- Research Article
- Open Access
On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications
Boundary Value Problems volume 2009, Article number: 808124 (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.
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:
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.
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 .
Function satisfies such conditions:
(iii) is nondecreasing.
Then for arbitrary the next estimate holds:
It follows from inequality (2.1)
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;
The following result is proved.
Suppose that the part (a) of Theorem 2.1 is valid and function belongs to the class Then for arbitrary such estimate holds:
By using the previous theorem we have . On the interval
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
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
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];
(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:
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:
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 ).
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 ).
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 ).
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:
(ii)The trivial solution of system (3.13) is stable by Lyapunov (uniformly, if ).
The trivial solution of system (3.13) is -stable by Chetaev (uniformly if is independent of ) and
Banov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
Borysenko SD, Iovane G, Giordano P: Investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(2):345–363. 10.1016/j.na.2005.03.031
Borysenko SD, Ciarletta M, Iovane G: Integro-sum inequalities and motion stability of systems with impulse perturbations. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(3):417–428. 10.1016/j.na.2005.03.032
Borysenko S, Iovane G: About some new integral inequalities of Wendroff type for discontinuous functions. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(10):2190–2203. 10.1016/j.na.2006.03.008
Hu SC, Lakshmikantham V, Leela S: Impulsive differential systems and the pulse phenomena. Journal of Mathematical Analysis and Applications 1989, 137(2):605–612. 10.1016/0022-247X(89)90266-7
Lakshmikantham V, Leela S: Differential and Integral Inequalities, Theory and Applications. Academic Press, New York, NY, USA; 1969.
Lakshmikantham V, Leela S, Mohan Rao Rama M: Integral and integro-differential inequalities. Applicable Analysis 1987, 24(3):157–164. 10.1080/00036818708839660
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.
Martyhyuk AA, Lakshmikantham V, Leela S: Stability of Motion: the Method of Integral Inequalities. Naukova Dumka, Kyiv, Russia; 1989.
Mitropolskiy YuA, Leela S, Martynyuk AA: Some trends in V. Lakshmikantham's investigations in the theory of differential equations and their applications. Differentsial'nye Uravneniya 1986, 22(4):555–572.
Mitropolskiy YuA, Samoilenko AM, Perestyuk N: On the problem of substantiation of overoging method for the second equations with impulse effect. Ukrainskii Matematicheskii Zhurnal 1977, 29(6):750–762.
Mitropolskiy YuA, Iovane G, Borysenko SD: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(10):2140–2165. 10.1016/j.na.2006.03.006
Samoilenko AM, Perestyuk N: Differential Equations with Impulse Effect. Visha Shkola, Kyiv, Russia; 1987.
Gallo A, Piccirillo AM: About new analogies of Gronwall-Bellman-Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(5):1550–1559. 10.1016/j.na.2006.07.038
Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489–493. 10.1016/S0893-9659(01)00163-X
Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications 1997, 205(2):423–433. 10.1006/jmaa.1997.5207
Nieto JJ: Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 2002, 51(7):1223–1232. 10.1016/S0362-546X(01)00889-6
Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(6):2248–2260. 10.1016/j.na.2008.03.004
Nieto JJ, O'Regan D: Variational approach to impulsive differential equations. Nonlinear Analysis: Real World Applications 2009, 10(2):680–690. 10.1016/j.nonrwa.2007.10.022
Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366.
Iovane G: Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(2):498–508. 10.1016/j.na.2005.11.043
Samoilenko A, Borysenko S, Cattani C, Matarazzo G, Yasinsky V: Differential Models: Stability, Inequalities and Estimates. Naukova Dumka, Kiev, Russia; 2001:328.
Borysenko DS, Gallo A, Toscano R: Integral inequalities Gronwall-Bellman type for discontinuous functions and estimates of solutions impulsive systems. Proc.DE@CAS, 2005, Brest 5–9.
About this article
Cite this article
Gallo, A., Piccirillo, A.M. On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications. Bound Value Probl 2009, 808124 (2009). https://doi.org/10.1155/2009/808124
- Integral Inequality
- Discontinuous Function
- Critical Point Theory
- Impulsive Differential Equation
- Impulsive System