Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 808124, 14 pages
doi:10.1155/2009/808124
Research Article
On Some Generalizations Bellman-Bihari Result
for Integro-Functional Inequalities for
Discontinuous Functions and Their Applications
Angela Gallo1 and Anna Maria Piccirillo2
1
Department of Mathematics and Applications, “R.Caccioppoli” University of Naples “Federico II”,
Claudio street 21, 80125 Naples, Italy
2
Department of Civil Engineering, Second University of Naples, Roma, street 21, 81100 Caserta, Italy
Correspondence should be addressed to Angela Gallo, angallo@unina.it
Received 22 December 2008; Revised 21 April 2009; Accepted 28 May 2009
Recommended by Juan J. Nieto
We present some new nonlinear integral inequalities Bellman-Bihari type with delay for discontinuous functions integro-sum inequalities; impulse integral inequalities. Some applications of the
results are included: conditions of boundedness uniformly, stability by Lyapunov uniformly,
practical stability by Chetaev uniformly for the solutions of impulsive differential and integrodifferential systems of ordinary differential equations.
Copyright q 2009 A. Gallo and A. M. Piccirillo. 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.
1. Introduction
The first generalizations of the Bihari result for discontinuous functions which satisfy
nonlinear impulse inequality integro-sum inequality are connected with such types of
inequalities:
a
vt ≤ c t
pτ vm τ dτ t0
βi vti − 0,
m > 0, m / 1,
1.1
t0 <ti <t
b
vt ≤ c t
t0
pτϕvτdτ t0 <ti <t
βi vti − 0,
1.2
2
Boundary Value Problems
Which are studied in the publications by Bainov, Borysenko, Iovane, Laksmikantham, Leela,
Martynyuk, Mitropolskiy, Samoilenko 1–13, and in many others. In these investigations
the method of integral inequalities for continuous functions is generalized to the case of
piecewise continuous one-dimensional inequalities and discontinuous multidimensional
inequalities functions.
For the generalization of the integral inequalities method for discontinuous functions
and for their applications to qualitative analysis of impulsive systems: existence, uniqueness,
boundedness, comparison, stability, and so forth. We refer to the results 2–5, 12, 14 and
for periodic boundary value problems we cite 15–17. More recently, a novel variational
approach appeared in 18. This approach to impulsive differential equations also used the
critical point theory for the existence of solutions of a nonlinear Dirichlet impulsive problem
and in 19 some new comparison principles and the monotone iterative technique to
establish a more general existence theorem for a periodic boundary value problem. Reference
20 is very interesting in that it gives a complete overview of the state-of-the-art of the
impulsive differential, inclusions.
In this paper, in Section 2, we investigate new analogies Bihari results for piece-wise
continuous functions and, in Section 3, the conditions of boundedness, stability, pract-ical
stability of the solutions of nonlinear impulsive differential and integro-differential systems.
2. General Bihari Theorems for Integro-Functional Inequalities
for Discontinuous Functions
Let us consider the class ℘ of continuous functions p : R → R, pt ≤ t, lim pt ∞ p |t| → ∞
pt is the delaying argument. The following holds.
Theorem 2.1. (a) Let one suppose that for x ≥ x0 the following integro-sum functional inequality
holds:
x
fτ W u pτ dτ βi um xi − 0,
2.1
ux ≤ ϕx qx
xi
x0 <xi <x
where qx ≥ 1, ϕx is a positive nondecreasing function, βi const ≥ 0, f : R → R , m const > 0; function ux is a nonnegative piecewise-continuous,with I-st kind of discontinuities in
the points xi : x0 < x1 < · · · limn → ∞ xn ∞, pt belongs to the class ℘.
(b) Function Wx satisfies such conditions:
i Wγβ ≤ WγWβ;
ii W : R → R , W0 0;
iii W is nondecreasing.
