4_269611_hristova.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 1, 2012, 30-44

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA

29 JUNE - 02 JULY 2011, ISTANBUL TURKEY

Practical Stability of Impulsive Differential Equations with

“Supremum” by Integral Inequalities

Snezhana Hristova 1∗, Kremena Stefanova 2

1 Department of Applied Mathematics and Modeling, Faculty of Mathematics and Informatics, Plov-

div University, Plovdiv, Bulgaria
2 Department of Computer Technologies, Faculty of Mathematics and Informatics, Plovdiv Univer-

sity, Plovdiv, Bulgaria

Abstract. The paper deals with some stability properties of the solutions of impulsive differential

equations with “supremum”. Initially several integro-summation inequalities for piecewise continuous

functions are solved. The main characteristic of the considered inequalities is the presence of the

supremum of the unknown function over a past time interval. These inequalities are generalizations

of Bihari’s integral inequality. They are base of studying the practical stability as well as the uniform

practical stability of the solutions of nonlinear impulsive differential equations with “supremum”.

2000 Mathematics Subject Classifications: 26D99, 34A37, 34D99

Key Words and Phrases: Integral Inequality, Supremum, Practical Stability, Impulses

1. Introduction

In the last few decades great attention has been paid to automatic control systems and

their applications to computational mathematics and modeling. Many problems in the con-

trol theory correspond to the maximal deviation of the regulated quantity. Such kind of real

world problems are adequately modeled by differential equations with “maxima” [16]. In

connection with many possible applications it is absolutely necessary to be developed qual-

itative theory of differential equations with “maxima” (see the monograph [3] and papers

[2, 4, 5, 6, 10, 11]). One of the main mathematical tools, employed successfully for studying

existence, uniqueness, continuous dependence, comparison results, perturbations, bounded-

ness, and stability of solutions of differential and integral equations is the method of integral

∗Corresponding author.

Email addresses: snehri�uni-plovdiv.bg (S. Hristova), kstefanova�uni-plovdiv.bg (K. Stefanova)

http://www.ejpam.com 30 c© 2012 EJPAM All rights reserved.



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 31

inequalities. Various types of integral inequalities are solved in the papers [1, 7, 9, 13, 15,

17, 20, 21, 22]. The involvement of maximum function in the equation requires applica-

tion of a new type of integral inequalities. Additionally, if the unknown function is piecewise

continuous then so called integro-summation inequalities with supremum have to be applied.

The purpose of the paper is studying some stability properties of the solutions of impulsive

differential equations with “supremum”. The main apparatus of investigation are integral in-

equalities which contain the supremum of the unknown scalar piecewise continuous function

over a past time interval. Some nonlinear inequalities are solved and applied to investigate

some properties of the solutions of the considered equation.

2. Mathematical Model

Let {t i}∞1 be a given sequence of points such that t i < t i+1, lim
i→∞

t i =∞.

Let the points t0, T be fixed, 0≤ t0 < T ≤∞, and the following condition be satisfied:

H1 The functions σ,τ ∈ C1([t0, T ),R+) are nondecreasing, τ(t) ≤ t for t ∈ [t0, T ) and

there exists a nonnegative constant h such that the inequalities 0 ≤ τ(t) − σ(t) ≤ h

hold for t ∈ [t0, T ).

Denote by Z(t0, T ) the set of all natural numbers k such that tk ∈ (t0, T ). Consider the

following impulsive differential equation with “supremum”

x ′ = f
�

t, x(t), sup
s∈[σ(t),τ(t)]

x(s)
�

, for t ∈ [t0, T ), t 6= t i , (1)

∆x
�

�

t=ti
= Ii

�

x(t i)
�

, for i ∈ Z(t0, T ), (2)

with initial condition

x(t) = φ(t), t ∈ [τ(t0)− h, t0] (3)

where x ∈ R, ∆x
�

�

t=ti
= x(t i + 0)− x(t i − 0) for i ∈ Z(t0, T ).

Let PC(Ω,R), Ω⊂ R, be the set of all functions u : Ω→ R which are piecewise continuous

in Ω, i.e. there exist limits lim
t↓tk

u(t) = u(tk + 0)<∞ and

lim
t↑tk

u(t) = u(tk − 0) = u(tk)<∞, tk ∈ Ω.

