Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 979705, 27 pages
doi:10.1155/2011/979705
Research Article
Lyapunov Stability of Quasilinear Implicit Dynamic
Equations on Time Scales
N. H. Du,
1
N. C. Liem,
1
C. J. Chyan,
2
and S. W. Lin
2
1
Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai,
Hanoi, Vietnam
2
Department of Mathematics, Ta mkang University, 151 Ying Chuang Road, Tamsui, Taipei County
25317, Taiwan
Correspondence should be addressed to N. H. Du, [email protected]
Received 29 September 2010; Accepted 4 February 2011
Academic Editor: Stevo Stevic
Copyright q 2011 N. H. Du et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper studies the stability of the solution x ≡ 0 for a class of quasilinear implicit dynamic
equationsontimescalesoftheformA
t
x
with A
.
being a given m × m-matrix function, has been an intensively discussed field in
both theory and practice. This problem can be seen in many real problems, such as in
electric circuits, chemical reactions, and vehicle systems. M
¨
arz in 1 has dealt with the
question whether the zero-solution of 1.1 is asymptotically stable in the Lyapunov sense
with ft, x
t,xt Bxtgt, x
t,xt,withA being constant a nd small perturbation
g.
Together with the theory of DAEs, there has been a great interest in singular difference
equation SDEalso referred to as descriptor systems, implicit difference equations
A
n
x
n 1
f
n, x
n 1
,x
t
,x
t
, 1.3
with t in time scale
and Δ being the derivative operator on .When , 1.3 is 1.1;if
, we have a similar equation to 1.2 if it is rewritten under the form A
n
xn1−xn
−A
n
xnfn, xn 1,xn; n ∈ .
The purpose of this paper is to answer the question whether results of stability for
1.1 and 1.2 can be extended and unified for the implicit dynamic equations of the form
1.3. The main tool to study the stability of this implicit dynamic equation is a generalized
direct Lyapunov method, and the results of this paper can be considered as a generalization
of 1.1 and 1.2.
The organization of this paper is as follows. In Section 2, we present shortly some
basic notions of the analysis on time scales and give the solvability of Cauchy problem for
quasilinear implicit dynamic equations
A
t
x
Δ
B
Journal of Inequalities and Applications 3
supplemented by inf ∅ sup
and ρtsup{s ∈ : s<t} supplemented by
sup ∅ inf
.Thegraininess μ : →
∪{0} is given by μtσt − t.Apointt ∈
is said to be right-dense if σtt, right-scattered if σt >t, left-dense if ρtt, left-scattered if
ρt <t,andisolated if t is right-scattered and left-scattered. For every a, b ∈
,bya, b,we
mean the set {t ∈
: a t b}.Theset
k
is defined to be if does not have a left-scattered
maximum; otherwise, it is
without this left-scattered maximum. Let f be a function defined
on
,valuedin
m
. We say that f is delta differentiable or simply: differentiable at t ∈
k
provided there exists a vector f
Δ
t ∈
m
, c alled the derivative of f,suchthatforall>0
there is a neighborhood V around t with fσt − fs − f
Δ
tσt − s |σt − s|
for all s ∈ V .Iff is differentiable for every t ∈
rd-continuous function. Then, for any t
0
∈
k
, the initial value problem (IVP)
x
Δ
A
t
x q
t
,x
t
0
x
0
2.1
has a unique solution x· defined on t
t
0
. Further, if A
t
is regressive, this solution exists on t ∈ .
The solution of the corresponding matrix-valued IVP X
Δ
A
t
X, XsI always
Δ
t
t, g
t
1
0
V
x
σ
t
,g
t
hμ
t
g
x
t, g
t
hμ
t
g
Δ
t
,g
Δ
t
dh,
2.2
where V
x
is the deriva tive in the second variable of the function V V t, x in normal
meaning and ·, · is the scalar product.
We refer to 12, 15 for more information on the analysis on time scales.
k
,
m×m
and ft, x is rd-continuous in t, x ∈ ×
m
.Inthecasewhere
the matrices A
t
are invertible for every t ∈ , we can multiply both sides of 2.3 by A
−1
t
to
obtain an ordinary dynamic equation
x
Δ
A
−1
t
B
t
x A
−1
t
f
t, x
,t∈ ,
2.5
k
,
m×m
.LetQ
t
be a projector onto ker A
t
satisfying Q
.
