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CUBO A Mathematical Journal
Vol.17, No¯ 03, (53–70). October 2015

Gronwall-Bellman type integral inequalities and
applications to global uniform asymptotic stability

Mekki Hammi and Mohamed Ali Hammami

University of Sfax,

Faculty of Sciences of Sfax, Department of Mathematics,

Route Soukra, BP 1171, 3000 Sfax, Tunisia,

mohamedali.hammami@fss.rnu.tn

ABSTRACT

In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral inequal-

ities are established. These inequalities can be used as handy tools to research stability

problems of perturbed dynamic systems. As applications, based on these new estab-

lished inequalities, some new results of practical uniform stability are also given. A

numerical example is presented to illustrate the validity of the main results.

RESUMEN

En este art́ıculo, establecemos algunas desigualdades integrales nolineales nuevas de

tipo Gronwall-Bellman. Estas desigualdades pueden ser usadas como herramientas

utiles para estudiar problemas de estabilidad de sistemas dinámicos perturbados. Como

aplicaciones, basados en las nuevas desigualdades establecidas, también damos algunos

resultados nuevos de estabilidad uniforme prácticos. Un ejemplo numérico es presen-

tado para ilustrar la validez de los resultados principales.

Keywords and Phrases: Gronwall-Bellman inequality, perturbed systems, stability.

2010 AMS Mathematics Subject Classification: 26D15, 26D20, 34A40, 34H15.



54 Mekki Hammi & Mohamed Ali Hammami CUBO
17, 3 (2015)

1 Introduction

In 1919, T.H. Gronwall [6] proved a remarkable inequality which has attracted and continues

to attract considerable attention in the literature. In the qualitative theory of differential, the

Gronwall type inequalities of one variable for the real functions play a very important role. The

first use of the Gronwall inequality to establish boundedness and uniqueness is due to R. Bellman

[1] . Gronwall-Bellman inequality, which is usually proved in elementary differential equations using

continuity arguments, is an important tool in the study of of qualitative behavior of solutions of

differential and stability.

The problem of stability analysis of nonlinear time-varying systems has attracted the attention

of several researchers and has produced a vast body of important results (see [2]-[15] and the

references therein). In this paper, we present a new generalization of the Gronwall- Bellman lemma.

This new generalization can develop a simple command to exponentially stabilize a large class of

nonlinear systems. In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral

inequalities are given. As applications, we give some new classes of time-varying perturbed systems

which are globally uniformly practically asymptotically stable. Moreover, we give an example to

illustrate the applicability of the results.

2 Definitions and notations

We consider the following system :

ẋ(t) = f(t, x(t)), x(t0) = x0, (1)

where t ∈ R+ is the time and x ∈ Rn is the state.

Definition 1. (uniform boundedness).

A solution of (1) is said to be globally uniformly bounded if for every α > 0 there exists c = c(α)

such that, for all t0 ≥ 0,
‖ x0 ‖≤ α ⇒‖ x(t) ‖≤ c, ∀t ≥ t0.

Let r ≥ 0 and Br = {x ∈ Rn/ ‖ x ‖≤ r}. First, we give the definition of uniform stability and
uniform attractivity of Br.

Definition 2. (uniform stability of Br).

i. Br is uniformly stable if for all ε > r, there exists δ = δ(ε) > 0 such that for all t0 ≥ 0,

‖ x0 ‖≤ δ ⇒‖ x(t) ‖≤ ε, ∀t ≥ t0.

ii. Br is globally uniformly stable if it is uniformly stable and the solutions of system (1) are globally

uniformly bounded.



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Gronwall-Bellman type integral inequalities and applications . . . 55

Definition 3. (uniform attractivity of Br).

Br is globally uniformly attractive if for all ε > r and c, there exists T(ε, c) > 0 such that for all

t0 ≥ 0,
‖ x(t) ‖≤ ε, ∀t ≥ t0 + T(ε, c), ‖ x0 ‖≤ c

The system (1) is globally uniformly practically asymptotically stable if there exists r ≥ 0
such that Br is globally uniformly stable and globally uniformly attractive.