Denote by x(t; t0,φ) the solution of the initial value problem (1)–(3)

and |φ|0 = max
s∈[τ(t0)−h,t0]

|φ(s)|.
Let the following conditions be satisfied:

H2 The function f ∈ C(R+×R×R,R), f (t, 0,0) = 0 and the inequality

| f (t, x , y)| ≤ A(t)|x |p+ B(t)|y|p for x , y ∈ R

holds, where the functions A, B ∈ C(R+,R+) and p = const > 0.



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 32

H3 The functions Ii : R→ R, Ii(0) = 0 and the inequalities

|Ii(x)| ≤ βi|x |p for x ∈ R,

hold, where βi = const > 0 for i ∈ Z(0,∞).
H4 For any point t0 ∈ R+ and any initial function φ ∈ C([τ(t0)− h, t0],R) the initial value

problem (1)–(3) has a solution x(t; t0,φ) ∈ PC([τ(t0)− h,∞),R).
The solution x(t) = x(t; t0,φ) of the initial value problem (1)–(3) satisfies the following

integral equation

x(t) = φ(t0) +
∑

t0<ti<t

Ii

�

x(t i)
�

+

∫ t

t0

f
�

s, x(s), sup
ξ∈[σ(s),τ(s)]

x(ξ)
�

ds, t ∈ [t0, T ). (4)

3. Integro-summation Inequalities with “Supremum”

We will solve some nonlinear integro-summation inequalities which contain the supremum

of the unknown scalar nonnegative piecewise continuous function over a past time interval.

In the proof of the main results we will use the following results:

Lemma 1 ([9, Corollary 1, p.16] ). Let the following conditions be satisfied:

1. The function v(t) ∈ PC([0,∞), [0,∞)).
2. The function u(t) ∈ PC([0,∞), [0,∞)) satisfies the inequality

u(t) ≤ c +
∑

0<ti<t

βiu(t i) +

∫ t

0

v(s)u(s)ds,

where c ≥ 0, βi ≥ 0, (i = 1,2, . . . ) are constants.

Then for t ≥ 0 the inequality

u(t) ≤ c

�

∏

0<ti<t

(1+ βi)

�

exp

�
∫ t

0

v(s)ds

�

holds.

Lemma 2 ([8, Corollary 2.2.]). Let the nonnegative piecewise continuous function V (t) at

t ≥ t0 ≥ 0, with discontinuities of the first kind in the points

tk (t0 < t1 < t2 < · · · < lim
i→∞

t i =∞) satisfies the inequality

V (t) ≤ c +
∑

t0<ti<t

aiV
m(t i − 0) +

∫ t

t0

q(s)V m(s)ds,

where q(s) ∈ C([t0,∞),R+) and m is a positive constant. Then for t ≥ t0 the following estimates

hold:



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 33

(i) for m ∈ (0,1)

V (t) ≤
∏

t0<ti<t

(1+ aic
m−1)

�

c1−m + (1−m)

∫ t

t0

q(τ)dτ

�
1

1−m

; (5)

(ii) for m> 1

V (t) ≤ c
∏

t0<ti<t

(1+aimcm−1)×
�

1−(m−1)

�

c
∏

t0<ti<t

(1+aimcm−1)

�m−1
∫ t

t0

q(τ)dτ

�− 1

m−1

,

(6)

where
∫ t

t0

q(τ)dτ≤ c1−m

m
, and
∏

t0<ti<t

(1+ aimcm−1)<

�

m

m− 1

�
1

m−1

.

In the case when the supremum of the unknown nonnegative scalar piecewise continuous

function is involved in the integrals we obtain the following result:

Theorem 1. Let the following conditions be fulfilled:

1. The function α ∈ C1([t0, T ),R+) is a nondecreasing function and α(t) ≤ t for t ∈ [t0, T ).

2. The functions a, b ∈ C([α(t0), T ),R+).

3. The function φ ∈ C([α(t0)− h, t0],R+), where h= const ≥ 0.

4. The function u ∈ PC([α(t0)− h, T ),R+) satisfies the following inequalities

u(t) ≤ γ+
∑

t0<ti<t

βiu
p(t i)+

∫ α(t)

α(t0)

�

a(s)
�

u(s)
�p
+b(s)
�

sup
ξ∈[s−h,s]

u(ξ)
�p
�

ds for t ∈ [t0, T ),

(7)

u(t) ≤ φ(t) for t ∈ [α(t0)− h, t0], (8)

where the constants p > 0, βi ≥ 0 for i ∈ Z(t0, T ) and γ≤ max
s∈[α(t0)−h,t0]

φ(s) = M.