Then for arbitrary x ∈ x0 , ∞ the next estimate holds:
ux ≤
ϕxqxG−1
i
x
x
fτ
W ϕ pτ q pτ dτ
ϕτ
xi
for x ∈ xi , xi1 fτ ,
W ϕ pτ q pτ dτ ∈ Dom G−1
i
xi ϕτ
2.2
Boundary Value Problems
3
G0 u Gi u ci 1 βi ϕ
m−1
xi q xi − 0
m
u
dσ
,
1 Wσ
u
dσ
,
ci Wσ
G−1
i−1
x
i 1, 2, . . . ,
2.3
2.4
fτ W ϕ pτ q pτ dτ ,
xi−1 ϕτ
i
i 1, 2, . . . if m ∈ 0, 1, ∀x ≥ x0 ,
x
m
i
fτ m−1
m
−1
ci 1 βi ϕ xi q xi − 0 Gi−1
W ϕ pτ q pτ dτ
,
xi−1 ϕτ
2.5
i 1, 2, . . . if m ≥ 1, ∀x ≥ x0 .
Proof. It follows from inequality 2.1
ux
≤ 1 qx
ϕx
fτ W u pτ
um xi − 0
dτ βi
ϕτ
ϕx
x0
x0 <xi <x
x
fτ
uxi − 0 m
m−1
.
W u pτ dτ ≤ qx 1 βi ϕ xi − 0
ϕxi − 0
x0 ϕτ
x0 <xi <x
x
2.6
Denoting by
x
fτ uxi − 0 m
m−1
u x 1 W u pτ dτ βi ϕ xi − 0
,
ϕxi − 0
x0 ϕτ
x0 <xi <x
∗
2.7
u∗ x 1 for x x0 ,
then
uxi − 0 ≤ ϕxi − 0 qxi − 0u∗ xi − 0,
ux ≤ ϕx qxu∗ x,
u pτ ≤ ϕ pτ q pτ u∗ pτ ≤ ϕ pτ q pτ u∗ τ,
x
fτ W ϕ pτ q pτ Wu∗ τdτ
⇒ u∗ x ≤ 1 ϕτ
x0
x0 <xi <x
βi ϕm−1 xi − 0qm xi − 0u∗ m xi − 0.
2.8
4
Boundary Value Problems
Let us consider the interval I1 x0 , x1 . Then
∗
u x ≤
G−1
0
x
fτ W ϕ pτ q pτ dτ ,
x0 ϕτ
x
if only
where G0 ξ ξ
1
fτ W ϕ pτ q pτ dτ ∈ Dom G−1
,
0
x0 ϕτ
2.9
dτ/Wτ. So it results in
ux ≤
ϕxqxG−1
0
x
fτ W ϕ pτ q pτ dτ ,
x0 ϕτ
2.10
and estimate 2.2 is valid in I1 .
Let us suppose that for x ∈ Ik xk−1 , xk , k 2, 3, . . . estimate 2.2 is fulfilled. Then
for every x ∈ Ik1 we have
∗
u x ≤
G−1
k
x
fτ
W ϕ pτ q pτ dτ
xk ϕτ
x
fτ
with
W ϕ pτ q pτ dτ ∈ Dom G−1
k ,
xk ϕτ
2.11
where Gk ξ is determined from 2.3–2.5.
Taking into account such inequality
ux ≤ ϕxqxu∗ x,
2.12
we obtain estimate 2.2 for every x ∈ x0 , ∞.
Let us consider the class I of functions f such that
i fx-positive, continuous, nondecreasing for x > 0;
ii ∀u ≥ 1, v > 0 ⇒ u−1 fv < fu−1 v;
iii f0 0.
The following result is proved.
Theorem 2.2. Suppose that the part (a) of Theorem 2.1 is valid and function W : 0 , ∞ → 0 , ∞
belongs to the class I. Then for arbitrary x0 ≤ x ≤ x∗ such estimate holds:
ux ≤
ϕxqxG∗i −1
x
fτ q pτ dτ
xi
for Ii xi , xi1 , i 0, 1, . . . ,
2.13
Boundary Value Problems
5
where
G∗0 η η
G∗i η dσ
,
1 Wσ
η
dσ
i 1, 2, . . . ,
ci∗ Wσ
x
i
∗
m−1
m
∗ −1
ci 1 βi ϕ xi q xi Gi−1
fτq pτ dτ
if m ∈ 0, 1,
xi−1
ci∗
1 βi ϕ
m−1
m
x
i
∗ −1
fτq pτ dτ
xi q xi Gi−1
m
2.14
if m ≥ 1,
xi−1
x
and x∗ supx { xi−1 fτqpτdτ ∈ DomG∗i−1 −1 }, i 1, 2, . . . .