Definition 4. A continuous function α : [0, a) → [0, +∞) is said to belong to class K if it is
strictly increasing and α(0) = 0. It is said to belong to class K∞ if a = +∞ and α(r) → +∞ as
r → +∞.

Definition 5. A continuous function β : [0, a) × [0, +∞) → [0, +∞) is said to belong to class KL
if, for each fixed s, the mapping β(r, s) belongs to class K with respect to r and, for each r, the

mapping β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → +∞.

Proposition 1. If there exists a class K-function α, and a constant r > 0 such that, given any

initial state x0, the solution satisfies

‖ x(t) ‖≤ α(‖ x0 ‖) + r ∀t ≥ t0,

then the system (1) is globally uniformly practically stable.

Proposition 2. If there exist a class KL-function β, a constant r > 0 such that, given any initial

state x0, the solution satisfies

‖ x(t) ‖≤ β(‖ x0 ‖, t − t0) + r ∀t ≥ t0,

then the system (1) is globally uniformly practically asymptotically stable.

The next definition concerns a special case of practical global uniform asymptotic stability,

namely, if the class KL in the above proposition consists of functions β(r, s) = kre−γs.

Definition 6. Br is globally uniformly exponentially stable if there exist γ > 0 and k ≥ 0 such
that for all t0 ∈ R+ and x0 ∈ Rn,

‖ x(t) ‖≤ k ‖ x0 ‖ exp(−γ(t − t0)) + r ∀t ≥ t0.

System (1) is globally practically uniformly exponentially stable if there exist r > 0 such that

Br is globally uniformly exponentially stable.

3 Basic results

Lemma 1. Let u, v and w nonnegative piecewise continuous functions on [0, +∞) for which the
inequality

u(t) ≤ c +
∫t

a

(uv + w) ∀t ≥ a (2)



56 Mekki Hammi & Mohamed Ali Hammami CUBO
17, 3 (2015)

holds, where a and c are nonnegative constants. Then,

u(t) ≤ ce

∫t

a

v
+ re

∫t

a

(v +
w

r
)

∀t ≥ a, ∀r > 0. (3)

Proof

It follows from (2) and the classic inequality

ex > x + 1 ∀x > 0

that for all r > 0 and t ≥ a
0 ≤ u(t) < (c + re

∫
t
a

w
r ) +

∫t

a

uv (4)

which implies that
u(t)

c + re
∫
t
a

w
r +

∫t
a
uv

≤ 1.

Since v ≥ 0, we obtain
u(t)v(t) + w(t)e

∫
t
a

w
r

c + re
∫
t
a

w
r +

∫t
a
uv

≤ v(t) + w(t)e
∫
t
a

w
r

c + re
∫
t
a

w
r

(5)

then we take

f(t) =

∫t

a

v + log(c + re
∫
t
a

w
r ) − log(c + re

∫
t
a

w
r +

∫t

a

uv) ∀t ≥ a.

It is clear that f is defined, continuous and piecewise continuously differentiable on [a, +∞). Con-
sequently, we get for all b > a, a sequence {a0, ..., an} of [a, b] verifying

f′(t) = v(t) +
w(t)e

∫
t
a

w
r

c + re
∫
t
a

w
r

−
w(t)e

∫
t
a

w
r + u(t)v(t)

c + re
∫
t
a

w
r +

∫t
a
uv

∀t ∈ [a, b] − {a0, ..., an}.

By using the inequality (5), we obtain

f′(t) ≥ 0.

Thus, f is increasing on the intervals [a, a0), ...(an, b]. Since f is continuous on [a, b], then f is

increasing on [a, b]. Consequently, we get

f(b) ≥ f(a)

however, f(a) = 0 which implies that

f(b) ≥ 0 ∀b ≥ a.

Consequently

log(c + re
∫
t
a

w
r +

∫t

a

uv) ≤
∫t

a

v + log(c + re
∫
t
a

w
r ) ∀t ≥ a



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Gronwall-Bellman type integral inequalities and applications . . . 57

hence

c + re
∫
t
a

w
r +

∫t

a

uv ≤ (c + re
∫
t
a

w
r )e

∫
t
a
v

by using the inequality (4), we have finally

u(t) ≤ ce
∫
t
a
v + re

∫
t
a
(v+ w

r
).

