution of equation (16) satisfying
xðaÞ ¼ x D ðaÞ ¼ 0;
x DD ðaÞ . 0:
The claim is that x(t) . 0, x D(t) . 0, and x DD(t) . 0 on (s(a),1)T. Assume this is not the
case. Then there is a first b [ (s(a),v)T such that x DD(b) # 0. From Lemma 8, using
x D(a) ¼ 0, we get
ðb
g
D
s
rðtÞðx ðtÞx ðtÞÞ Dt ¼
ðb
a
a
¼
ðb
b
ðx DD ðtÞÞ2 2 pðtÞðx Ds ðtÞÞ2 Dt 2 x D ðtÞx DD ðtÞ a
ðx DD ðtÞÞ2 2 pðtÞðx Ds ðtÞÞ2 Dt 2 x D ðbÞx DD ðbÞ:
a
Since equation (16) is right-disfocal, it is easy to see that there is a solution v of equation
(16) satisfying
v D ðtÞ . 0;
vðtÞ . 0;
t [ ½a; bT :
Then by Lemma 9, with y(t):x D(t), we have that
ðb
ðb
DD 2
g
D
s
rðtÞx ðtÞx ðtÞÞ Dt ¼
ðx ðtÞÞ 2 pðtÞðx Ds ðtÞÞ2 Dt 2 x D ðbÞx DD ðbÞ
a
a
¼
ðb
ðy D ðtÞÞ2 þ
a
¼
ðb
F 2 ðtÞDt þ
a
$
¼
ðx D Þ2 D
v
v
b
v DD ðtÞ s 2
ðy
ðtÞÞ
Dt 2 x D ðbÞx DD ðbÞ
v s ðtÞ
y2 D
v
v
b
2 x D ðbÞx DD ðbÞ
a
2 x D ðbÞx DD ðbÞ
a
ðx D ðbÞÞ2 D
v ðbÞ 2 x D ðbÞx DD ðbÞ:
vðbÞ
Since the left hand side is strictly negative to get a contradiction it suffices to show that the
right hand side
D :¼
ðx D ðbÞÞ2 D
v ðbÞ 2 x D ðbÞx DD ðbÞ $ 0:
vðbÞ
Monotone solutions
355
We know that x DD(b) # 0. If x DD(b) ¼ 0, then
D¼
ðx D ðbÞÞ2 D
v ðbÞ $ 0:
vðbÞ
Next assume that x DD(b) , 0. In this case r(b) , b and since x DD(r(b)) , 0 implies
x (b) . 0 we get that
D
D :¼
ðx D ðbÞÞ2 D
v ðbÞ 2 x D ðbÞx DD ðbÞ $ 0
vðbÞ
and this is the desired contradiction.
A
Acknowledgement
This research was supported by the Australian Research Council’s Discovery Project
DP0450752.
References
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equations, Annali di Mat. Pura ed Applicata, 100, 373–391.
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