We present some new nonlinear integral inequalities BellmanBihari type with delay for discontinuous functions (integrosum 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.
1. Introduction
The first generalizations of the Bihari result for discontinuous functions which satisfy nonlinear impulse inequality (integrosum inequality) are connected with such types of inequalities:
(a)
(b)
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 (onedimensional 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 stateoftheart 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 integrodifferential systems.
2. General Bihari Theorems for IntegroFunctional Inequalities for Discontinuous Functions
Let us consider the class of continuous functions ( is the delaying argument). The following holds.
Theorem 2.1.
(a) Let one suppose that for the following integrosum functional inequality holds:
where is a positive nondecreasing function, function is a nonnegative piecewisecontinuous,with Ist kind of discontinuities in the points , belongs to the class .
(b) Function satisfies such conditions:
(i)
(ii)
(iii) is nondecreasing.
Then for arbitrary the next estimate holds:
Proof.
It follows from inequality (2.1)
Denoting by
then
Let us consider the interval Then
where So it results in
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;
(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:
where
and
Proof.
By using the previous theorem we have . On the interval
Then
Taking into account estimate (2.16), we obtain
Then in we have
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:
where functions are real nonnegative for, function satisfies conditions (i),…,(iii) of Theorem 2.1.
Then for it results in
where
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:
(d), then one obtains the result [21, Theorem 2.1] and estimate (2.2) are as follows:
(e) then one obtains the analogy of Bihari result for discontinuous functions [23, Lemma 2] and estimate (2.2) reduces as follows are reduced:
such that
(f) W(u) = u^{m,}, m> 0, then estimate (2.2) reduces as follows (see [21, Theorem 2.2]):
(g)Suppose that in Theorem 2.3 then estimates (2.20), (2.21) reduce as shown:
which coincide with result of [21, Theorem 3.1] for.
3. Applications
Let us consider the following system of differential equations
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
from which it follows
By using the result of Theorem 2.1 and estimate (2.2) we obtain
where
Let us consider some particular cases of .
If , estimate (3.4) is reduced in such form
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:
(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
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
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
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 nonLipschitz type of discontinuities are investigated in [23, see Proposition 1, Proposition 2].
Let us consider the following impulsive system of integrodifferential equations:
where and defined in the domain , .
We suppose that such conditions are valid:
(i)
(ii)
(iii).
It is easy to see that
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:
(ii)The trivial solution of system (3.13) is stable by Lyapunov (uniformly, if ).
(iii) The trivial solution of system (3.13) is stable by Chetaev (uniformly if is independent of ) and
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