Then for t ∈ [t0, T ) the following inequalities are fulfilled:

(i) for p = 1

u(t) ≤ M

�

∏

t0<ti<t

�

1+ βi

�

�

exp
�

Q(t)
�

, (9)

where

Q(t) =

∫ α(t)

α(t0)

h

a(s) + b(s)
i

ds; (10)



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 34

(ii) for p ∈ (0,1)

u(t) ≤
�

∏

t0<ti<t

�

1+ βi M
p−1
�

�

h

M1−p + (1− p)Q(t)
i

1

1−p
; (11)

(iii) for p > 1

u(t) ≤ M

�

∏

t0<ti<t

�

1+pβi M
p−1
�

�

×
�

1−(p−1)

�

M

�

∏

t0<ti<t

�

1+pβi M
p−1
�

��p−1

Q(t)

�
1

p−1

,

(12)

where

Q(t)≤ M1−p

p
, (13)

∏

t0<ti<t

�

1+ pβi M
p−1
�

<

�

p

p− 1

� 1

p−1

. (14)

Proof. Define a function z : [α(t0)− h, T )→ [M ,∞) by the equalities

z(t) =











M +
∑

t0<ti<t βiu
p(t i)

+
∫ α(t)

α(t0)

�

a(s)
�

u(s)
�p
+ b(s)
�

supξ∈[s−h,s] u(ξ)
�p
�

ds
t ∈ [t0, T )

M t ∈ [α(t0)− h, t0]

From the definition of the function z(t) and the choice of the constant M it follows the validity

of the inequalities

u(t) ≤ z(t), t ∈ [α(t0)− h, T ) (15)

sup
ξ∈[s−h,s]

u(ξ)≤ sup
ξ∈[s−h,s]

z(ξ) = z(s), s ∈ [α(t0), T ). (16)

Then from (7), (15), (16) and the definition of the function z(t) we get

z(t) ≤ M +
∑

t0<ti<t

βiz
p(t i) +

∫ α(t)

α(t0)

h

a(s) + b(s)
i

�

z(s)
�p

ds, t ∈ [t0, T ). (17)

Consider the following three cases:

Case (i): Let p = 1. Then inequality (17) reduces to the following inequality

z(t) ≤ M +
∑

t0<ti<t

βiz(t i) +

∫ α(t)

α(t0)

h

a(s) + b(s)
i

z(s)ds, t ∈ [t0, T ). (18)

From inequality (18) according to Lemma 1 it follows

z(t) ≤ M

�

∏

t0<ti<t

�

1+ βi

�

�

exp

�
∫ α(t)

α(t0)

h

a(s) + b(s)
i

�

, t ∈ [t0, T ). (19)

Inequalities (19) and (15) imply the validity of the required inequality (9).



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 35

Case (ii): Let p ∈ (0,1). From inequality (17) according to Lemma 2 we obtain for t ∈ [t0, T )

z(t) ≤
�

∏

t0<ti<t

�

1+ βi M
p−1
�

�

h

M1−p + (1− p)Q(t)
i

1

1−p
, (20)

where the function Q(t) is defined by equality (10).

Substitute the bound (20) for the function z(t) into the right hand-side of (15) and get

the required inequality (11).

Case (iii): Let p > 1. As in the case (ii) from inequality (17) according to Lemma 2 we obtain for

t ∈ [t0, T )

z(t) ≤ M

�

∏

t0<ti<t

�

1+ pβi M
p−1
�

�

×

×
�

1− (p− 1)

�

M

�

∏

t0<ti<t

�

1+ pβi M
p−1
�

��p−1

Q(t)

�
1

p−1

,

(21)

where the function Q(t) is defined by equality (10) and inequalities (13) and (14) hold.

Substitute the bound (21) for the function z(t) into the right hand-side of (15) and get

the required inequality (12).