∈ C
rd
k
,
m×m
.Wecan
find such operators T
t
and Q
t
by the following way: let matrix A
t
possess a singular value
decomposition
A
t
U
t
Σ
t
, we can choose the matrix V
t
to be in C
rd
k
,
m×m
see 16. Hence, by putting Q
t
V
t
diagO, I
m−r
V
t
and T
t
V
ρt
V
−1
t
,weobtainQ
t
and V
t
− B
t
T
t
Q
t
is nonsingular;
iii
m
kerA
ρt
⊕ S
t
, for all t ∈ .
Journal of Inequalities and Applications 5
Proof. i⇒ii Let t ∈
and x ∈
m
such that A
t
− B
t
T
t
Q
t
x 0 ⇔ B
t
T
t
.Thus,x Q
t
x
0, that is, the matrix G
t
A
t
− B
t
T
t
Q
t
is nonsingular.
ii⇒iii It is obvious that x I T
t
Q
t
G
−1
t
B
t
x−T
t
Q
t
G
−1
t
t
T
t
Q
t
G
−1
t
B
t
x A
t
G
−1
t
B
t
x A
t
G
−1
t
B
t
x ∈ imA
t
.
Thus, I T
t
Q
t
z A
t
P
t
z and since x ∈ ker A
ρt
, T
−1
t
x ∈ ker A
t
. Therefore, T
−1
t
x Q
t
T
−1
t
x.
Hence, A
t
− B
t
T
t
Q
t
T
t
G
−1
t
A
t
, 2.8
2
Q
t
−G
−1
t
B
t
T
t
Q
t
, 2.9
3
Q
t
: −T
t
t
P
ρt
, 2.11
b
Q
t
G
t
−1
B
t
Q
t
G
−1
t
B
t
P
ρt
− T
t
−1
Q
ρt
, 2.12
P
t
A
t
,weget2.8.
2 From B
t
T
t
Q
t
A
t
− G
t
, it follows G
−1
t
B
t
T
t
Q
t
P
t
− I −Q
t
. Thus, we have 2.9.
3
−1
t
B
t
−T
t
Q
t
G
−1
t
B
t
Q
t
and A
ρt
Q
t
−A
ρt
T
t
Q
t
G
G
−1
t
B
t
Q
ρt
P
t
G
−1
t
B
t
T
t
T
−1
t
Q
ρt
−P
t
G
−1
t
A
t
− B
so we have 2.11. Finally,
Q
t
G
−1
t
B
t
Q
t
G
−1
t
B
t
P
ρt
Q
t
G
−1
t
B
t
T
t
Q
t
T
−1
−1
t
Q
ρt
Q
t
G
−1
t
B
t
P
ρt
− Q
t
T
−1
t
Q
ρt
Q
t
G
−1
t
B
t
P
ρt
− T
t
.DenoteG
t
A
t
−
B
t
T
t
Q
t
. It is easy to see that
T
t
Q
t
G
−1
t
G
t
T
t
Q
t
T
t
Q
t
T
t
Q
t
T
t
Q
t
T
t
Q
t
. 2.16
Therefore, T
t
Q
t
G
−1
t
− f
t, w
L
t
w − w
, ∀w, w
∈
m
, 2.17
where
γ
t
: L
t
T
t
and
Q
ρt
Δ
is rd-continuous. For each t ∈
k
,wehaveP
ρt
xt
Δ
P
ρσt
x
Δ
tP
ρt
Δ
xt.
Therefore,
A
t
x
Δ
t
A
t
, 2.19
and the implicit equation 2.3 canberewrittenas
A
t
P
ρt
x
Δ
A
t
P
ρt
Δ
B
t
x f
t, x
,t∈
k
. 2.20
Thus, we should look for solutions of 2.3 from the space C
is differentiable at every t ∈
k
. 2.21
Note that C
1
N
does not depend on the choice of the projector function sinc e the relations
P
t
P
t
P
t
and P
t
P
t
P
t
are true for each two projectors P
t
and P
t
along the space ker A
t
.