Proof. By using the previous theorem we have ux ≤ ϕxgxu∗ x, u∗ x 1 x x0 . On
the interval I1
du∗ x fx W u px .
dx
ϕx
2.15
Then
u px ≤ ϕ px q px u∗ px ≤ ϕxq px u∗ x,
du∗ x fx
≤
W q px ϕxu∗ x
dx
ϕx
fxq px
≤
W q px ϕxu∗ x
ϕxq px
2.16
≤ fxq px Wu∗ x.
Taking into account estimate 2.16, we obtain
x
u∗ σ
dσ ≤
∗
x0 Wu σ
x
u∗ σ
dσ ∗
x0 Wu σ
u∗ x0 1,
x
x0
u∗ x
du
G∗0 u∗ x − G∗0 u∗ x0 ,
u∗ x0 Wu
u∗ x ≥ 1,
G∗0 u∗ x ≤
fτq pτ dτ,
x
x0
G∗0 u∗ x0 G∗0 1 0,
fτ q pτ dτ.
2.17
6
Boundary Value Problems
Then in I1 we have
ux ≤ ϕxqx
G∗0 −1
x
x
fτq pτ dτ
if only
x0
x0
fτq pτ dτ ∈ Dom G∗0 −1 .
2.18
As in the previously theorem, the proof is completed by using the inductive method.
The following result is easily to obtain
Theorem 2.3. Suppose that for x ≥ x0 the next inequality holds:
ux ≤ u0 qx
x
fs u ps ds fs
x0
x
x
x
x0
hs Wuσsds x0
gτu pτ dτ ds
x0
2.19
βi um xi − 0,
x0 <xi <x
where functions ux, fx, qx, gx, hx, px, σx are real nonnegative for x ≥ x0
0, px, σx ∈ I, qx ≥ 1, βi ≥ 0, function W satisfies conditions (i),. . .,(iii) of Theorem 2.1.
Then for x ≥ x0 it results in
ux ≤
1 βi q
x0 <xi <x
⎛
·
ψ0−1 ⎝
x
m
xi um−1
0
⎡
hτ W ⎣
x0
x
exp
q pτ fτ gτ dτ
>
x0
⎤
1 βi q
m
xi um−1
0
⎦W
x0 <xi <στ
× qστ exp
στ
q ps fs gs ds dτ ,
if m ∈ 0, 1
x0
x
⎡
hτW ⎣
x0
⎤
⎦W
1 βi qm xi um−1
0
x0 <xi <στ
× qστ exp
στ
x0
q ps fs gs ds dτ ∈ Dom ψ0−1 ,
2.20
Boundary Value Problems
u
where ψ0 u u0
dv/Wv;
ux ≤
7
1 βi q
m
x0 <xi <x
⎛
· ψ0−1 ⎝
x
xi um−1
0
⎡
x0
⎤
hτ ⎣
x0
x
exp m q pτ fτ gτ
⎦
1 βi qm xi um−1
0
x0 <xi <στ
στ
·W qστ exp m
q ps fs gs ds dτ ,
if m ≥ 1,
2.21
x0
x
⎡
hτW ⎣
x0
⎤
1 βi q
m
xi um−1
0
⎦W
x0 <xi <στ
στ
× qστ exp m
q ps fs gs ds dτ ∈ Dom ψ0−1 .
x0
The proof the same procedure as that of Iovane 21, Theorems 2.1 and 3.1.