Similarly to the proof of Theorem 1 we can obtain the following result:

Theorem 2. Let the following conditions be fulfilled:

1. The functions α j ∈ C1([t0, T ),R+) are nondecreasing and the inequalities α j(t) ≤ t hold

for t ∈ [t0, T ), j = 1,2, . . . , m.

2. The functions a j, b j ∈ C([A, T ),R+) for j = 1,2, . . . , m, where A= min
1≤ j≤n

α j(t0).

3. The function φ ∈ C([A− h, t0],R+), where h= const ≥ 0.

4. The function u ∈ PC([A− h, T ),R+) satisfies the following inequalities

u(t) ≤ γ+
∑

t0<ti<t

βiu
p(t i)+

m
∑

j=1

∫ α j(t)

α j(t0)

�

a j(s)
�

u(s)
�p
+b j(s)
�

sup
ξ∈[s−h,s]

u(ξ)
�p
�

ds, t ∈ [t0, T ),

(22)

u(t) ≤ φ(t), t ∈ [A− h, t0], (23)

where the constants p > 0, βi ≥ 0 for i ∈ Z(t0, T ) and γ ≤ max
s∈[A−h,t0]

φ(s) = M̃ .

Then for t ∈ [t0, T ) the following inequalities are fulfilled:



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 36

(i) for p = 1

u(t) ≤ M̃

�

∏

t0<ti<t

�

1+ βi

�

�

exp
�

Q̃(t)
�

, (24)

where

Q̃(t) =

m
∑

j=1

∫ α j(t)

α j(t0)

h

a j(s) + b j(s)
i

ds; (25)

(ii) for p ∈ (0,1)

u(t) ≤
�

∏

t0<ti<t

�

1+ βi M̃
p−1
�

�

h

M̃1−p + (1− p)Q̃(t)
i

1

1−p
; (26)

(iii) for p > 1

u(t) ≤ M̃

�

∏

t0<ti<t

�

1+pβi M̃
p−1
�×
�

1−(p−1)

�

M̃

�

∏

t0<ti<t

�

1+pβi M̃
p−1
�

��p−1

Q̃(t)

�
1

p−1

,

(27)

where

Q̃(t)≤ M̃1−p

p
, (28)

∏

t0<ti<t

�

1+ pβi M̃
p−1
�

<

�

p

p− 1

� 1

p−1

. (29)

4. Practical Stability

Now we will use the solved above inequalities to investigate some stability properties of

the solutions of impulsive differential equation with “supremum” (1), (2). Note that stability

properties of solutions of various types of differential equations are very intensively studied

because of its applications to many models of real world problems [14, 18, 19]. The main

object of the paper is practical stability. We will extend the concept of boundedness as well as

practical stability to the considered nonlinear system of impulsive differential equation with

“supremum” (1), (2), based on the definitions for ordinary differential equations given in

[14].

Definition 1. We will say that the solution x(t; t0,φ) of the initial value problem (1), (2), (3)

is bounded if for any number α > 0 there exists β = β(α, t0) > 0 such that inequality |φ|0 < α
implies |x(t; t0,φ)| < β , t ≥ t0, where φ ∈ C([τ(t0)− h, t0],R).

Definition 2. We will say that the solutions of the initial value problem (1), (2), (3) are uni-

formly bounded if for any number α > 0 there exists β = β(α)> 0 such that inequality |φ|0 < α
implies |x(t; t0,φ)| < β , t ≥ t0 for all t0 ∈ R+, where φ ∈ C([τ(t0)− h, t0],R).



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 37

Let the constants λ,Λ : 0< λ < Λ be given.

Definition 3. We will say that the system of impulsive differential equation with “supremum”

(1), (2) is

- practically stable with respect to (λ,Λ) if the inequality |φ|0 < λ implies |x(t; t0,φ)| < Λ,

t ≥ t0 for some t0 ∈ R+, where φ ∈ C([τ(t0)− h, t0],R);

- uniformly practically stable with respect to (λ,Λ) if the inequality |φ|0 < λ implies

|x(t; t0,φ)| < Λ, t ≥ t0 for all t0 ∈ R+, where φ ∈ C([τ(t0)− h, t0],R).

Now will obtain some stability properties of the solutions of the impulsive differential

equation with “supremum” (1), (2). We will consider the case when the right part of the

equations satisfy the conditions H2 and H3 for different values of the power p.