Lemma 2. Let φ ∈ Lp(R+, R+) where p ∈ [1, +∞]. We denote by ‖ φ ‖p the p-norm of φ. Then,
for all t0 ≥ 0, s ≥ 0 and t ≥ t0 ∫t

t0

φ ≤ N + L(t − t0) (6)

where N =
∫s
0
φ + Ms

p
and L = p−1

p
Ms with Ms =‖ φ|[s,+∞) ‖p .

Proof

We first consider the case p ∈ (1, +∞). By using Hölder inequality to the function φ, we obtain
for all t ≥ t0 :

∫t

t0

φ ≤ (
∫t

t0

φp)
1
p (

∫t

t0

1)
p−1
p

≤ (t − t0)
p−1
p (

∫+∞

t0

φp)
1
p .

We put

f(x) =
1

p
+

p − 1

p
x − x

p−1
p ∀x > 0

then, f is differentiable on (0, +∞) and verifying

f′(x) =
p − 1

p
(1 − x

− 1
p ).

Hence, f is decreasing on (0, 1] and increasing on [1, +∞). Since f(1) = 0, we conclude that f is
positive on (0, +∞) which means that

x
p−1
p ≤ 1

p
+

p − 1

p
x ∀x > 0

consequently, we have

(t − t0)
p−1
p ≤

1

p
+

p − 1

p
(t − t0) ∀t ≥ t0

then

0 ≤
∫t

t0

φ ≤ Mt0[
1

p
+

p − 1

p
(t − t0)] (7)

where Mt0 =‖ φ|[t0,+∞) ‖p . This inequality holds also for p ∈ {1, +∞}.
Now, for all t0 ≥ 0, s ≥ 0 and t ≥ t0, we distingue three cases :



58 Mekki Hammi & Mohamed Ali Hammami CUBO
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• s ≤ t0 ≤ t
In view of (7), we obtain

∫t

t0

φ ≤ Mt0[
1

p
+

p − 1

p
(t − t0)]

≤ Ms
p

+
p − 1

p
(t − t0)Ms.

Now, since
∫s
0
φ ≥ 0, we obtain

∫t

t0

φ ≤ (
∫s

0

φ +
Ms

p
) +

p − 1

p
(t − t0)Ms.

• t0 < s ≤ t
We can write by using (7)

∫t

t0

φ ≤
∫s

t0

φ +

∫t

s

φ

≤
∫s

0

φ + [
1

p
+

p − 1

p
(t − s)]Ms.

then ∫t

t0

φ ≤
∫s

0

φ +
Ms

p
+

p − 1

p
(t − s)Ms

however, s ∈ (t0, t] then
∫t

t0

φ ≤ (
∫s

0

φ +
Ms

p
) +

p − 1

p
(t − t0)Ms.

• t0 ≤ t < s
It is clear that

∫t

t0

φ ≤
∫s

0

φ

≤ (
∫s

0

φ +
Ms

p
) +

p − 1

p
(t − t0)Ms.

then the lemma is proved.

Lemma 3. Consider the differential system

ẋ(t) = A(t)x(t) + h(t, x(t)) (8)

where :

i. A is an n× n matrix whose entries are all real-valued piecewise-continuous functions of t ∈ R+.

ii. The function h is defined on R+ × Rn, piecewise continuous in t, and locally Lipshitz in x.



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Gronwall-Bellman type integral inequalities and applications . . . 59

iii. There exist φ and ε piecewise continuous functions, positives and verifying

‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ R+. (9)

Then, for all (t0, x0) ∈ R+×Rn, there exist a unique maximal solution x of (8) such that x(t0) = x0.
Moreover, x is defined on [t0, +∞).

Proof

It is clear that the system (8) can be written

ẋ(t) = f(t, x(t))

where

f(t, x) = A(t)x + h(t, x).