Corollary 2.4. Suppose that
a m 1, then the result of Theorem 2.1 coincides with the result [22, Theorem 3.7.1, page
232];
b m 1, ϕx c, qx 1, pt t, then the result of Theorem 2.1 coincides with result
[12, Proposition 2.3, page 2143];
c qx 1, Wu u, pt t, then one obtains the analogy of Gronwall- Bellman result
for discontinuous functions [23, Lemma 1] and estimate 2.2 reduces in the following form:
ux ≤ ϕx
1 βi ϕ
x
xi exp
fτ dτ
if m ∈ 0, 1, ∀x ≥ x0 ,
x0
x0 <xi <x
ux ≤ ϕx
m−1
x
m−1
1 βi ϕ xi exp m fτ dτ
if m ≥ 1, ∀x ≥ x0 .
2.22
x0
x0 <xi <x
d qx 1, Wu u, then one obtains the result [21, Theorem 2.1] and estimate 2.2 are
as follows:
1 βi ϕ
ux ≤ ϕx
m−1
xi exp
x0 <xi <x
ux ≤ ϕx
x0 <xi <x
1 βi ϕ
m−1
x
ϕ pτ
dτ ,
fτ
ϕτ
x0
x
ϕ pτ
dτ
xi exp m fτ
ϕτ
x0
if m ∈ 0, 1, ∀x ≥ x0 ;
if m ≥ 1, ∀x ≥ x0 .
2.23
8
Boundary Value Problems
e qx 1, Wu um , m > 0, pt t, then one obtains the analogy of Bihari result for
discontinuous functions [23, Lemma 2] and estimate 2.2 reduces as follows are reduced:
ux ≤ ϕx
1 βi ϕ
m−1
xi 1 1 − m
1/1−m
x
m−1
ϕ
τfτdτ
,
x0
x0 <xi <x
if 0 < m < 1, ∀x ≥ x0 ,
ux ≤ ϕx
1 βi mϕ
m−1
⎡
xi ⎣1 − m − 1
m−1
1 βi mϕ
x0 <xi <x
m−1
xi x0 <xi <x
×
− 1/m−1
x
m−1
ϕ
∀x ≥ x0 ,
τ fτ dτ
x0
2.24
such that
x
ϕm−1 τfτ dτ ≤
x0
1
,
m
1 βi ϕm−1 xi < 1 m > 1,
x0 <xi <x
1
m−1
1/m−1
2.25
f W(u) = um, , m> 0, then estimate 2.2 reduces as follows (see [21, Theorem 2.2]):
ux ≤ ϕx qx
1 βi ϕm−1 xi qm xi x0 <xi <x
× 1 1 − m
x
m−1
ϕ
τfτq
m
m 1/1−m
ϕ pτ
dτ
pτ
ϕτ
x0
if 0 < m < 1, ∀x ≥ x0 ,
ux ≤ ϕx qx
1 βi mϕm−1 xi qm xi x0 <xi <x
×
⎧
⎨
1 − m − 1
⎩
×
1 βi mϕ
m−1
xi q xi m−1
m
x0 <xi <x
x
m−1
ϕ
x0
τ fτq
m
m − 1/m−1
ϕ pτ
dτ
∀x ≥ x0
pτ
ϕτ
.
Boundary Value Problems
9
such that
x
m−1
ϕ
τ fτq
m
x0
m
ϕ pτ
1
dτ ≤ ,
pτ
ϕτ
m
1 βi mϕ
m−1
xi q xi < 1 m
x0 <xi <x
1
m−1
m > 1,
−1/m−1
.
2.26
g Suppose that in Theorem 2.3 qx 1, Wu u, σs ps s, then estimates 2.20,
2.21 reduce as shown:
ux ≤ u0
1
x0 <xi <x
ux ≤ u0
1
x0 <xi <x
βi um−1
xi 0
βi um−1
xi 0
x
exp
fξ gξ hξ dξ
if m ∈ 0, 1, ∀x ≥ x0 ;
x0
x
exp m
fξ gξ hξ dξ
if m ≥ 1, ∀x ≥ x0 ,
x0
2.27
which coincide with result of [21, Theorem 3.1] for ht u0 .