Theorem 3. Let the following conditions be fulfilled:

1. The conditions H1–H4 are satisfied for p = 1.

2. For any t0 ∈ R+ there exist lim
t→∞Ψ(t0, t) = η1(t0) and lim

t→∞Φ(t0, t) = η2(t0) where the

functions Ψ(t0, t) and Φ(t0, t) are defined by the equalities

Ψ(t0, t) =
∏

t0<ti<t

�

1+ βi

�

, (30)

Φ(t0, t) =

∫ t

t0

h

A(s) + B(s)
i

ds, (31)

and the functions η1,η2 ∈ C(R+,R+).

Then:

(i) any solution of the impulsive differential equation with “supremum” (1), (2) is bounded;

(ii) if the functions η1(t) and η2(t) are bounded, i.e. there exist constants µ1,µ2 > 0 such that

ηk(t) ≤ µk, (k = 1,2) for t ∈ R+, then all solutions of the impulsive differential equation

with “supremum” (1), (2) are uniformly bounded;

(iii) if for the given constants 0< λ < Λ there exists a point t0 ∈ R+ such that

λη1(t0)e
η2(t0) < Λ, (32)

then the trivial solution of the impulsive differential equation with “supremum” (1), (2) is

practically stable with respect to (λ,Λ);

(iv) if the functions η1(t) and η2(t) are bounded, i.e. there exist constants µ1,µ2 > 0 such

that ηk(t) ≤ µk, (k = 1,2) for t ∈ R+, and

λµ1eµ2 < Λ, (33)

then the impulsive differential equation with “supremum” (1), (2) is uniformly practically

stable with respect to (λ,Λ).



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 38

Proof. According to conditions H1–H4 from integral equation (4) we get

|x(t)| ≤|φ|0 +
∑

t0<ti<t

�

�Ii

�

x(t i)
�
�

�+

∫ t

t0

�

A(s)|x(s)|p+ B(s)

�

�

� sup
ξ∈[σ(s),τ(s)]

x(ξ)

�

�

�

p
�

ds

≤|φ|0 +
∑

t0<ti<t

βi|x(t i)|p +
∫ t

t0

A(s)|x(s)|pds

+

∫ τ(t)

τ(t0)

B(τ−1(η))(τ−1(η))′ sup
ξ∈[η−h,η]

�

�x(ξ)
�

�

p
dη, t ∈ [t0, T )

(34)

|x(t)| ≤ |φ|0, t ∈ [τ(t0)− h, t0], (35)

where x(t) = x(t; t0,φ).

From inequalities (34), (35) according to Theorem 2 for m = 2, α1(t) ≡ t, α2(t) ≡ τ(t),
u(t) = |x(t)|, M̃ = |φ|0, a1(t) ≡ A(t), a2(t) ≡ 0, b1(t) ≡ 0, b2(t) ≡ B(τ−1(η))(τ−1(η))

′
for

t ∈ [τ(t0), T ), p = 1 and t ∈ [t0, T ) we obtain

|x(t)| ≤ |φ|0
�

∏

t0<ti<t

�

1+ βi

�

�

exp

�
∫ t

t0

h

A(s) + B(s)
i

ds

�

= |φ|0Ψ(t0, t)eΦ(t0 ,t).

(36)

Since the functions Ψ(t0, t) and Φ(t0, t) are nondecreasing in their second arguments, from

inequality (36) and condition 2 of Theorem 3 it follows

|x(t)| ≤ |φ|0η1(t0)e
η2(t0). (37)

The inequality (37) proves the claim of Theorem 3.

Remark 1. If the conditions 1 and 2 of Theorem 3 are satisfied, then the trivial solution of the

impulsive differential equation with “supremum” (1), (2) is stable in the sense of Lyapunov.

Theorem 4. Let the following conditions be fulfilled:

1. The conditions H1–H4 are satisfied for p ∈ (0,1).

2. For any t0 ∈ R+ there exist lim
t→∞Ψ(t0, t) = η1(t0) and lim

t→∞Φ(t0, t) = η2(t0) where the

functions Ψ(t0, t) and Φ(t0, t) are defined by (30) and (31), correspondingly, and the

functions η1,η2 ∈ C(R+,R+).

