BOUNDEDNESS IN NONLINEAR DIFFERENTIAL
EQUATIONS
YOUSSEF N. RAFFOUL
Abstract. Non-negative definite Lyapunov functions are employed
to obtain sufficient conditions that guarantee boundedness of solutions of a nonlinear differential system. The theory is illustrated
with several examples.
1. Introduction
In this paper, we make use of non-negative definite Lyapunov functions
and obtain sufficient conditions that guarantee the boundedness of all
solutions of the initial value problem
ẋ = f (t, x), t ≥ 0,
(1.1)
x(t0 ) = x0 , t0 ≥ 0
where x ∈ Rn , f : R+ × Rn → Rn is a given nonlinear continuous
function in t and x, where t ∈ R+ . Here Rn is the n-dimensional
Euclidean vector space; R+ is the set of all non-negative real numbers;
kxk is the Euclidean norm of a vector x ∈ Rn . Our interest in studying
the boundedness of solutions of (1.1) arises from the fact that in the
study of the stability of the zero solution of (1.1) when f (t, 0) = 0, one
may have to assume the existence of solutions for all positive time. For
more on the stability, we refer the interested reader to [1] and [6]. In
[2], the authors obtained , using positive definite Lyapunov functionals,
interesting results regarding the boundedness of solutions of systems
that are similar to (1.1). For more results regarding the stability of
the zero solution or boundedness of solutions of (1.1), we refer the
interested reader to [4], [5], [7] and [9]. In the spirit of the work in
[3] and [6], in this investigation, we establish sufficient conditions that
yield all solutions of (1.1) are bounded. We achieve this by assuming
the existence of a Lyapunov function that is bounded below and above
and that its derivative along the trajectories of (1.1) to be bounded by
a negative definite function, plus a positive constant.
1991 Mathematics Subject Classification. 34C11, 34C35, 34K15.
Key words and phrases. Nonlinear differential system, Boundedness; Lyapunov
functions.
1
2
YOUSSEF N. RAFFOUL
2. Boundedness of Solutions
In this section we use non-negative Lyapunov type functions and establish sufficient conditions to obtain boundedness results on all solutions
x(t) of (1.1). From this point forward, if a function is written without
its argument, then the argument is assumed to be t.
Definition 2.1 We say that solutions of system (1.1) are bounded, if
any solution x(t, t0 , x0 ) of (1.1) satisfies
³
´
||x(t, t0 , x0 )|| ≤ C ||x0 ||, t0 , for all t ≥ t0 ,
where C : R+ × R+ → R+ is a constant that depends on t0 and x0 .
We say that solutions of system (1.1) are uniformly bounded if C is
independent of t0 .
If x(t) is any solution of system (1.1), then for a continuously differentiable function
V : R+ × Rn → R+ ,
we define the derivative V 0 of V by
n
V 0 (t, x) =
∂V (t, x) X ∂V (t, x)
+
fi (t, x).
∂t
∂x
i
i=1
Theorem 2.2 Let D be a set in Rn . Suppose there exist a continuously
differentiable Lyapunov function V : R+ × D → R+ that satisfies
λ1 ||x||p ≤ V (t, x) ≤ λ2 ||x||q ,
(2.1)
V 0 (t, x) ≤ −λ3 ||x||r + L
(2.2)
and
for some positive constants λ1 , λ2 , λ3 , p, q, r and L. Moreover, if for
some constant γ ≥ 0 the inequality
V (t, x) − V r/q (t, x) ≤ γ
(2.3)
holds, then all solutions of (1.1) that stay in D are uniformly bounded.
r/q
Proof Let M = λ3 /λ2 . For any initial time t0 , let x(t) be any
solution of (1.1) with x(t0 ) = x0 . Then,
´ h
i
d³
V (t, x(t))eM (t−t0 ) = V 0 (t, x(t)) + M V (t, x(t)) eM (t−t0 ) .
dt
BOUNDED SOLUTIONS
3
For x(t) ∈ Rn , using (2.2) we get
´ h
i
d³
M (t−t0 )
r
V (t, x(t))e
≤ − λ3 ||x|| + L + M V (t, x(t)) eM (t−t0 ) .(2.4)
dt
From condition (2.1) we have ||x||q ≥ V (t, x)/λ2 .
Consequently, −||x||r ≤ −[ V (t,x(t))
]r/q . In addition, if we use (2.3) inλ2
equality (2.4) becomes
i
´
h
d³
r/q
M (t−t0 )
r/q
V (t, x(t))e
≤
− (λ3 /λ2 )V (t, x(t)) + M V (t, x(t)) eM (t−t0 )
dt
h
i
= M (V (t, x(t)) − V r/q (t, x(t))) + L eM (t−t0 )
³
´
= M γ + L eM (t−t0 )
= : KeM (t−t0 ) .
Integrating the above inequality from t0 to t we obtain,
K M (t−t0 ) K
e
−
M
M
K
≤ λ2 ||x0 ||q + eM (t−t0 ) .