3. Applications
Let us consider the following system of differential equations
dx
Ft, x,
dt
t/
ti ,
3.1
Δx|tti Ii x
where x ∈ Rn , F ∈ Rn , Ii x ∈ Rn i 1, 2, . . ., t ≥ t0 ≥ 0, limi → ∞ ti ∞, ti−1 < ti for all i 1, 2, . . ..
Let us assume that Ft, x and Ii x are defined in the domain D {t, x : t ∈ I t0 , T , T ≤ ∞, x ≤ h } and satisfy such conditions:
a Ft, x ≤ ftWx, f : R → R ,
W satisfies conditions i–iii of Theorem 2.1;
b Ii x ≤ βi xm , βi const > 0, m > 0.
Consider xt x t, t0 , x0 the solution of Cauchy problem for system 3.1. Then
xt, t0 , x0 x0 t
t0
Fτ, xτ, t0 , x0 dτ t0 <ti <t
Ii xti − 0, t0 , x0 ,
3.2
10
Boundary Value Problems
from which it follows
xt, t0 , x0 ≤ x0 t
fτ Wxτ, t0 , x0 dτ t0
βi xti − 0, t0 , x0 m .
3.3
t0 <ti <t
By using the result of Theorem 2.1 and estimate 2.2 we obtain
xt, t0 , x0 ≤
x0 G−1
i
x
Wx0 fτ
dτ
x0 xi
for x ∈ xi , xi1 ,
3.4
x
Wx0 ,
fτ
dτ ∈ Dom G−1
i
x0 xi
where
u
dσ
G0 u ,
Wσ
1
ci 1 βi x0 Gi u m−1
G−1
i−1
ci 1 βi x0 m−1
dσ
,
Wσ
ci
x
G−1
i−1
i 1, 2, ...,
Wx0 fτ
dτ ,
x0 xi−1
i 1, 2, . . .
u
i
3.5
if m ∈ 0, 1, ∀x ≥ x0 ,
x
Wx0 fτ
dτ
x0 xi−1
i
i 1, 2, . . .
m
,
if m ≥ 1, ∀x ≥ x0 .
Let us consider some particular cases of W.
If Wu u, m 1, estimate 3.4 is reduced in such form
xt, t0 , x0 ≤ x0 1 βi exp
t0 <ti <t
t
fτdτ .
t0
Then such result holds.
Proposition 3.1. Let the following conditions be fulfilled for system 3.1 :
i Ft, x ≤ ftx;
ii Ii x ≤ βi x;
iii ∃m1 t0 const. > 0 :
iv ∃m2 t0 const. > 0 :
t
t0 <ti <t 1
t0
βi ≤ m1 t0 < ∞;
fτ dτ ≤ m2 t0 < ∞, ∀t ≥ t0 .
3.6
Boundary Value Problems
11
Then one has:
a All solutions of system 3.1 are bounded (uniformly, if mi t0 are independent of t0 ) and
such estimate is valid:
xt, t0 , x0 ≤ m1 t0 expm2 t0 x0 .
3.7
b The trivial solution of system 3.1 is stable by Lyapunov (uniformly stable relative t0 , if
mi t0 mi , i 1, 2).
Remark 3.2. If conditions I–IV of Proposition 3.1 are valid and λ/Λ < m1 t0 expm2 t0 −1 ,
then the trivial solution is λ, Λ, I-stable by Chetaev uniformly λ, Λ, I-stable, if mi t0 ,
i 1, 2 is independent of t0 .
1, m 1 the estimate 3.4 is reduced in such form
If Wu ul , l /
xt, t0 , x0 ≤
1 βi
1−l
x0 1 − l
∀t ≥ t0 , if 0 < l < 1,
fτdτ
3.8
t0
t0 <ti <t
xt, t0 , x0 ≤ x0 1/1−l
t
1 βi
t0 <ti <t
⎡
×⎣1 − l − 1 x0 l−1
·
1 βi
l−1 t
⎛
fτdτ < ⎝l − 1 x0 t0
fτdτ ⎦
3.9
∀t ≥ t0 ,
t0
t0 <ti <t
t
⎤− 1/l−1
1 βi
l−1 ⎞−1
⎠ ,
if l > 1.