Then for the given constants 0< λ < Λ such that λ ∈ (0,1) and

(i) there exists a point t0 ∈ R+ such that

η1(t0)
h

λ1−p + (1− p)η2(t0)
i

1

1−p
< Λ,

then the impulsive differential equation with “supremum” (1), (2) is practically stable with

respect to (λ,Λ);



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 39

(ii) if the functions η1(t) and η2(t) are bounded, i.e. there exist constants µ1,µ2 > 0 such

that ηk(t) ≤ µk, (k = 1,2) for t ∈ R+, and

µ1

h

λ1−p + (1− p)µ2

�
1

1−p < Λ,

then the impulsive differential equation with “supremum” (1), (2) is uniformly practically

stable with respect to (λ,Λ).

Proof. From inequalities (34), (35) according to Theorem 2 for m = 2, α1(t) ≡ t,

α2(t) ≡ τ(t), u(t) = |x(t)|, M̃ = |φ|0, a1(t) ≡ A(t), a2(t)≡ 0, b1(t) ≡ 0,

b2(t) ≡ B(τ−1(t))(τ−1(t))
′
for t ∈ [τ(t0), T ), p ∈ (0,1) and t ∈ [t0, T ) we obtain

|x(t)| ≤
�

∏

t0<ti<t

�

1+ βi|φ|p−1

0

�

�

×
�

|φ|1−p

0
+ (1− p)

∫ t

t0

h

A(s)+ B(s)
i

ds

� 1

1−p

. (38)

Let |φ|0 < λ. Then from 1+ βi|φ|p−1

0 ≤ 1+ βiλ
p−1 ≤ 1+ βi, the monotonic property of

functions Ψ(t0, t) and Φ(t0, t), condition 2 of Theorem 4 and inequality (38) we get

|x(t)| ≤ Ψ(t0, t)
h

λ1−p + (1− p)Φ(t0, t)
i

1

1−p
. (39)

From inequality (39) according to condition 2 of the theorem it follows

|x(t)| ≤ η1(t0)
h

λ1−p + (1− p)η2(t0)
i

1

1−p
. (40)

The inequality (40) proves the claim of Theorem 4.

5. Applications

Now we will apply some of the obtained sufficient conditions for special types of impulsive

differential equations with “supremum” (1), (2).

Theorem 5. Let the following conditions be fulfilled:

1. The conditions H1 and H4 are satisfied.

2. The function f ∈ C(R+×R×R,R), f (t, 0,0) = 0 and

| f (t, x , y)| ≤ e−t
h

|x |+ |y|
i

for x , y ∈ R.

3. The functions Ii : R→ R, Ii(0) = 0 and

|Ii(x)| ≤
1

4i2 − 1
|x | for x ∈ R, i ∈ Z(t0, T ).



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 40

Then:

(i) all solutions of the system of impulsive differential equation (1), (2) are uniformly bounded;

(ii) if, additionally, the given positive constants λ and Λ are such that λπe2 < 2Λ, then the

impulsive differential equation with “supremum” (1), (2) is uniformly practically stable

with respect to (λ,Λ).

Proof. According to the notations in Theorem 3 we have

Φ(t0, t) =

∫ t

t0

h

e−s + e−s
i

ds = 2(e−t0 − e−t)

and

Ψ(t0, t) =
∏

t0<ti<t

�

1+
1

4i2 − 1

�

≤
∞
∏

i=1

�

1+
1

4i2 − 1

�

.

Using the convergence of Wallis product
∞
∏

n=1

�

1+ 1

4n2−1

�

= π

2
, it follows Ψ(t0, t) ≤ π

2
for any

t, t0 ∈ R+. Also lim
t→∞Φ(t0, t) = 2e−t0 ≤ 2. Therefore the conditions of Theorem 3 are satisfied

for µ1 =
π

2
and µ2 = 2.

According to claim (ii) of Theorem 3 all solutions of the system (1), (2) are uniformly

bounded, i.e. for any number α > 0 the inequality |φ|0 < α implies |x(t; t0,φ)| < e2απ
2

for

all t0 ∈ R+.

If, additionally, the inequality λπe2 < 2Λ holds, then according to claim (iv) of Theorem 3

the impulsive differential equation with “supremum” (1), (2) is uniformly practically stable

with respect to (λ,Λ).