Journal of Inequalities and Applications 7
We now describe shortly the decomposition technique for 2.3 as follows.
Since 2.3 has index-1 and by virtue of Lemma 2.2, we see that the matrices G
t
f
t, x
,
0 Q
t
G
−1
t
B
t
x Q
t
G
−1
t
f
t, x
.
2.22
Therefore, by using the results of Lemma 2.3,weget
P
ρt
x
P
ρt
Δ
T
t
Q
t
G
−1
t
P
t
G
−1
t
f
t, x
,
Q
ρt
x T
t
Q
P
ρt
Δ
I T
t
Q
t
G
−1
t
B
t
u P
t
G
−1
t
B
t
u
P
ρt
Δ
T
t
G
−1
t
f
t, u v
.
2.25
For fixed u ∈
m
and t ∈
k
, we consider a mapping C
t
:imQ
ρt
→ im Q
ρt
given by
C
t
v
: T
t
Q
t
T
t
Q
t
G
−1
t
f
t, u v
− f
t, u v
T
t
Q
t
G
−1
t
B
t
u T
t
Q
t
G
−1
t
f
t, u g
t
u
,
2.28
and it is easy to see that g
t
u is rd-continuous in t.
u − u
T
t
Q
t
G
−1
t
f
t, u g
t
u
u − u
L
t
T
t
Q
t
G
−1
t
u − u
g
γ
t
1 − γ
t
−1
L
−1
t
L
t
B
t
u − u
.
2.30
Thus, g
t
Q
t
G
−1
t
B
t
u P
t
G
−1
t
B
t
u
P
ρt
Δ
T
t
Q
t
G
−1
t
P
t
Q
t
G
−1
t
B
t
P
t
G
−1
t
B
t
L
t
1 δ
t
P
,x
0
, t t
0
.
Thus, we get the following theorem.
Theorem 2.7. Let Hypothesis 2.5 and the assumptions on the projector Q
t
be satisfied. Then, 2.3
with the initial condition
P
ρt
0
x
t
0
− x
0
0 2.33
has a unique solution. This solution is expressed by
x
t
x
where utut; t
0
,x
0
is the solution of 2.31 with ut
0
P
ρt
0
x
0
.
We now describe the solution space of the implicit dynamic equation 2.3.Denote
Ł
t
x ∈
m
: Q
ρt
x T
t
Q
t
G
−1
t
B
.
2.35
Lemma 2.8. There hold the following statements:
iŁ
t
Ω
t
,
ii If ft, 00 for all t ∈
then Ω
t
∩ ker A
ρt
{0}.
Proof. i Let y ∈ Ł
t
,thatis,Q
ρt
y T
t
Q
t
G
−1
t
B
t
P
ρt
G
−1
t
f
t, y
. 2.36
Journal of Inequalities and Applications 9
Hence,
B
t
y f
t, y
B
t
I T
t
Q
t
G
−1
t
B
t
P
t
P
ρt
y
I B
t
T
t
Q
t
G
−1
t
f
t, y
I B
t
T
t
Q
t
G
−1
t
−1
t
A
t
G
−1
t
,
2.38
it yields
B
t
y f
t, y
A
t
G
−1
t
B
t
P
ρt
y f
t, y
t
Q
t
G
−1
t
f
t, y
,
2.40
or equivalently,
y T
t
Q
t
G
−1
t
f
t, y
T
t
Q
t
G
−1
P
ρt
y P
ρt
y
T
t
Q
t
G
−1
t
f
t, y
T
t
Q
t
G
−1
t
B
t
y − T
t
Q
t
G
−1
t
B
t
Q
ρt
y P
ρt
y
T
t
Q
t
G
−1
t
A
t
z − T
t
Q
t
G
−1
t
B
t
Q
ρt
y P
t
Q
ρt
y P
ρt
y Q
ρt
y P
ρt
y y,
2.42
where we have already used a result of Lemma 2.3 that
Q −T
t
Q
t
G
−1
t
B
t
is a projector onto
ker A
ρt
.SoŁ
t
Ω
t
.
