Open Access

On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications

Boundary Value Problems20092009:808124

DOI: 10.1155/2009/808124

Received: 22 December 2008

Accepted: 28 May 2009

Published: 29 June 2009

Abstract

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 integro-differential systems of ordinary differential equations.

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:
  1. (a)
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ1_HTML.gif
    (11)
     
  1. (b)
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ2_HTML.gif
    (12)
     

Which are studied in the publications by Bainov, Borysenko, Iovane, Laksmikantham, Leela, Martynyuk, Mitropolskiy, Samoilenko ([113]), 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 [25, 12, 14] and for periodic boundary value problems we cite [1517]. 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 piecewise continuous functions and, in Section 3, the conditions of boundedness, stability, practical 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq1_HTML.gif of continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq2_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq3_HTML.gif is the delaying argument). The following holds.

Theorem 2.1.
  1. (a)
    Let one suppose that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq4_HTML.gif the following integro-sum functional inequality holds:
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ3_HTML.gif
    (21)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq5_HTML.gif is a positive nondecreasing function, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq6_HTML.gif function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq7_HTML.gif is a nonnegative piecewise-continuous,with I-st kind of discontinuities in the points https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq9_HTML.gif belongs to the class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq10_HTML.gif .
  1. (b)

    Function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq11_HTML.gif satisfies such conditions:

     

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq12_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq13_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq14_HTML.gif is nondecreasing.

Then for arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq15_HTML.gif the next estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ4_HTML.gif
(22)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ5_HTML.gif
(23)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ6_HTML.gif
(24)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ7_HTML.gif
(25)

Proof.

It follows from inequality (2.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ8_HTML.gif
(26)
Denoting by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ9_HTML.gif
(27)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ10_HTML.gif
(28)
Let us consider the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq16_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ11_HTML.gif
(29)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq17_HTML.gif So it results in
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ12_HTML.gif
(210)

and estimate (2.2) is valid in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq18_HTML.gif .

Let us suppose that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq19_HTML.gif estimate (2.2) is fulfilled. Then for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq20_HTML.gif we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ13_HTML.gif
(211)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq21_HTML.gif is determined from (2.3)–(2.5).

Taking into account such inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ14_HTML.gif
(212)

we obtain estimate (2.2) for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq22_HTML.gif .

Let us consider the class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq23_HTML.gif of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq24_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq25_HTML.gif positive, continuous, nondecreasing for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq26_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq27_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq28_HTML.gif

The following result is proved.

Theorem 2.2.

Suppose that the part (a) of Theorem 2.1 is valid and function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq29_HTML.gif belongs to the class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq30_HTML.gif Then for arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq31_HTML.gif such estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ15_HTML.gif
(213)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ16_HTML.gif
(214)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq32_HTML.gif

Proof.

By using the previous theorem we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq33_HTML.gif . On the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq34_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ17_HTML.gif
(215)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ18_HTML.gif
(216)
Taking into account estimate (2.16), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ19_HTML.gif
(217)
Then in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq35_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ20_HTML.gif
(218)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq36_HTML.gif the next inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ21_HTML.gif
(219)

where functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq37_HTML.gif are real nonnegative for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq38_HTML.gif , function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq39_HTML.gif satisfies conditions (i),,(iii) of Theorem 2.1.

Then for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq40_HTML.gif it results in

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ22_HTML.gif
(220)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq41_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ23_HTML.gif
(221)

The proof the same procedure as that of (Iovane [21, Theorems 2.1 and 3.1]).

Corollary 2.4.

Suppose that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq42_HTML.gif , then the result of Theorem 2.1 coincides with the result [22, Theorem 3.7.1, page 232];

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq43_HTML.gif then the result of Theorem 2.1 coincides with result [12, Proposition 2.3, page 2143];

(c) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq44_HTML.gif , then one obtains the analogy of Gronwall- Bellman result for discontinuous functions [23, Lemma 1] and estimate (2.2) reduces in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ24_HTML.gif
(222)
(d) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq45_HTML.gif , then one obtains the result [21, Theorem 2.1] and estimate (2.2) are as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ25_HTML.gif
(223)
(e) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq46_HTML.gif then one obtains the analogy of Bihari result for discontinuous functions [23, Lemma 2] and estimate (2.2) reduces as follows are reduced:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ26_HTML.gif
(224)
such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ27_HTML.gif
(225)
  1. (f)
    W(u) = um,, m> 0, then estimate (2.2) reduces as follows (see [21, Theorem 2.2]):
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ28_HTML.gif
    (226)
     
(g)Suppose that in Theorem 2.3 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq47_HTML.gif then estimates (2.20), (2.21) reduce as shown:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ29_HTML.gif
(227)

which coincide with result of [21, Theorem 3.1] for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq48_HTML.gif .