M
V (t, x(t))eM (t−t0 ) ≤ V (t0 , x0 ) +
Consequently,
V (t, x(t)) ≤ λ2 ||x0 ||q e−M (t−t0 ) +
K
.
M
From condition (2.1) we have λ1 ||x||p ≤ V (t, x(t)), which implies that
1 1/p h
K i p1
}
λ2 ||x0 ||q +
for all t ≥ t0 .
λ1
M
This completes the proof.
||x|| ≤ {
♦
Example 2.3 Consider the semi-linear differential equation
x0 = σx + Rx1/3 .
If
2σ +
4
= −α for some α > 0,
3
(2.5)
4
YOUSSEF N. RAFFOUL
then all solutions of (2.5) are uniformly bounded.
To see this, let V (t, x) = x2 . Then along solutions of (2.5) we have
V 0 (t, x) = 2xx0
= 2σx2 + 2Rx4/3
≤ 2σx2 + 2|R|x4/3 .
(2.6)
To further simplify (2.6), we make use of Young’s inequality, which
says for any two nonnegative real numbers w and z, we have
wz ≤
we z f
+ , with 1/e + 1/f = 1.
e
f
Thus, for e = 3/2 and f = 3, we get
h1
(x4/3 )3/2 i
2|R|x4/3 ≤ 2 |R|3 +
3
3/2
4 2 4 3
x + |R| .
=
3
3
By substituting the above inequality into (2.6), we arrive at
³
4´ 2 2 3
x + |R|
3
3
2
= −αx2 + |R|3 .
3
V 0 (t, x) ≤
2σ +
One can easily check that conditions (2.1)-(2.3) of Theorem 2.2 are
satisfied with λ1 = λ2 = 1, λ3 = α, L = 32 |R|3 and p = q = r = 2.
Hence all solutions of (2.5) are uniformly bounded.
♦
In Example 2.3, condition (2.3) did not come into play, which was due
to the fact that r = q = 2. In the next example, we consider a nonlinear
system in which condition (2.3) naturally comes into play.
Example 2.4 Let D = {x ∈ R : ||x|| ≥ 1} and consider the nonlinear
differential equation
5
x0 = − x3 + Rx1/3 , t ≥ 0,
6
x(0) = 1.
BOUNDED SOLUTIONS
5
Consider the Lyapunov functional V (t, x) : R+ × D → R+ such that
V (t, x) = x2 . Then along solutions of the differential equation we have
V 0 = 2xx0
5
= − x4 + 2Rx4/3
3
5
≤ − x4 + 2|R|x4/3
3
Using Young’s inequality with e = 3 and f = 3/2, we get
|x|4/3 |R| ≤
(2.8)
x4 2 3/2
+ |R| .
3
3
Hence
4
V 0 (t, x) ≤ −x4 + |R|3/2 .
3
4
3/2
We take λ1 = λ2 = 1, λ3 = 1 , L = 3 |R| , p = q = 2 and r = 4.
For x ∈ D
V (t, x) − V r/q (t, x) = x2 (1 − x2 ) ≤ 0.
Thus, condition (2.3) is satisfied for γ = 0. An application of Theorem
2.2 yields
1 1/p h
K i p1
q
|x(t)| ≤ { }
λ2 ||x0 || +
λ1
M
L 1
= (1 + ) 2
M
1
4
= (1 + |R|3/2 ) 2 for all t ≥ 0.
3
Hence, every solution x with x(t) ∈ D satisfies
1
4
1 ≤ |x(t)| ≤ (1 + |R|3/2 ) 2 , for t ≥ 0.
3
♦
In the next theorem we show that solutions of (1.1) are bounded.
Theorem 2.5 Suppose there exist a continuously differentiable Lyapunov function V : R+ × Rn → R+ that satisfies
λ1 (t)||x||p ≤ V (t, x) ≤ λ2 (t)||x||q ,
(2.9)
and
V 0 (t, x) ≤ −λ3 (t)||x||r + L
(2.10)
for some positive constants p, q, r, L, and positive continuous functions
λ1 (t), λ2 (t), and λ3 (t), where λ1 (t) is nondecreasing. Moreover, if for
some constant γ ≥ 0 the inequality
6
YOUSSEF N. RAFFOUL
V (t, x) − V r/q (t, x) ≤ γ
holds, then all solutions of (1.1) are bounded.
(2.11)
Proof Let
λ3 (t)
.
t∈R [λ2 (t)]r/q
For any initial time t0 , let x(t) be any solution of (1.1) with x(t0 ) = x0 .
By calculating
´
d³
V (t, x(t))eM (t−t0 )
dt
and then by a similar argument as in Theorem 2.2 we obtain,
K
V (t, x(t)) ≤ λ2 (t0 )||x0 ||q e−M (t−t0 ) + .
(2.12)
M
Since λ1 (t) is nondecreasing and by using (2.7), we arrive at
M = inf+
V (t, x(t)) 1/p
V (t, x(t)) 1/p
} ≤{
} .