Q
t
G
−1
t
ft, yT
t
Q
t
G
−1
t
ft, Q
ρt
y.Fromthe
assumption ft, 00, it follows that Q
ρt
y L
t
T
t
Q
t
G
−1
t
Q
ρt
y γ
t
P
ρt
0
and A
ρt
0
P
ρt
0
A
ρt
0
, the initial condition 2.33 is
equivalent to the condition A
ρt
0
xt
0
A
ρt
0
x
0
. This implies that the initial condition is
xP
ρt
x
0
g
t
P
ρt
x
0
x
0
. This means that there exists a solution of 2.3 passing
x
0
∈ Ł
t
.
2.3. Quasilinear Implicit Dynamic Equations
Now we consider a quasilinear implicit dynamic equation of the form
A
t
x
Δ
f
t, x
,
2.43
x
t, x
T
t
Q
t
2.44
is invertible for every t ∈
and x ∈
m
.
Denote
S
t, x
z ∈
m
,f
x
t, x
z ∈ imA
t
for x ∈ Ω
t
, 2.47
where T
x
is the tangent space of Ω
t
at the point x.
Suppose that 2.43 is of index-1. Then, by Lemma 2.2, this condition is equivalent to
one of the following conditions:
Journal of Inequalities and Applications 11
1 St, x ⊕ N
ρt
m
,
2 St, x ∩ N
ρt
{0}.
3 Let B
t
∈
m×m
be a matrix such that the matrix G
t
A
t
− B
t
T
t, x
T
t
Q
t
G
t
B
t
− f
x
t, x
T
t
Q
t
I
B
t
T
t
Q
t
G
−1
t
2.49
is invertible.
Lemma 2.10. Suppose that the bounded linear operator triplet:
: X → Y, : Y → Z, : Z →
X is given, where X, Y, Z are Banach spaces. Then the operator I −
is invertible if and only if
I −
is invertible.
Proof . See 17, Lemma 1.
By virtue of 2.49 and Lemma 2.10,wegetthat
I T
t
Q
t
G
−1
t
B
t
− f
Δ
u v
P
t
G
−1
t
f
t, u v
,
0 T
t
Q
t
G
−1
t
f
t, u v
.
2.51
Consider the function
k
x
t, u v
h,
2.53
where h ∈ Q
ρt
m
.
12 Journal of Inequalities and Applications
Let h ∈ Q
ρt
m
be a vector satisfying T
t
Q
t
G
−1
t
f
x
t, u vh 0. Paying attention to
T
t
Q
t
G
− f
x
t, u v
h. 2.54
Therefore, by 2.50 we get h 0. This means that ∂k/∂vt, u, v|
Q
ρt
m
is an isomorphism
of Q
ρt
m
. By the implicit function theorem, equation kt, u, v0hasauniquesolutionv
g
t
u.Moreover,thefunctionv g
t
u is continuous in t, u and continuously differentiable
in u. Its derivative is
∂g
t
u
∂u
−1
t
f
x
t, u g
t
u
|
P
ρt
m
.
2.55
Then, by substituting v g
t
u into the first equation of 2.51 we come to
u
Δ
P
ρt
Δ
x
0
2.57
is locally uniquely solvable and the solution xt; t
0
,x
0
of 2.43 with the initial condition
2.33 can be expressed by xt; t
0
,x
0
ut; t
0
,x
0
g
t
ut; t
0
,x
0
.
Now suppose further that ft, x satisfies the Lipschitz condition in x and we can find
amatrixB
t
such that
T
|
P
ρt
m
2.58
is bounded for all t ∈
and x ∈
m
. Then, the right-hand side of 2.56 also satisfies the
Lipschitz condition. Thus, from the global existence theorem see 12, 2.56 with the initial
condition 2.57 has a unique solution defined on t
0
, sup .
Therefore, we have the following theorem.
Theorem 2.11. Given an index-1 quasilinear implicit dynamic equation 2.43, then there holds the
following.
(1) Equation 2.43 is locally solvable, that is, for any t
0
∈
k
, x
0
∈
m
, there exists a unique
solution xt; t
0
,x
0
T
t
Q
t
G
−1
t
f
x
t, x
|
P
ρt
m
2.59
Journal of Inequalities and Applications 13
is bounded, then this solution is defined on t
0
, sup and we have the expression
x
t; t
0
,x
0
u
x
0
.