3. Applications

Let us consider the following system of differential equations

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ30_HTML.gif
(31)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq49_HTML.gif .

Let us assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq51_HTML.gif are defined in the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq52_HTML.gif and satisfy such conditions:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq53_HTML.gif

W satisfies conditions (i)–(iii) of Theorem 2.1;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq54_HTML.gif .

Consider https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq55_HTML.gif the solution of Cauchy problem for system (3.1). Then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ31_HTML.gif
(32)

from which it follows

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ32_HTML.gif
(33)

By using the result of Theorem 2.1 and estimate (2.2) we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ33_HTML.gif
(34)

where

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ34_HTML.gif
(35)

Let us consider some particular cases of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq56_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq57_HTML.gif , estimate (3.4) is reduced in such form

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ35_HTML.gif
(36)

Then such result holds.

Proposition 3.1.

Let the following conditions be fulfilled for system (3.1) :

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq58_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq59_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq60_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq61_HTML.gif

Then one has:

(a)All solutions of system (3.1) are bounded (uniformly, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq62_HTML.gif are independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq63_HTML.gif ) and such estimate is valid:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ36_HTML.gif
(37)

(b)The trivial solution of system (3.1) is stable by Lyapunov (uniformly stable relative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq64_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq65_HTML.gif ).

Remark 3.2.

If conditions I–IV of Proposition 3.1 are valid and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq66_HTML.gif then the trivial solution is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq67_HTML.gif -stable by Chetaev (uniformly https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq68_HTML.gif -stable, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq70_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq71_HTML.gif ).

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq72_HTML.gif the estimate (3.4) is reduced in such form

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ37_HTML.gif
(38)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ38_HTML.gif
(39)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ39_HTML.gif
(310)

From estimate (3.8) the next propositions follow.

Proposition 3.3.

Suppose that such conditions occur:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq73_HTML.gif

(b)estimates ii–iv of Proposition 3.1 be fulfilled.

Then all the solutions of system (3.1) are bounded (uniformly if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq74_HTML.gif ).

Remark 3.4.

Suppose that conditions (a), (b) of Proposition 3.3 are valid and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ40_HTML.gif
(311)

Then trivial solution of system (3.1) is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq75_HTML.gif -stable by Chetaev (uniformly if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq76_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq77_HTML.gif ).

Proposition 3.5.

Let conditions ii–iv of Proposition 3.1 be fulfilled for system (3.1), inequality (3.10) holds and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ41_HTML.gif
(312)

Then trivial solution of system (3.1) is stable by Lyapunov (uniformly if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq78_HTML.gif ).

Remark 3.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq79_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq80_HTML.gif the conditions of boundedness, stability, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq81_HTML.gif -stability 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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ42_HTML.gif
(313)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq82_HTML.gif and defined in the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq84_HTML.gif .

We suppose that such conditions are valid:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq85_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq86_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq87_HTML.gif .

It is easy to see that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ43_HTML.gif
(314)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ44_HTML.gif
(315)

From estimate (3.15) such result follows.

Proposition 3.7.

Let one suppose that for system (3.13) conditions (i)–(iii) take place for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq88_HTML.gif and the following estimates are fulfilled:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq89_HTML.gif ;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq90_HTML.gif

Then we have:

(i)All solutions of system (3.13) are bounded and satisfy the estimate:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ45_HTML.gif
(316)
(ii)The trivial solution of system (3.13) is stable by Lyapunov (uniformly, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq91_HTML.gif ).
  1. (iii)

    The trivial solution of system (3.13) is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq92_HTML.gif -stable by Chetaev (uniformly if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq93_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq94_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq95_HTML.gif

     

Authors’ Affiliations

(1)
Department of Mathematics and Applications, "R.Caccioppoli" University of Naples "Federico II"
(2)
Department of Civil Engineering, Second University of Naples

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© A. Gallo and A.M. Piccirillo 2009

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