3.10
t0 <ti <t
From estimate 3.8 the next propositions follow.
Proposition 3.3. Suppose that such conditions occur:
a Ft, x − Ft, y ≤ ft x − yl , 0 < l < 1 for all x, y ∈ D
b estimates ii–iv of Proposition 3.1 be fulfilled.
Then all the solutions of system 3.1 are bounded (uniformly if mi t0 mi , i 1, 2).
Remark 3.4. Suppose that conditions a, b of Proposition 3.3 are valid and
1−l
λ
Λ
1 − l m2 t0 <
m1 t0 1−l
.
3.11
12
Boundary Value Problems
Then trivial solution of system 3.1 is λ, Λ, I-stable by Chetaev uniformly if mi t0 is
independent of t0 .
Proposition 3.5. Let conditions ii–iv of Proposition 3.1 be fulfilled for system 3.1, inequality 3.10
holds and
Ft, x ≤ ftx l ,
3.12
l > 1.
Then trivial solution of system 3.1 is stable by Lyapunov (uniformly if mi t0 mi , i 1, 2).
Remark 3.6. If Wu u1 , 1 > 0, and m / 1 the conditions of boundedness, stability, λ, Λ, Istability is investigated in 14, see Theorems 3.4–3.6; the estimates of the solutions of system
3.1 with non-Lipschitz type of discontinuities are investigated in 23, see Proposition 1,
Proposition 2.
Let us consider the following impulsive system of integro-differential equations:
dx
Ft, x, Kxt,
dt
t/
ti ,
3.13
Δx|tti Ii x,
where
x ∈ Rn , F ∈ Rn , Ii x ∈ Rn i 1, 2, . . . and defined in the domain D, Kxt t
kt, τ, xτ dτ.
t0
We suppose that such conditions are valid:
i Ft, x, y ≤ ftx y for all x, y ∈ D, f : R → R ;
ii kt, s, x ≤ gtx for all s ∈ t0 , t, g : R → R ;
iii Ii x ≤ βi xm for all x, y ∈ D, βi const > 0, m > 0 m /
1.
It is easy to see that
xt, t0 , x0 ≤ x0 t
fτxτ, t0 , x0 dτ
t0
t
τ
fτ
t0
⇒ xt, t0 , x0 ≤ x0 gξxξ, t0 , x0 dξ dτ t0
1 βi x0 m−1 exp
t0 <ti <t
βi xti − 0, t0 , x0 m
t0 <ti <t
t
fξ gξ dξ,
t0
if 0 < m ≤ 1, t ≥ t0
t
m−1
1 βi x0 exp m
fξ gξ dξ ,
xt ≤ x0 t0 <ti <t
t0
if 0 m ≥ 1, t ≥ t0 .
From estimate 3.15 such result follows.
3.14
3.15
Boundary Value Problems
13
Proposition 3.7. Let one suppose that for system 3.13 conditions (i)–(iii) take place for m > 1 and
the following estimates are fulfilled:
a ∃ m3 t0 const. > 0 : t0 <ti <t 1 βi x0 m−1 ≤ m3 t0 < ∞;
t
b ∃ m4 t0 const. > 0 : t0 fξ gξ dξ ≤ m4 t0 < ∞ for all t ≥ t0 .
Then we have:
i All solutions of system 3.13 are bounded and satisfy the estimate:
xt ≤ m3 t0 expm4 t0 x0 .
3.16
ii The trivial solution of system 3.13 is stable by Lyapunov (uniformly, if mi t0 mi , i 3, 4).
iii The trivial solution of system 3.13 is λ, Λ, I-stable by Chetaev (uniformly if mi t0 is
independent of t0 ) and m3 t0 expm4 t0 < Λ/λ.