Remark 2.12. 1 We note that the expression T
t
Q
t
G
−1
t
B
t
depends only on choosing the
matrix B
t
.
2 The assumption that T
t
Q
t
G
−1
t
f
x
t, x|
Q
ρt
m
0
.Hence,T
t
Q
t
G
−1
t
ft, x
0
0.
Therefore, by the same argument as in Section 2 .2, we can prove that for every x
0
∈ Ω
t
,there
is a unique solution passing through x
0
.
3. Stability Theorems of Implicit Dynamic Equations
For the reason of our purpose, in this section we suppose that is an upper unbounded time
scale, that is, sup
∞.Forafixedτ ∈ ,denote
τ
{t ∈ ,t τ}.
Consider an implicit dynamic equation of the form
A
t
x
Δ
τ
, 3.1 with the initial condition
A
ρt
0
x
t
0
− x
0
0 3.2
has a unique solution defined on
t
0
. The condition ensuring the existence of a unique
solution can be refered to Section 2. We denote the solution with the initial condition 3.2
by xtxt; t
0
,x
0
. Remember that we look for the solution of 3.1 in the space C
1
N
k
, there exists a positive δ
δt
0
, such that A
ρt
0
x
0
<δresp., P
ρt
0
x
0
<δ implies xt; t
0
,x
0
<for all
t
t
0
,
2 A-uniformly resp., P-uniformly stable if it is A-stable resp., P -stable and the
number δ mentioned in the part 1. of this definition is independent of t
0
,
3 A-asymptotically resp., P-asymptotically stable if it is stable and for each t
0
4 A-uniformly globally asymptotically resp., P-uniformly globally asymptotically
stable if for any δ
0
> 0 there exist functions δ·, T· such that A
ρt
0
x
0
<δ
resp., P
ρt
0
x
0
<δ implies xt; t
0
,x
0
<for a ll t t
0
and if A
ρt
0
x
0
<δ
0
xt
0
− x
0
0satisfiesxt; t
0
,x
0
NP
ρt
0
x
0
e
−α
t, t
0
,t
t
0
,t∈
τ
. If the constant N can be chosen independent of t
0
, then this solution is
called P-uniformly exponentially stable.
Remark 3.2. From G
−1
t
,φ
0
0,φis strictly increasing; a>0
, 3.3
and
φ is the domain of definition of φ.
Proposition 3.3. The trivial solution x ≡ 0 of 3.1 is A-uniformly (resp., P -uniformly) stable if and
only if there exists a function ϕ ∈
such that for each t
0
∈
k
τ
and any solution xt; t
0
,x
0
of 3.1
the inequality
x
t; t
0
,x
0
0
x
0
∀t
t
0
, 3.4
holds, provided A
ρt
0
x
0
∈ ϕ (resp., P
ρt
0
x
0
∈ ϕ).
Proof. We only need to prove the proposition for the A-uniformly stable case.
Sufficiency. Suppose there exists a function ϕ ∈
satisfying 3.4 for each >0; we
take δ δ > 0suchthatϕδ <,thatis,ϕ
−1
>δ.Ifxt; t
ρt
0
x
0
<δ
implies xt; t
0
,x
0
<,forallt t
0
. For the sake of simplicity in computation, we choose
δ <.Denote
γ
sup
δ
: δ
has such a property
. 3.6
Journal of Inequalities and Applications 15
By putting
β
:
1
0
γ
t
dt, 3.7
it is seen that
β ∈
, 0 <β
<γ
. 3.8
Let ϕ : 0, sup β →
,x
0
0 does not imply that x·; t
0
,x
0
≡
0. Consider the case where
t
> 0. If A
ρt
0
x
0
<β
t
, then by the relations 3.6 and 3.8
we have xs; t
0
,x
0
<
t
, ∀s t
0
.Inparticular,xt; t
0
,x
0
x
0
.
The proposition is proved.
Similarly, we have the following proposition.
