On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications
© A. Gallo and A.M. Piccirillo 2009
Received: 22 December 2008
Accepted: 28 May 2009
Published: 29 June 2009
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.
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 . 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  some new comparison principles and the monotone iterative technique to establish a more general existence theorem for a periodic boundary value problem. Reference  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.
- (a)Let one suppose that for the following integro-sum functional inequality holds:(21)
Function satisfies such conditions:
(iii) is nondecreasing.
and estimate (2.2) is valid in .
Let us suppose that for estimate (2.2) is fulfilled. Then for every we have
where is determined from (2.3)–(2.5).
Taking into account such inequality
we obtain estimate (2.2) for every .
Let us consider the class of functions such that
(i) positive, continuous, nondecreasing for ;
The following result is proved.
As in the previously theorem, the proof is completed by using the inductive method.
The following result is easily to obtain
where functions are real nonnegative for , function satisfies conditions (i),…,(iii) of Theorem 2.1.
Then for it results in
The proof the same procedure as that of (Iovane [21, Theorems 2.1 and 3.1]).
(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];
- (f)W(u) = um,, m> 0, then estimate (2.2) reduces as follows (see [21, Theorem 2.2]):(226)
which coincide with result of [21, Theorem 3.1] for .
Let us consider the following system of differential equations
Let us assume that and are defined in the domain and satisfy such conditions:
W satisfies conditions (i)–(iii) of Theorem 2.1;
Consider the solution of Cauchy problem for system (3.1). Then
from which it follows
By using the result of Theorem 2.1 and estimate (2.2) we obtain
Let us consider some particular cases of .
If , estimate (3.4) is reduced in such form
Then such result holds.
Let the following conditions be fulfilled for system (3.1) :
Then one has:
(a)All solutions of system (3.1) are bounded (uniformly, if are independent of ) and such estimate is valid:
(b)The trivial solution of system (3.1) is stable by Lyapunov (uniformly stable relative , if ).
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
From estimate (3.8) the next propositions follow.
Suppose that such conditions occur:
(b)estimates ii–iv of Proposition 3.1 be fulfilled.
Then all the solutions of system (3.1) are bounded (uniformly if ).
Then trivial solution of system (3.1) is -stable by Chetaev (uniformly if is independent of ).
Then trivial solution of system (3.1) is stable by Lyapunov (uniformly if ).
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:
where and defined in the domain , .
We suppose that such conditions are valid:
It is easy to see that
From estimate (3.15) such result follows.
Let one suppose that for system (3.13) conditions (i)–(iii) take place for and the following estimates are fulfilled:
Then we have:
(i)All solutions of system (3.13) are bounded and satisfy the estimate:
The trivial solution of system (3.13) is -stable by Chetaev (uniformly if is independent of ) and
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