References
1 D. Banov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
2 S. D. Borysenko, G. Iovane, and P. Giordano, “Investigations of the properties motion for essential
nonlinear systems perturbed by impulses on some hypersurfaces,” Nonlinear Analysis: Theory, Methods
& Applications, vol. 62, no. 2, pp. 345–363, 2005.
3 S. D. Borysenko, M. Ciarletta, and G. Iovane, “Integro-sum inequalities and motion stability of
systems with impulse perturbations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no.
3, pp. 417–428, 2005.
4 S. Borysenko and G. Iovane, “About some new integral inequalities of Wendroff type for
discontinuous functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 10, pp. 2190–
2203, 2007.
5 S. C. Hu, V. Lakshmikantham, and S. Leela, “Impulsive differential systems and the pulse
phenomena,” Journal of Mathematical Analysis and Applications, vol. 137, no. 2, pp. 605–612, 1989.
6 V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications,
Academic Press, New York, NY, USA, 1969.
7 V. Lakshmikantham, S. Leela, and M. Mohan Rao Rama, “Integral and integro-differential
inequalities,” Applicable Analysis, vol. 24, no. 3, pp. 157–164, 1987.
8 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
9 A. A. Martyhyuk, V. Lakshmikantham, and S. Leela, Stability of Motion: the Method of Integral
Inequalities , Naukova Dumka, Kyiv, Russia, 1989.
10 Yu. A. Mitropolskiy, S. Leela, and A. A. Martynyuk, “Some trends in V. Lakshmikantham’s
investigations in the theory of differential equations and their applications,” Differentsial’nye
Uravneniya, vol. 22, no. 4, pp. 555–572, 1986.
11 Yu. A. Mitropolskiy, A. M. Samoilenko, and N. Perestyuk, “On the problem of substantiation of
overoging method for the second equations with impulse effect,” Ukrainskii Matematicheskii Zhurnal,
vol. 29, no. 6, pp. 750–762, 1977.
12 Yu. A. Mitropolskiy, G. Iovane, and S. D. Borysenko, “About a generalization of Bellman-Bihari type
inequalities for discontinuous functions and their applications,” Nonlinear Analysis: Theory, Methods
& Applications, vol. 66, no. 10, pp. 2140–2165, 2007.
13 A. M. Samoilenko and N. Perestyuk, Differential Equations with Impulse Effect, Visha Shkola, Kyiv,
Russia, 1987.
14
Boundary Value Problems
14 A. Gallo and A. M. Piccirillo, “About new analogies of Gronwall-Bellman-Bihari type inequalities
for discontinuous functions and estimated solutions for impulsive differential systems,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1550–1559, 2007.
15 J. J. Nieto, “Impulsive resonance periodic problems of first order,” Applied Mathematics Letters, vol. 15,
no. 4, pp. 489–493, 2002.
16 J. J. Nieto, “Basic theory for nonresonance impulsive periodic problems of first order,” Journal of
Mathematical Analysis and Applications, vol. 205, no. 2, pp. 423–433, 1997.
17 J. J. Nieto, “Periodic boundary value problems for first-order impulsive ordinary differential
equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 51, no. 7, pp. 1223–1232, 2002.
18 Z. Luo and J. J. Nieto, “New results for the periodic boundary value problem for impulsive integrodifferential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2248–2260,
2009.
19 J. J. Nieto and D. O’Regan, “Variational approach to impulsive differential equations,” Nonlinear
Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.
20 M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of
Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA,
2006.
21 G. Iovane, “Some new integral inequalities of Bellman-Bihari type with delay for discontinuous
functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 2, pp. 498–508, 2007.
22 A. Samoilenko, S. Borysenko, C. Cattani, G. Matarazzo, and V. Yasinsky, Differential Models: Stability,
Inequalities and Estimates, Naukova Dumka, Kiev, Russia, 2001.
23 D. S. Borysenko, A. Gallo, and R. Toscano, “Integral inequalities Gronwall-Bellman type for
discontinuous functions and estimates of solutions impulsive systems,” in Proc.DE@CAS, pp. 5–9,
Brest, 2005.