Proposition 3.4. The trivial solution x ≡ 0 of 3.1 is A-stable (resp., P -stable) if and only if for each
t
0
∈
k
τ
and any solution xt; t
0
,x
0
of 3.1 there exists a function ϕ
t
0
∈ such that there holds the
following:
x
t; t
0
,x
0
ϕ
t
0
x
0
∀t
t
0
, 3.9
provided A
ρt
0
x
0
∈ ϕ
t
0
(resp., P
ρt
0
x
0
∈ ϕ
t
0
).
t, A
ρt
x
1
0
V
x
σ
t
,A
ρt
x hμ
t
A
ρt
x
Δ
and t ∈
τ
,
2 V
Δ
3.10
t, A
ρt
x c/1 − cμtV t, A
ρt
x, for any x ∈ Ω
t
and t ∈
k
τ
.
16 Journal of Inequalities and Applications
Assume further that 3.1 is locally solvable. Then, 3.1 is globally solvable, that is, every solution
with the initial condition 3.2 is defined on
t
0
.
Proof. Denote
W
t, x
V
t, x
t
,t
0
− cV
t, A
ρt
x
e
−c
t, t
0
c
1 − cμ
t
V
t, A
ρt
x
1 − cμ
x
t
− W
t
0
,A
ρt
0
x
t
0
t
t
0
W
Δ
3.10
τ, A
ρτ
W
t
0
,A
ρt
0
x
t
0
V
t
0
,A
ρt
0
x
t
0
3.14
or
−1
V
t
0
,A
ρt
0
x
t
0
e
c/1−cμt
t, t
0
.
3.15
The last inequality says that the solution xt can be lengthened on
t
0
,thatis,3.1 is globally
solvable.
.
Assume further that 3.1 is locally solvable. Then the trivial solution of 3.1 is stable.
Proof. By virtue of Theorem 3.6 and the conditions 2 and 3, it follows that 3.1 is globally
solvable. Suppose on the contrary that the trivial solution x ≡ 0of3.1 is not stable. Then,
there exists an
0
> 0suchthatforallδ>0thereexistsasolutionxt of 3.1 satisfying
A
ρt
0
xt
0
<δand xt
1
; t
0
,xt
0
0
for some t
1
t
0
.Put
1
ψ
0
.
0
for some t
1
t
0
.
Since xt ∈ Ω
t
and by the condition 3 ,
t
1
t
0
V
Δ
3.10
t, A
ρt
x
t
Δt V
t
1
,A
xt
1
V t
0
,A
ρt
0
xt
0
<
1
.Further,xt
1
∈ Ω
t
1
and by the condition
2 we have V t
1
,A
ρt
1
xt
1
ψxt
1
ψ
x → 0
V t, x0 uniformly in t ∈
τ
,
2 ψx
V t, A
ρt
x for all x ∈ Ω
t
and t ∈
τ
,
3 V
Δ
3.10
t, A
ρt
x −δtφA
ρt
x for any x ∈ Ω
t
and t ∈
k
τ
.
Further, 3.1 is locally solv able. Then the trivial solution of 3.1 is asymptotically stable.
Proof. Also from Theorem 3.6 and the conditions 2 and 3, it implies that 3.1 is globally
solvable.
And since V
Δ
t
V
t
0
,A
ρt
0
x
t
0
t
t
0
V
Δ
3.10
s, A
ρs
x
A
ρs
x
s
Δs V
t
0
,x
t
0
− φ
r
t
t
0
δ
s
Δs −→ − ∞ ,
s
V
Δ
3.10
τ, A
ρτ
x
τ
Δτ 0.
3.19
18 Journal of Inequalities and Applications
This means that V t, A
ρt
xt is a decreasing function. Consequently,
lim
t →∞
V
t, A
ρt
x
t
inf
τ
and ax V t, A
ρt
x for all x ∈ Ω
t
and
t ∈
τ
,
2 V
Δ
3.10
t, A
ρt
x 0, for any x ∈ Ω
t
and t ∈
k
τ
.
Assume further that 3.1 is locally solvable. Then, the trivial solution of 3.1 is A-uniformly stable.
Proof. The proof is similar to the one of Theorem 3.7 with a remark that since lim
x → 0
V t, x
0 uniformly in t ∈
τ
,wecanfindδ
0
> 0suchthatify <δ
0
, a defined on 0, ∞, and a function V ∈ C
rd
τ
×
m
,
satisfying
1 ax
V t, A
ρt
x bA
ρt
x for all x ∈ Ω
t
and t ∈
τ
,
2 V
Δ
3.10
t, A
ρt
x −cA
ρt
x for any x ∈ Ω
t
and t ∈
k
. 3.21
T is not necessary in
.