(2.13)
λ1 (t)
λ1 (t0 )
Combining (2.10) and (2.11) we get
h
1
K i p1
||x|| ≤ {
}1/p λ2 (t0 )||x0 ||q +
for all t ≥ t0 .
λ1 (t0 )
M
||x|| ≤ {
This completes the proof.
♦
The next theorem does not require an upper bound on the Lyapunov
function.
Theorem 2.6 Suppose there exist a continuously differentiable Lyapunov function V : R+ × Rn → R+ that satisfies
λ1 ||x||p ≤ V (t, x),
(2.14)
and
V 0 (t, x) ≤ −λ2 V (t, x) + L
for some positive constants λ1 , λ2 , p and L.
Then, all solutions of (1.1) are bounded.
(2.15)
Proof For any initial time t0 , let x(t) be any solution of (1.1) with
x(t0 ) = x0 . By calculating
´
d³
V (t, x(t))eλ2 t ,
dt
and making use of (2.15) we obtain
BOUNDED SOLUTIONS
´
d³
λ2 t
V (t, x)e
≤ Leλ2 t
dt
An integration of the above inequality from t0 to t yields
L
V (t, x) ≤ V (t0 , x0 )e−λ2 (t−t0 ) + .
λ2
Using (2.14) into the above inequality, we have
h
1
L i p1
||x|| ≤ { }1/p V (t0 , x0 ) +
for all t ≥ t0 .
λ1
λ2
This completes the proof.
7
♦
In the next theorem our Lyapunov function satisfies a different type of
condition on its derivative along the solutions.
Theorem 2.7 Suppose there exist a continuously differentiable Lyapunov function V : R+ × Rn → R+ that satisfies
λ1 ||x||p ≤ V (t, x), V (t, x) 6= 0 if x 6= 0
(2.16)
and
V 0 (t, x) ≤ −λ2 (t)V q (t, x)
(2.17)
for some positive constants λ1 , p, q > 1 where λ2 (t) is a positive continuous function such that
c1 = inf λ2 (t) > 0.
t≥t0 ≥0
(2.18)
Then all solutions of (1.1) are bounded.
Proof For any initial time t0 ≥ 0, let x(t) be any solution of (1.1)
with x(t0 ) = x0 . By calculating
d
V (t, x(t))
dt
along the solutions of (1.1) and making use of (2.17) and (2.18), we
obtain
V 0 (t, x(t)) ≤ −λ2 (t)V q (t, x(t))
≤ −c1 V q (t, x(t)).
Denoting u(t) = V (t, x(t)), we have that this function satisfies the
following differential inequality
u̇(t) ≤ −c1 [u(t)]q ,
and, therefore its positive solutions satisfy
u̇(t)
≤ −c1 .
[u(t)]q
8
YOUSSEF N. RAFFOUL
An integration of the above inequality from t0 to t yields
h
i−1/(q−1)
1−q
V (t, x(t)) ≤ V
(t0 , x0 ) + c1 (q − 1)(t − t0 )
.
By invoking condition (2.16) we arrive at
nh
i−1/(q−1) o1/p
1/p
1−q
||x|| ≤ 1/λ1
V
(t0 , x0 ) + c1 (q − 1)(t − t0 )
.
This completes the proof.
♦
As an application of the previous theorem, we furnish the following
example.
Example 2.8 To illustrate the application of Theorem 2.7, we consider
the following two dimensional system
y2
y10 = y2 − y1 |y1 | −
1 + y12
y1
y20 = −y1 − y2 |y2 | +
1 + y12
y1 (t0 ) = y1 (0), y2 (0) = y2 (0) for t0 ≥ 0.
Let us take V (t, y1 , y2 ) = 12 (y12 + y22 ). Then
y2 y1
y1 y2
+
V 0 (t, y1 , y2 ) = −y12 |y1 | − y22 |y2 | −
2
1 + y1 1 + y12
¡ 3
¢
= − |y1 | + |y2 |3
h |y |3 |y |3 i
1
2
= −2
+
2
2
h (|y |2 )3/2 (|y |2 )3/2 i
2
1
= −2
+
2
2
¡ 2
2 ¢3/2 −3/2
≤ −2 |y1 | + |y2 |
2
= −2V 3/2 (t, y1 , y2 )
¡ ¢l
l
l
where we have used the inequality a+b
≤ a2 + b2 , a, b > 0, l > 1.
2
Hence, by Theorem 2.7 all solutions of the above two dimensional system are uniformly bounded.
♦
References
[1] Burton, T., Volterra Integral and Differential Equations. Academic Press, New
York, 1983.
BOUNDED SOLUTIONS
9
[2] Burton, A., Quichang, H, and Mahfoud, W. E., Liapunov functionals of convolution type, J. Math. Analy. Appl. 106(1985), 249-272.
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Berlin, (1977).
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Department of Mathematics, University of Dayton, Dayton, OH
45469-2316 USA
E-mail address: [email protected]