The function f is piecewise continuous in t and locally Lipshitz in x, then we have : For all

(t0, x0) ∈ R+ × Rn, there exist a unique maximal solution x of (8) such that x(t0) = x0.
We will prove that x is defined on [t0, +∞). Supposed that is not true, then there exist Tmax ∈
(t0, +∞) such that x is defined on [t0, Tmax). Then, for all t ∈ [t0, Tmax)

‖ ẋ(t) ‖≤ (M1 + M2) ‖ x(t) ‖ +M3

where

M1 = sup
[t0,Tmax]

‖ A(t) ‖,

M2 = sup
[t0,Tmax]

‖ φ(t) ‖

and

M3 = sup
[t0,Tmax]

‖ ε(t) ‖ .

It is clear that M1, M2 and M3 ∈ R+, therefore

‖
∫t

t0

ẋ(s)ds ‖ ≤
∫t

t0

‖ ẋ(s) ‖ ds

≤
∫t

t0

[(M1 + M2) ‖ x(s) ‖ +M3]ds

then

‖ x(t) ‖≤‖ x(t0) ‖ +
∫t

t0

[(M1 + M2) ‖ x(s) ‖ +M3]ds

By using the lemma 1, we obtain for all t ∈ [t0, Tmax)

‖ x(t) ‖ ≤ ‖ x(t0) ‖ e
∫
t
t0

(M1+M2)ds
+ e

∫
t
t0

(M1+M2+M3)ds

≤ M4



60 Mekki Hammi & Mohamed Ali Hammami CUBO
17, 3 (2015)

where

M4 =‖ x(t0) ‖ e(M1+M2)Tmax + e(M1+M2+M3)Tmax.

Consequently, x remains within the compact BM4, which is impossible. So, we conclude that

Tmax = +∞.

Theorem 1. Consider the following time-varying :

ẋ(t) = A(t)x(t) + h(t, x(t)) (10)

where :

(1) A is an n × n matrix whose entries are all real-valued piecewise-continuous functions of
t ∈ R+.

(2) The transition matrix for the system

ẋ = A(t)x

satisfies :

‖ R(t, s) ‖≤ ke−γ(t−s) ∀(t, s) ∈ R2+ (11)

for some k > 0 and γ > 0.

(3) The function h is defined on R+ × Rn, piecewise continuous in t, and locally Lipshitz in x.

(4) There exist φ and ε piecewise continuous functions, positives and verifying

‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ R+. (12)

(5) φ ∈ Lp(R+, R+) for some p ∈ [1, +∞).

(6) There exist a constant M > 0 such that

ε(t) ≤ Me−γt. (13)

Then for all (t0, x0) ∈ R+ × Rn, the maximal solution x of (10) such that x(t0) = x0, is verifying :

i. The function x is defined on [t0, +∞).

ii. For all t ≥ t0
‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + Ne−θt

where N, L > 0 and δ, θ ∈ (0, γ].



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Gronwall-Bellman type integral inequalities and applications . . . 61

Proof of theorem 1

i. By using the lemma 3, we proved that the system (10) has a unique maximal solution x such

that x(t0) = x0. Moreover, x is defined on [t0, +∞).
ii. We can write the solution x of (10) as

x(t) = R(t, t0)x(t0) +

∫t

t0

R(t, s)h(s, x(s))ds

where R(t, t0) is the transition matrix of the system

ẋ = A(t)x.

Further, we have :

‖ x(t) ‖ ≤ ‖ R(t, t0) ‖‖ x(t0) ‖ +
∫t

t0

‖ R(t, s) ‖‖ h(s, x(s)) ‖ ds

≤ ke−γ(t−t0) ‖ x0 ‖ +
∫t

t0

ke−γ(t−s) ‖ h(s, x(s)) ‖ ds.

From the inequalities (11) and (12), we deduce that

u(t) ≤ ku(t0) +
∫t

t0

[kφ(s)u(s) + keγsε(s)]ds

where

u(t) = eγt ‖ x(t) ‖ .

