I nternat. J. Mh. & Mh. Sci.
(1982) 105-112
Vol. 5 No.
105
STABILITY IMPLICATIONS ON THE ASYMPTOTIC
BEHAVIOR OF NONLINEAR SYSTEMS
KUO-LIANG CHIOU
Department of Mathematics
Wayne State University
Detroit, Michigan 48202
(Received July 20, 1978)
ABSTRACT
In this paper we generalize Bownds’ Theorems (I) to the systems
A(t) Y(t) and dX(t)
dt
A(t) Xt) + F(t X(t))
always exists a solution X(t)-of
o (=
t
Moreover
_
dY.(t)
dt
we also show that there
A(t)X + B(t) for which
sup
oo) if there exists a solution Y(t) for which lira
t
limt ooSUpl IX(t) II
Iy(t)
>
+
> o (= oo)
KEY WORDS AND PHRASES. stable, norm, linear systems, null solution, SchauderTychcoff Theorem, uniformly converges, equicontnu
7980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
Secondary 34AL50, 34C15, 4D20.
Primary 4D05, 54D, 4C10,
I. INTRODUCTION.
In this paper we shall study the stability behavior of the following systems
dY(t)
dt
0 < t <
A(t)Y(t)
(I.i)
and
dX(t)
dt
A(t)X(t) + F(t,X(t)),
n
where A(t) is a continuous matrix on R
valued continuous n-vector defined on
0 -< t <
for all 0 < t <
, F(t,X(t))
(1.2)
is a real
n
[0, ) X R and X(t) and Y(t) are n-vectors.
Consider special equations of (i.i) and
y" + a(t)y
0,
(1.2)
0 < t <
(1.3)
and
x" + a(t)x
g(t,x,x’),
0 -< t <
(1.4)
106
K.L. CHIOU
where
a(t) C[0,)
and
g(t,x,x’) is continuous on [0,)R
R.
From some theorems
of stability theory, Bownds [i] showed that (1.3) has a solution y(t) with property
lim sup
(ly(t) + lY’(t)I)
t_oo
> 0
(i 5)
He also established that (1.4) has the property (1.5) provided that the zero solution of (1.3) is stable and there exists a function y(t)
Ig(t,x,x’)
<
y(t)
(Ixl +
6L[0,) such that
Ix’ I)
for (t,x,x’) E [0, oo) x R x R.
Thus in the following section we shall extend the above results to systems
(i.i) and (1.2).
In section 3 we shall consider a nonhomogeneous system
dX(t)
A(t)X(t) + B(t)
dt
0 < t <
(I 6)
where B(t) is a continuous vector for 0 _< t < oo.
We shall prove that there always
lim sup
exists a solution X(t) of (i 6) for which
t+
lim sup
a solution Y(t)of (] i) for which
lly(t)
t+oo
llX(t) ll
ll
> 0(= )
> 0(= oo)
Here
if there exists
II
II
is an
appropriate vector (or matrix) norm.
2.
ASYMPTOTIC BEHAVIOR FOR (i. I) AND (i. 2).
Before stating main theorems, let us recall a theorem from Coppel
THEOREM 2.1.
(Hartman [2, p. 60]).
[2, p. 60].
Suppose that, for every solution Y(t) of
(i.i), the limit
lim
t
exists and is finite.
IY(t)
_o
(2 i)
If there exists a nontrivial solution Y(t) of (i.i) for
which the limit (2.1) is zero, then
t
t
t
r
A(s)ds
+
as t +
o
From the above theorem we will obtain the following corollary which is a generalization of Theorem i in [i].
COROLLARY 2.1.
Suppose that
t
t
r
A(s)ds <
o
Then there exists a nontrivial solution Y (t) of (I.i) for which
ASYMTOTIC BEHAVIOR OF NONLINEAR SYSTEMS
t
PROOF.
107
>0
_+o
Suppose, to the contrary, that all solutions Y(t) of (i.i) satisfy
lim sup
t
From Theorem 2.1 we obtain
t
t
rA(s)ds
/-
as t +
This leads to a contra-
o
The corollary then follows.
diction.
Throughout this paper we shall denote $(t), the fundamental matrix of (i.i)
with initial condition (0)
I (identity matrix).
Now we shall prove the following theorem via the Schauder-Tychonoff Theorem
[2, p. 9].
THEOREM 2.2.