Let xt be a solution of 3.1 with A
ρt
0
xt
0
<δ. From the condition 2,wesee
that
V
t, A
ρt
x
t
− V
t
0
,A
ρt
0
x
t
V
t, A
ρt
x
t
V
t
0
,A
ρt
0
x
t
0
b
∗
∈ t
0
,t
0
T such that A
ρt
∗
xt
∗
<δ. Assume that such a t
∗
does not
exist, that is A
ρt
xt δ for all t ∈ t
0
,t
0
T. From the condition 2,weget
V
t
0
T
,A
Δs
V
t
0
,A
ρt
0
x
t
0
b
A
ρt
0
x
t
0
b
δ
0
c
δ
, 3.25
which contradicts the definition of T in 3.21. The proof is complete.
When A
ρt
is not differentiable, one supposes that there exists a Δ-differentiable
projector Q
t
onto ker A
t
and Q
ρt
Δ
is rd-continuous on
k
τ
;moreover,Q
ρt
Q
t
and ker A
ρt
and the matrix G
t
A
t
− B
t
T
t
Q
t
is invertible. Define
V
Δ
3.26
t, P
ρt
x
V
Δ
t
t, P
ρt
x
x
Δ
dh,
3.26
where P
ρt
x
Δ
P
ρt
Δ
x P
t
G
−1
t
ft, xsee 2.51.
From now on we remain following the above assumptions on the operators Q
t
,T
t
,B
t
whenever V
Δ
3.26
t, P
k
τ
.
Assume further that 3.1 is locally solvable. Then, 3.1 is globally solvable.
20 Journal of Inequalities and Applications
Theorem 3.13. Assume that 3.1 is locally solvable. Then, the trivial solution x ≡ 0 of 3.1 is stable
if there exist a function V :
τ
×
m
→
being rd-continuous and a function ψ ∈ , ψ defined on
0, ∞ such that
1 V t, 0 ≡ 0 for all t ∈
τ
,
2 V t, P
ρt
y ψy for all y ∈ Ω
t
and t ∈
τ
,
3 V
Δ
3.26
t, P
ρt
x 0 for all x ∈ Ω
0
,forsomet
1
t
0
.Let
1
ψ
0
.SinceV t
0
, 00, it is possible to find
a δ δ
0
,t
0
> 0 satisfying V t
0
,P
ρt
0
z <
1
when P
ρt
0
z <δ,z∈
t, P
ρt
x
t
Δt V
t
1
,P
ρt
1
x
t
1
− V
t
0
,P
ρt
0
ψ
x
t
1
ψ
0
1
. 3.28
We get a contradiction because
1
>Vt
0
,P
ρt
0
x
0
when P
ρt
0
k
τ
,
then the trivial solution of 3.1 is P-uniformly stable.
Proof. The proof is similar to the one of Theorem 3.9.
Theorem 3.15. If there exist functions a, b, c ∈ , a defined on 0, ∞ and a function V ∈ C
rd
τ
×
m
,
satisfying
1 ax
V t, P
ρt
x bP
ρt
x for all x ∈ Ω
t
and t ∈
τ
,
2 V
Δ
3.10
t, P
ρt
x −cP
0
,x
0
of 3.1,wehave
x
t; t
0
,x
0
ϕ
P
ρt
0
x
0
∀t
t
0
, 3.29
provided P
where xs; t, z is the unique solution of 3.1 satisfying the initial condition P
ρt
xtP
ρt
z.
It is seen that V is defined for all z satisfying P
ρt
0
z∈ ϕ, V t, 0 ≡ 0, and V t, x ∈
C
rd
τ
×
m
,
.
Let y ∈ Ω
t
. By the definition, Vt, P
ρt
ysup
s t
xs; t, P
ρt
y xt; t, P
ρt
y.
ysup
s t
xs; t, P
ρt
y and
V
σ
t
,P
ρσt
x
σ
t
,t,P
ρt
y
sup
s σt
x
s; σ
ρt
y
.