Now by the lemma 1, we get

u(t) ≤ ku(t0)e

∫t

t0

kφ
+ re

∫t

t0

[kφ(s) +
keγsε(s)

r
]ds

∀t ≥ t0, ∀r > 0

since

‖ x(t) ‖= e−γtu(t)

we obtain the estimation

‖ x(t) ‖≤ k ‖ x0 ‖ e

∫t

t0

kφ − γ(t − t0)
+ re

∫t

t0

[kφ(s) +
keγsε(s)

r
]ds − γt

. (14)

Let us denote

M = sup
t≥0

[eγtε(t)] and Ms = (

∫+∞

s

φp)
1
p

we deduce from the assumptions 5 and 6, that

M, Ms ∈ R+



62 Mekki Hammi & Mohamed Ali Hammami CUBO
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it follows that ∫t

t0

keγsε(s)

r
ds ≤

kM

r
t ∀t ≥ t0. (15)

Moreover φ ∈ Lp(R+, R+), then ∫+∞

t

φp −−−−→
t→+∞

0

and so there exist s ≥ 0 such that
Ms <

γ

k

p

p − 1
.

By using the lemma 2, we find for all t ≥ t0
∫t

t0

φ ≤
∫s

0

φ +
Ms

p
+ Ms

p − 1

p
(t − t0) (16)

from (15) and (16), we get :

∫t

t0

kφ − γ(t − t0) ≤ k(
∫s

0

φ +
Ms

p
) + [kMs

p − 1

p
− γ](t − t0)

and ∫t

t0

[kφ(s) +
keγsε(s)

r
]ds − γt ≤ [−γ + kMs

p − 1

p
+

kM

r
]t + k(

∫s

0

φ +
Ms

p
).

Thus, (14) yields

‖ x(t) ‖≤ kek(
∫
s
0
φ+

Ms
p

) ‖ x0 ‖ e−[γ−kMs
p−1
p

](t−t0) + re
−[γ−kMs

p−1
p

− kM
r

]t+k(
∫
s
0
φ+

Ms
p

)
.

Taking

r >
M

γ
k
−

p−1
p

Ms

L = ke
k(

Ms
p

+
∫
s
0
φ)

N = re
k(

Ms
p

+
∫
s
0
φ)

=
r

k
L

δ = γ − k
p − 1

p
Ms ∈ (0, γ]

θ = γ − k
p − 1

p
Ms −

kM

r
∈ (0, δ).

Finally, we obtain

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + Ne−θt ∀t ≥ t0.

Corollary 1. Under the same assumptions of theorem 1, we get

∀r > 0, ∀t ≥ t0, ∀x0 ∈ Rn \ Br :

‖ x(t) ‖≤ P ‖ x0 ‖ e−θ(t−t0)

where P > 0 and θ ∈ (0, γ).



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Gronwall-Bellman type integral inequalities and applications . . . 63

Proof

Due to theorem 1, we have

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + Ne−θt ∀t ≥ t0.

Let r > 0, then for all x0 ∈ Rn \ Br

‖ x(t) ‖ ≤ L ‖ x0 ‖ e−δ(t−t0) +
N

r
re−θ(t−t0)

≤ (L + N
r
) ‖ x0 ‖ e−θ(t−t0).

Taking

P = L +
N

r
> 0,

we obtain

‖ x(t) ‖≤ P ‖ x0 ‖ e−θ(t−t0).

Remark 1. Take limit as r →
M

γ
k
−

p−1
p

Ms
in theorem 1, we obtain

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + N ∀t ≥ t0 (17)

with

N =
M

γ
k
−

p−1
p

Ms
e
k(

Ms

p
+

∫s

0

φ)
.

In particular, if we choose p = 1, we find

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + N ∀t ≥ t0 (18)

with

L = kek‖φ‖1

and

N =
kM

γ
ek‖φ‖1.

The estimation (17) and (18) implies that the system (10) is globally uniformly practically asymp-

totically stable in the sense that the ball BN is globally uniformly asymptotically stable.