Suppose that the null solution of (I.I) is stable and that there
exists a solution Y(t) of (i.i) for which
lim sup
t/oo
Suppose also that there exists y(t) E
stant
,
llY(t) ll
> 0
Ll[to’)
such that for some positive con-
llF(t,x) ll
<
Y(t)
llxl
(2.2)
E
(2.3)
Then there exists a nontrivial solution X(t) of (1.2) for which
lim sup
t+oo
PROOF.
llX(t) ll
> 0
Since the null solution of (i.i) is stable, there exists a positive
constant k such that
(2.4)
for all 0 < t < s and there exists a nontrivial solution Y(t) of (i.i) for which
(2.2) holds and
IIY(t) ll
for t
t
o
-< 1
(2.5)
and for given small positive constant g (<i).
Since y(t)
E
Ll[t o,),
there exists
(s)ds <
k
t
To (> to)
such that
for all t
T
o
(2.6)
Via the Schauder-Tychonoff Theorem we shall establish the existance of a solution
108
K.L. CHIOU
of the integral equation
X(t)
I
(t)
Y(t)
$-l(s)
f(s,X(s))ds
t
-> T
t
(2.7)
o
Consider the set
{U; U(t)
F
Jo [To’ )
X(t) is continuous on
IIU(t) II
and
I for t e TO }
;
and define the operator T by
Y(t)
TU(t)
-l(s)
(t)
(2.8)
f(s,U(s))ds.
t
First, we shall show that TF
Taking the norm to both sides of (2.8) and
F.
c
using (2.3), (2.4), (2.5), and (2.6), we obtain for U 6 F
IITU(t) II
IIY(t) ll
<
< i
-< 1
< 1
I
+
I
+ k
E
+
e
f
f
k
e + k
lib(t)
-l(s) f(s’U(s))llds
t
lf(s’U(s))l Ids
t
y(s)
[IU(s) llds
t
y(s)ds
t
It is clear that TU(t) is continuous on J
This proves TF
O
F.
Suppose that the sequence {U
n
We claim
in F converges uniformly to U in F on every compact subinterval of J
Second, we shall show that t is continuous.
O
that TU
n
converges uniformly to TU on every compact subinterval of J
small positive number satisfying e
T
I
>
To
so that for t e T
I
y(s)ds <
k
t
U (s))
Then using (2.8)
(2 9)
llU(t) ll-<
O
i for T
f(s
n
< t <
(2 i0)
oo,
U(s))II
(2 3)
Ll[to’)’
<
(2 4)
i
2kTl
Let
i
be a
there exists
(2.9)
such that if n -> N, then
N(I, T I)
By the uniform convergence, there is an N
lf(s
-
Since y(t)
< i.
I
o
T
< s
O
< TI
and the fact that
(2.10)
[]Un (t)l[
we obtain the following inequalities
<
I and
109
ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS
t
.T I
It
<
l(t)
(s)
I
(t)-" (s)f(s’Un(S))ds
I[
lTUn(t)- TU(t)II
l(t)-l(s)ll lf(s’Un(S))
ll lf(S,Un(S))llds
If(s,Un(S))
<k
t
f(s,U(s)>
f(s0U(s))l]ds
l(t)-1 (s) ll
+
T
(t)-l(s)f(s’U(s)dsl
1
Ids
+ 2k
7(s)ds
T
1
for n >- N.
This shows that TU
converges uniformly to TU on every compact subinterval of J o
n
Hence T is continuous.
Third, we claim that the functions in the image set TF are equicontinuous and
bounded at every point of J
o
TU(t) is a solution of the linear system
For each U F, the function z(t)
dV
dt
Since
IITU(t) II
lz(t)[i
F, it is clear that the functions in TF
Since TF
Now we show that they are equicontinuous at eact Doint of
are uniformly bonded.
J
o
1 and
any finite t interval, we see that
A(t)V + f(t,U(t))
lf(t,U(t))ll
dz
is uniformly bounded for U
E F on
is uniformly bounded on any finite interval.
Therefore, the set of all such z is equicontinuous at each point of J o (see [2, p.6]).
All of the hypotheses of the Schauder-Tychonoff Theorem are satisfied.
there exists a U
F such that U(t)
Thus
TU(t); that is, there exists a solution X(t)
of
X(t)
Y(t)
-l(s) f(s,x(s_))ds
(t)
t
Thus, from the hypotheses and the above equality, we obtain
lira sup
t-oo
Since
lim .sup
t
lly(t)
ll
> 0
’,
IX(t)
Y(t)[I
(2 Ii) implies that
(2.11)
0
lim sup
t
l[X(t) ll
the theorem.
