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

  • Angela Gallo1Email author and

    Affiliated with

    • Anna Maria Piccirillo2

      Affiliated with

      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)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ1_HTML.gif
        (11)
         
      1. (b)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq1_HTML.gif of continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq2_HTML.gif ( http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ3_HTML.gif
        (21)
         
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq5_HTML.gif is a positive nondecreasing function, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq6_HTML.gif function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq9_HTML.gif belongs to the class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq10_HTML.gif .
      1. (b)

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

         

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

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

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

      Then for arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq15_HTML.gif the next estimate holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ4_HTML.gif
      (22)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ5_HTML.gif
      (23)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ6_HTML.gif
      (24)
      http://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)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ8_HTML.gif
      (26)
      Denoting by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ9_HTML.gif
      (27)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ10_HTML.gif
      (28)
      Let us consider the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq16_HTML.gif Then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ11_HTML.gif
      (29)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq17_HTML.gif So it results in
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq18_HTML.gif .

      Let us suppose that for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq20_HTML.gif we have

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

      where http://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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq22_HTML.gif .

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

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

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

      (iii) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq29_HTML.gif belongs to the class http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq30_HTML.gif Then for arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq31_HTML.gif such estimate holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ15_HTML.gif
      (213)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ16_HTML.gif
      (214)

      and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq33_HTML.gif . On the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq34_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ17_HTML.gif
      (215)
      Then
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ19_HTML.gif
      (217)
      Then in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq35_HTML.gif we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq36_HTML.gif the next inequality holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ21_HTML.gif
      (219)

      where functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq37_HTML.gif are real nonnegative for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq38_HTML.gif , function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq40_HTML.gif it results in

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ22_HTML.gif
      (220)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq41_HTML.gif
      http://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) http://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) http://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) http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ24_HTML.gif
      (222)
      (d) http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ25_HTML.gif
      (223)
      (e) http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ26_HTML.gif
      (224)
      such that
      http://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]):
        http://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 http://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:
      http://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 http://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

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

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

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

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

      Consider http://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

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

      from which it follows

      http://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

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

      where

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq56_HTML.gif .

      If http://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

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

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

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

      (iv) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq62_HTML.gif are independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq63_HTML.gif ) and such estimate is valid:

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq64_HTML.gif , if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq66_HTML.gif then the trivial solution is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq67_HTML.gif -stable by Chetaev (uniformly http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq68_HTML.gif -stable, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq70_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq71_HTML.gif ).

      If http://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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ37_HTML.gif
      (38)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ38_HTML.gif
      (39)
      http://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) http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq75_HTML.gif -stable by Chetaev (uniformly if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq76_HTML.gif is independent of http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq78_HTML.gif ).

      Remark 3.6.

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

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq82_HTML.gif and defined in the domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq83_HTML.gif , http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq85_HTML.gif

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

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

      It is easy to see that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_Equ43_HTML.gif
      (314)
      http://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 http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F808124/MediaObjects/13661_2008_Article_879_IEq89_HTML.gif ;

      (b) http://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:

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