Theorem 2. Consider the following time-varying :

ẋ(t) = A(t)x(t) + h(t, x(t)) (19)

where :

(1) A is an n × n matrix whose entries are all real-valued piecewise-continuous functions of
t ∈ R+.



64 Mekki Hammi & Mohamed Ali Hammami CUBO
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(2) The transition matrix for the system

ẋ = A(t)x

satisfies :

‖ R(t, s) ‖≤ ke−γ(t−s) ∀(t, s) ∈ R2+ (20)

for some k > 0 and γ > 0.

(3) The function h is defined on R+ × Rn, piecewise continuous in t, and locally Lipshitz in x.

(4) There exist φ and ε piecewise continuous functions, positives and verifying

‖ h(t, x) ‖≤ φ(t) ‖ x ‖ +ε(t) ∀t ∈ R+. (21)

(5) sup
[s,+∞)

φ <
γ

k
for some s ∈ [0, +∞).

(6) There exist a constant M > 0 such that

ε(t) ≤ Me−γt. (22)

Then for all (t0, x0) ∈ R+ × Rn, the maximal solution x of (10) such that x(t0) = x0, is verifying :

i. The function x is defined on [t0, +∞).

ii. For all t ≥ t0
‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + Ne−θt

where N, L > 0 and δ, θ ∈ (0, γ].

Proof of theorem 2

i. By using the lemma 3, we proved that the system (19) has a unique maximal solution x such

that x(t0) = x0. Moreover, x is defined on [t0, +∞).
ii. Similar to the proof of theorem 1, it can be shown that :

‖ x(t) ‖≤ k ‖ x0 ‖ e

∫t

t0

kφ − γ(t − t0)
+ re

∫t

t0

[kφ(s) +
keγsε(s)

r
]ds − γt

∀t ≥ t0, ∀r > 0.

Let us denote

M = sup
t≥0

[eγtε(t)] ∈ R+

it follows that ∫t

t0

keγsε(s)

r
ds ≤

kM

r
t ∀t ≥ t0.

Hence, there exist s ∈ R+ such that
sup

[s,+∞)

φ <
γ

k



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Gronwall-Bellman type integral inequalities and applications . . . 65

then we can apply the lemma 2, we deduce that

∫t

t0

φ ≤
∫s

0

φ + ( sup
[s,+∞)

φ)(t − t0) ∀t ≥ t0

consequently, we obtain

‖ x(t) ‖≤ ke
k

∫s

0

φ
‖ x0 ‖ e

−[γ − k sup
[s,+∞)

φ](t − t0)

+ re

−[γ − k sup
[s,+∞)

φ −
kM

r
]t + k

∫s

0

φ

.

Taking

r >
M

γ

k
− sup

[s,+∞)

φ

L = kek
∫
s
0
φ

N = rek
∫
s
0
φ)

=
r

k
L

δ = γ − k sup
[s,+∞)

φ ∈ (0, γ]

θ = γ − k sup
[s,+∞)

φ −
kM

r
∈ (0, δ).

Finally, we obtain

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) + Ne−θt ∀t ≥ t0.

Corollary 2. Under the assumptions (1),(2),(3),(4) and (6) of theorem 2, and by replacing the

condition (5) by

(5′) : φ(t) −−−−→
t→+∞

0

then, we obtain the same consequences of theorem 2.

Proof

Since lim
t→+∞

φ(t) = 0 , then there exist s ≥ 0 such that

∀t ≥ s φ(t) ≤ γ
2k

therefore

sup
[s,+∞)

φ <
γ

k
.

Thus, we can apply theorem 2 to prove the result.

Remark 2. It is clear that if we choose ε = 0 in theorem 1 or 2, we obtain due to M = 0 :

θ = δ = γ − k
p − 1

p
Ms



66 Mekki Hammi & Mohamed Ali Hammami CUBO
17, 3 (2015)

L = ke
k(

Ms
p

+
∫
s
0
φ)

N = re
k(

Ms
p

+
∫
s
0
φ) ∀r > 0

as r → 0+, we get the classic result:

‖ x(t) ‖≤ L ‖ x0 ‖ e−δ(t−t0) ∀t ≥ t0.

