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)
    (11)
     
  1. (b)
    (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 of continuous functions ( is the delaying argument). The following holds.

Theorem 2.1.
  1. (a)
    Let one suppose that for the following integro-sum functional inequality holds:
    (21)
     
where is a positive nondecreasing function, function is a nonnegative piecewise-continuous,with I-st kind of discontinuities in the points , belongs to the class .
  1. (b)

    Function satisfies such conditions:

     

(i)

(ii)

(iii) is nondecreasing.

Then for arbitrary the next estimate holds:
(22)
(23)
(24)
(25)

Proof.

It follows from inequality (2.1)
(26)
Denoting by
(27)
then
(28)
Let us consider the interval Then
(29)
where So it results in
(210)

and estimate (2.2) is valid in .

Let us suppose that for estimate (2.2) is fulfilled. Then for every we have

(211)

where is determined from (2.3)–(2.5).

Taking into account such inequality

(212)

we obtain estimate (2.2) for every .

Let us consider the class of functions such that

(i) positive, continuous, nondecreasing for ;

(ii)

(iii)

The following result is proved.

Theorem 2.2.

Suppose that the part (a) of Theorem 2.1 is valid and function belongs to the class Then for arbitrary such estimate holds:
(213)
where
(214)

and

Proof.

By using the previous theorem we have . On the interval
(215)
Then
(216)
Taking into account estimate (2.16), we obtain
(217)
Then in we have
(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 the next inequality holds:
(219)

where functions are real nonnegative for , function satisfies conditions (i),,(iii) of Theorem 2.1.

Then for it results in

(220)
where
(221)

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

Corollary 2.4.

Suppose that

(a) , then the result of Theorem 2.1 coincides with the result [22, Theorem 3.7.1, page 232];

(b) then the result of Theorem 2.1 coincides with result [12, Proposition 2.3, page 2143];

(c) , then one obtains the analogy of Gronwall- Bellman result for discontinuous functions [23, Lemma 1] and estimate (2.2) reduces in the following form:
(222)
(d) , then one obtains the result [21, Theorem 2.1] and estimate (2.2) are as follows:
(223)
(e) then one obtains the analogy of Bihari result for discontinuous functions [23, Lemma 2] and estimate (2.2) reduces as follows are reduced:
(224)
such that
(225)
  1. (f)
    W(u) = um,, m> 0, then estimate (2.2) reduces as follows (see [21, Theorem 2.2]):
    (226)
     
(g)Suppose that in Theorem 2.3 then estimates (2.20), (2.21) reduce as shown:
(227)

which coincide with result of [21, Theorem 3.1] for .

3. Applications

Let us consider the following system of differential equations

(31)

where .

Let us assume that and are defined in the domain and satisfy such conditions:

(a)

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

(b) .

Consider the solution of Cauchy problem for system (3.1). Then

(32)

from which it follows

(33)

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

(34)

where

(35)

Let us consider some particular cases of .

If , estimate (3.4) is reduced in such form

(36)

Then such result holds.

Proposition 3.1.

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

(i)

(ii)

(iii)

(iv)

Then one has:

(a)All solutions of system (3.1) are bounded (uniformly, if are independent of ) and such estimate is valid:

(37)

(b)The trivial solution of system (3.1) is stable by Lyapunov (uniformly stable relative , if ).

Remark 3.2.

If conditions I–IV of Proposition 3.1 are valid and then the trivial solution is -stable by Chetaev (uniformly -stable, if , is independent of ).

If the estimate (3.4) is reduced in such form

(38)
(39)
(310)

From estimate (3.8) the next propositions follow.

Proposition 3.3.

Suppose that such conditions occur:

(a)

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

Then all the solutions of system (3.1) are bounded (uniformly if ).

Remark 3.4.

Suppose that conditions (a), (b) of Proposition 3.3 are valid and
(311)

Then trivial solution of system (3.1) is -stable by Chetaev (uniformly if is independent of ).

Proposition 3.5.

Let conditions ii–iv of Proposition 3.1 be fulfilled for system (3.1), inequality (3.10) holds and
(312)

Then trivial solution of system (3.1) is stable by Lyapunov (uniformly if ).

Remark 3.6.

If , and the conditions of boundedness, stability, -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:

(313)

where and defined in the domain , .

We suppose that such conditions are valid:

(i)

(ii)

(iii) .

It is easy to see that

(314)
(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 and the following estimates are fulfilled:

(a) ;

(b)

Then we have:

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

(316)
(ii)The trivial solution of system (3.13) is stable by Lyapunov (uniformly, if ).
  1. (iii)

    The trivial solution of system (3.13) is -stable by Chetaev (uniformly if is independent of ) and

     

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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