Example 1. Consider the initial value problem for the scalar impulsive differential equation







x ′ = e−t x for t 6= n,

x(n+ 0)− x(n− 0) = 1

4n2−1
x(n− 0) for n ∈ Z(t0,∞),

x(t0) = x0

(41)

where x ∈ R and t0 ∈ R+.

The solution of the initial value problem (41) is

x(t; t0, x0) =
�

k
∏

i= j

4i2

4i2 − 1

�

x0ee−t0−e−t

for t ∈ (k, k+ 1],

where j is a natural number such that j− 1≤ t0 < j and k = j, j = 1, j+ 2, . . . . It is easy to see

the solution is uniformly bounded and stable.



S. Hristova, K. Stefanova / Eur. J. Pure Appl. Math, 5 (2012), 30-44 41

Now we will perturb the equation (41) by the maximum function of the unknown function,

i.e. consider the following impulsive differential equation with “supremum”











x ′ = e−t

�

x + sups∈[t−h,t] x(s)

�

for t ≥ t0, t 6= n,

x(n+ 0)− x(n− 0) = 1

4n2−1
x(n− 0) for n ∈ Z(t0,∞),

x(t) = ϕ(t) for t ∈ [t0 − h, t0],

(42)

where x ∈ R and h> 0 is a given constant.

The initial value problem (42) is not possible to be solved in analytical form, but according

to Theorem 5 its solutions are uniformly bounded. i.e. the perturbation as well as impulsive

conditions could save the property boundedness.

Also, if the positive constants λ and Λ satisfy λπe2 < 2Λ, then the solution of the impulsive

differential equation with “supremum” (42) is uniformly practically stable with respect to (λ,Λ).

Theorem 6. Let the following conditions be fulfilled:

1. The conditions 1 and 2 of Theorem 5 are satisfied.

2. The functions Ii : R→ R, Ii(0) = 0 and

|Ii(x)| ≤
1

2i
|x | for x ∈ R, i ∈ Z(t0, T ).

Then:

(i) all solutions of the system of impulsive differential equation (1), (2) are uniformly bounded;

(ii) if, additionally, the given positive constants λ and Λ are such that λe3 < Λ, then the

impulsive differential equation with “supremum” (1), (2) is uniformly practically stable

with respect to (λ,Λ).

Proof. The proof of the claim follows by the fact that

∞
∏

i=1

�

1+
1

2i

�

≤ e
∑∞

i=1
1

2i = e

and Theorem 3.

Theorem 7. Let the following conditions be fulfilled:

1. The conditions H1 and H4 are satisfied.

2. The function f ∈ C(R+×R×R,R), f (t, 0,0) = 0 and

| f (t, x , y)| ≤ e−t
h

|x |p+ |y|p
i

for x , y ∈ R,

where the constant p ∈ (0,1).



REFERENCES 42

3. The functions Ii : R→ R, Ii(0) = 0 and

|Ii(x)| ≤
1

4i2 − 1
|x |p for x ∈ R, i ∈ Z(t0, T ).

Then if the given constants λ ∈ (0,1) and Λ > 0 are such that π
2

�

λ1−p + 2(1− p)
�

1

1−p < Λ,

then the impulsive differential equation with “supremum” (1), (2) is uniformly practically stable

with respect to (λ,Λ).

Proof. As in the proof of Theorem 5 we prove the conditions of Theorem 4 are satisfied

and therefore if λ ∈ (0,1) and π
2

�

λ1−p+2(1−p)
�

1

1−p < Λ, then according to claim (ii) of The-

orem 4 the impulsive differential equation with “supremum” (1), (2) is uniformly practically

stable with respect to (λ,Λ).

Example 2. Consider the initial value problem for the scalar impulsive differential equation with

“supremum”











x ′ = e−t

�p
x +
p

sups∈[t−h,t] x(s)

�

for t ≥ t0, t 6= n,

x(n+ 0)− x(n− 0) = 1

4n2−1

p

x(n− 0) for n ∈ Z(t0,∞),
x(t) = ϕ(t) for t ∈ [t0 − h, t0],

(43)

where x ∈ R, h> 0 is a given constant and ϕ ∈ C([t0 − h, t0],R+).