We can see that the claim of the theorem 1 is true by examining a specific example, where a

solution of the scalar equation can be found.

Example 1. Consider the stability of following system :






ẋ1 = −x1 − tx2 +
1

(1 + t2)2
x21

1 +
√

x2
1
+ x2

2

+
e−2t

1 + x2
1

ẋ2 = tx1 − x2 +
t

(1 + t2)2
x22

1 +
√

x21 + x
2
2

(23)

which can be writing as

ẋ = A(t)x + h(t, x)

where

X =

(

x1

x2

)

,

A(t) =

(

−1 −t

t −1

)

and

h(t, x) =

(

h1(t, x)

h2(t, x)

)

with 




h1(t, x) =
1

(1 + t2)2
x21

1 +
√

x21 + x
2
2

+
e−2t

1 + x2
1

h2(t, x) =
t

(1 + t2)2
x22

1 +
√

x2
1
+ x2

2

it is clear that the system

ẋ = A(t)x



CUBO
17, 3 (2015)

Gronwall-Bellman type integral inequalities and applications . . . 67

is globally uniformly asymptotically stable. Indeed, the transition matrix R(t, t0) satisfies :

R(t, t0) = e
(t−t0)A

= e−(t−t0)

(

cos t − sin t

sin t cos t

)

thus, we obtain

‖ R(t, t0) ‖= ke−γ(t−t0)

with γ = k = 1 and ‖ ‖ represents the euclidean norm. On the other hand,

‖ h(t, x) ‖ = h21(t, x) + h22(t, x)

≤ 1
(1 + t2)3

(x21 + x
2
2) + 2e

−2t.

By using the classic inequality

√

a2 + b2 ≤ a + b ∀a, b ≥ 0

we get

‖ h(t, x) ‖≤ φ(t) ‖ x(t) ‖ +ε(t) ∀t ≥ 0

where

φ(t) =
1

(1 + t2)
3
2

and

ε(t) =
√
2e−t.

It is easy to verify that φ and ε are continuous, positive and bounded on [0, +∞), in particular

φ ∈ Lp(R+, R+) ∀p ∈ [1, +∞].

To estimate ‖ φ ‖p, we use the inequality :

φp(t) ≤ φ(t) ∀t ≥ 0

since ‖ φ ‖∞= 1, then ∫+∞

0

φp ≤
∫+∞

0

φ

however
∫+∞
0

φ = 1, then

‖ φ ‖p≤ 1 ∀p ≥ 1.

Consequently ‖ φ ‖p< pp−1 ∀p ≥ 1, and we can apply theorem 1 to prove the following results :
• ∀(t0, x0) ∈ R+ × R2, there exist a unique maximal solution x of (8) such that x(t0) = x0.
Moreover, x is defined on [t0, +∞).
• ∀t ≥ t0, ∀p ≥ 1

‖ x(t) ‖≤ e
1
p ‖ x0 ‖ e−

1
p
(t−t0) + 2

√
2e

− 1
2p

t+ 1
p



68 Mekki Hammi & Mohamed Ali Hammami CUBO
17, 3 (2015)

by choosing r = 2
√
2p.

In particular

‖ x(t) ‖≤ e ‖ x0 ‖ e−(t−t0) + 2
√
2e. (24)

The estimation (24) implies that the system (23) is globally uniformly practically asymptotically

stable in the sense that the ball B
2
√
2e

is globally uniformly asymptotically stable.

0 5 10 15 20 25 30 35 40 45 50
−0.5

0

0.5

1

1.5

2

Time (s)

X
1

Figure 1:

Time evolution of the state x1(t) of system (23)



CUBO
17, 3 (2015)

Gronwall-Bellman type integral inequalities and applications . . . 69

0 5 10 15 20 25 30 35 40 45 50
−0.5

0

0.5

1

1.5

2

Time (s)

X
2

Figure 2:

Time evolution of the state x2(t) of system (23)

Received: September 2013. Accepted: February 2015.

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Netherlands, (1964).


	Introduction
	Definitions and notations
	Basic results