3.31
This implies
V
Δ
3.26
t, P
ρt
y
t
V
σ
t
,P
ρσt
x
σ
3.33
22 Journal of Inequalities and Applications
where A and B are constant matrices with ind A, B1, ft, 00 ∀t ∈
,andft, x
satisfing the Lipschitz condition
f
t, x
− f
t, y
<L
x − y
, 3.34
where L is sufficiently small. Let Q be defined by 2.9 with T
t
I and G A − BQ, P
I − Q.ByTheorem 2.7, we see that there exists a unique solution satisfying the condition
Pxt
0
−1
BP.
Note that the general solution of 3.35 is
x
t; t
0
,x
0
e
M
t, t
0
Px
t
0
exp
tM
⎛
⎝
s∈I
t,t
0
is denoted the set of right-scattered points of the interval t
0
,t.
Denote σA, B{λ :detλA−B0}. It is easy to show that the trivial solution x ≡ 0
of 3.35 is P-uniformly exponentially stable if and only if σA, B ⊂ S,whereS is the domain
of uniform exponential stability of
. On the exponential stable domain of a time scale, we
can refer to 10, 18, 19. By the definition of exponential stability, it implies that the graininess
function of the time scale
is upper bounded. Let μ
∗
sup
t∈
μt.
We denote the set
U
⎧
⎪
⎨
⎪
⎩
λ :
λ
1
∞
k0
I μ
∗
M
n
P
FP
I μ
∗
M
n
Q
FQ, 3.39
where the matrix F is supposed to be symmetric positive definite. It is clear that H is
symmetric positive definite.
Journal of Inequalities and Applications 23
Since σA, B ⊂ U, the above series is convergent. Further, for any k
0wehave
I μ
∗
I μ
∗
M
k1
P
FP
I μ
∗
M
k1
−
I μ
∗
M
k
I μ
μ
∗
I μ
∗
M
k
P
FP
×
I μ
∗
M
k
M μ
∗
M
I μ
∗
M
k
FP
I μ
∗
M
n
k0
μ
∗
I μ
∗
M
k
P
FP
I μ
∗
M
k
M
n
P
FPI μ
∗
M
n
0, we obtain
−P
FP
I μ
∗
M
HM M
H HM M
H μ
∗
M
HM. 3.42
In the case where μ
∗
3.33 is
V
Δ
3.26
Px
Px
Δ
H
Px
σ
Px
H
Px
Δ
Mx PG
−1
f
t, x
Mx PG
−1
f
t, x
HPx μ
t
Mx PG
−1
f
t, x
HPx μ
∗
Mx PG
−1
f
t, x
H
Mx PG
−1
f
t, x
Px
H
× H
Px μ
∗
Mx μ
∗
PG
−1
f
t, x
Px
H
I μ
∗
M
PG
−1
f
Px
H
I μ
∗
M
PG
−1
f
t, x
−
Px
FPx
PG
−1
f
PG
−1
f
t, x
.
3.44
From the Lipschitz condition and 2.25, it is seen that Qx
KPx where K QG
−1
B
LQG
−1
/1 − LQG
−1
. Therefore,
f
t, x
L
1 K
Px
x f
t, x
,
3.47
with
A
t
t 1
10
00
,B
t
−t − 20
0 −t − 1
,f
t, x
sin x
00
. Let us choose T
t
I.Weseethat
G
t
A
t
− B
t
T
t
Q
t
t 1
10
01
. 3.49
Since t
0, det G
t
t 1
2
/
0
∈ ,
3.47 with the initial condition P
ρt
0
xt
0
P
ρt
0
x
0
has the unique solution.
It is easy to compute, G
−1
t
1/t 1
10
01
, T
t
Q
t
G
−1
t
t
B
t
−1/t1
t20
00
,andP
t
G
−1
t
ft, x
0, 0
.
Therefore, utP
ρt
xt satisfies u
Δ
−1/t 1
t20
00
u. M oreover, we have
Ł
t
∈ Ł
t
,wehaveV t, P
ρt
x2P
ρt
x 2|x
1
| and
x
x
2
1
x
2
2
1/2
x
2
1
V
t, P
ρt
x
2
P
ρt
x
, ∀x ∈ Ł
t
,t∈ . 3.52
We have for any solution xt of 3.47 and t ∈
noting that t 0,
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