The conditions of Theorem 5 are satisfied for p = 1

2
. Then if the positive constants

λ ∈ (0,1) and Λ satisfy π

2

�p
λ + 1
�2
< Λ , then the solution of the impulsive differential

equation with “supremum” (43) is uniformly practically stable with respect to (λ,Λ).

ACKNOWLEDGEMENTS: Research was partially supported by Fund “Scientific Research”

MU11FMI005/29.05.2011, Plovdiv University.

References

[1] R Agarwal, S Deng and W Zhang. Generalization of a retarded Gronwall-like inequality

and its applications. Appl. Math. Comput. 165(3):599–612, 2005.

[2] V Angelov and D Bainov. On the functional differential equations with “maximums”.

Appl. Anal. 16:187–194, 1983.

[3] D Bainov and S Hristova. Differential Equations with Maxima. Chapman and Hall/CRC,

USA,ISBN-10: 1439867577, 2011.

[4] D Bainov and S Hristova. Monotone-iterative techniques of Lakshmikantham for a

boundary value problem for systems of differential equations with “maxima”. J. Math.

Anal. Appl. 190(2):391–401, 1995.



REFERENCES 43

[5] D Bainov, V Petrov and V Proytcheva. Existence and asymptotic behavior of nonoscilla-

tory solutions of second-order neutral differential equations with “maxima”. J. Comput.

Appl. Math., 83(2):237–249, 1997.

[6] D Bainov, V Petrov and V Proytcheva. Asymptotic behavior of second order neutral

differential equations with “maxima”. Tamkang J. Math., 26(3):267–275, 1995.

[7] W Cheung. Some new nonlinear inequalities and applications to boundary value prob-

lems. Nonlinear Analysis, 64:2112–2128, 2006.

[8] A Gallo and A Piccirillo. About new analogies of Gronwall–Bellman–Bihari type in-

equalities for discontinuous functions and estimated solutions for impulsive differential

systems. Nonlinear Analysis, 67:1550–1559, 2007.

[9] S Hristova. Qualitative investigations and Approximate Methods for Impulsive Equations.

Nova Science Publ., New York, 2009.

[10] S Hristova and D Bainov. Application of the monotone-iterative techniques of V. Laksh-

mikantham to the solution of the initial value problem for impulsive differential equa-

tions with “supremum”. J. Math. Phys. Sci., 25(1):69–80, 1991.

[11] S Hristova and L Roberts. Boundedness of the solutions of differential equations with

“maxima”. Int. J. Appl. Math. 4(2):231–240, 2000.

[12] S Hristova and K Stefanova. Some integral inequalities with maximum of the unknown

functions. Adv. Dyn. Sys. Appl., 6(1):57–69, 2011.

[13] Y Kim. On some new integral inequalities for functions in one and two variables. Acta

Math. Sin., English Series, 21(2):423–434, 2005.

[14] V Lakshmikantham, S. Leela and A. Martynyuk. Practical stability of nonlinear systems.

World Scientific, 1990.

[15] Q Ma and J Pecaric. On certain new nonlinear retarded integral inequalities for func-

tions in two variables and their applications. J. Korean Math. Soc., 45(1):121–136,

2008.

[16] E Popov. Automatic Regulation and Control. Moscow 1966 (in Russian).

[17] D Snow. Gronwall’s inequality for systems of partial differential equations in two inde-

pendent variables. Proc. Amer. Math. Soc., 33(1):46–54, 1972.

[18] C Tunc. On the instability of solutions of some fifth order nonlinear delay differential

equations. Appl. Math. Inf. Sci., 5(1):112-121, 2011.

[19] C Tunc. Further results on the instability of solutions of certain nonlinear vector differ-

ential equations of fifth order. Appl. Math. Inf. Sci., 2(3):51–60, 2008.



REFERENCES 44

[20] W Wang. A generalized retarded Gronwall-like inequality in two variables and applica-

tions to BVP. Appl. Math. Comput., 191:144-154, 2007.

[21] W Wang and C Shen. On a generalized retarded integral inequality with two variables.

J. Ineq. Appl., Article ID 518646, 9 p, 2008.

[22] K Zheng. Some retarded nonlinear integral inequalities in two variables and applica-

tions. J. Ineq. Pure Appl. Math., 9, Iss. 2, Article 57, 11 pp, 2008.