It is clear that (1.4) can be written as the form (1.2) with
> 0
This proves
II0
K.L. CHIOU
A(t)
where X
F(t,X)
and
-a(t)
colum(x,x’).
g(t,x,x’)
0
Thus we can apply Theorem 2.2 to (1.4) to obtain the fol-
lowing corollary which is a generalization of Theorem 2 in [i].
COROLLARY 2.2.
Suppose that the null solution of (1.3) is stable and that
Ll[to’)
there exists y(t)
such that for some positive constant
Then there exists a nontrivial solution x(t) of (1.4) for which
lim sup
t-
PROOF.
Since t
r
A(t)
(Ixl +
x
I)
> 0
0 for Corollary 2.1, we know that there exists a sol-
ution Y(t)of (I.I) for which
lim sup
t
]IXI]
If we take
3.
]x[ + Ix’l,
IIY(t)II > 0
then the corollary follows from Theorem 2.2.
ASYMPTOTIC BEHAVIOR FOR (1.6)
In this section we shall show that if there exists a solution Y(t) of (I.I)
for which
for which
lim sup
t
lim sup
t
THEOREM 3.1.
ly(t)ll
> 0 (= o)
]IX(t)]i
> 0 (= )
then there exists a solution X(t) of (I 6)
Suppose that there exists a solution Y(t) of (I.i) for which
I im sup
t
Y (t)
> 0
(3.1)
Then there exists a solution X(t) of (1.6) for which
lim sup
PROOF.
:’X(t)’l > 0
From the variation of constants formula we know that any solution
X(t) of (1.6) can be written as the form below
X(t)
(t)c + (t)
Hence we shall choose c so that Y(t)
-l(s)
B(s)ds
(t)c satisfies (3.1).
First, let us suppose
lim sup
t
II
*(t)
-l(s)
g(s)dsll
> O.
(3.3)
ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS
Let
Xl(t)
X(t)
Y(t).
It is clear that
Xl(t)
iii
is a solution of
(1.6).
Thus
from (3.3) and (3.4) we obtain
lim sup
t
lim sup
llXl(t)ll
lim sup
t/
Xl(t)
Thus there exists a solution
t
iX(t)
l(t)
-I (s)
ooiim II
ll
B(s)ds[[
> 0
of (1.6) for which (3.2) holds.
Second, suppose that
t
y(t)
I
(t)
-l(s) B(s)dsll
(3.5)
0
Taking the norm to both sides of (3.3) and using (3.1) and (3.5) we obtain
t
limt sup
lX(t)ll
>_
limt oosup
(I IY(t)l]
I(t)
fO
-l(s) B(s)dsl I)
t
> lim sup
t
_> lim sup
t
lly(t)
ll
ly(t)ll
This shows that X(t) satisfies (3.2).
lim sup
t
I(t)
fO -i
(s)
> 0
The theorem then follows.
Using the same argument as Theorem 3.1 we also can obtain the following theorem.
THEOREM 3.2.
Suppose that there exists a solution Y(t) of (i.I) for which
lim sup
t
iy(t)ll
Then there exists a solution X(t) cf (1.6) for which
lim sup
t
PROOF.
Ix(t)II
Since the proof is almost the same as Theorem 3.1, we shall omit the
detail.
REMARKS.
It is interesting to note that Hatvani and Pintr [3] have studied
this type of problem for equation (1.4).
ACKNOWLEDGEMENTS.
This work was supported by the U.S. Army Research Office
Grant DAAG29-78-G-0042 at Research Triangle Park, N.C.
112
K.L. CHIOU
REFERENCES
i.
Bownds, J.M.
Stability Implications on the Asymptotic Behavior of Second
Order Differential Equations, Proc. Amer. Math. Soc. 39 (1973), 169-172,
MR 47 #2150.
2.
Coppel, W. A. Stability and Asymptotic Behavior of Differential Equations,
Heath, Boston, 1965. MR 32 7875.
3.
Hatvani, L. and L. Pintr. On Pertubation of Unstable Second Order Linear
Differential Equations, Proc. Amer. Math. Soc. 61 (1976), 36-38.