Abstract and Applied AnalysisVolume 2011 (2011), Article ID 792689, 13 pagesdoi:10.1155/2011/792689

Research Article

Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

Alexander Boichuk,¹ Martina Langerová,² and Jaroslava Škoríková²

¹Institute of Mathematics, National Academy of Science of Ukraine, 01601 Kyiv, Ukraine²Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia

Received 31 December 2010; Revised 1 July 2011; Accepted 1 July 2011

Academic Editor: Josef Diblík

Copyright © 2011 Alexander Boichuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The weakly perturbed linear nonhomogeneous impulsive systems in the form , and are considered. Under the assumption that the generating system (for ) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter with finitely many terms with negative powers of , and we suggest an algorithm of construction of these solutions.

1. Introduction

In this contribution we study the problem of existence and construction of solutions of weakly perturbed linear differential systems with impulsive action bounded on the entire real axis. The application of the theory of differential systems with impulsive action (developed in [1–3]), the well-known results on the splitting index by Sacker [4] and by Palmer [5] on the Fredholm property of bounded solutions of linear systems of ordinary differential equations [6–9], the theory of pseudoinverse matrices [10] and results obtained in analyzing boundary-value problems for ordinary differential equations (see [10–12]), enables us to obtain existence conditions and to propose an algorithm for the construction of solutions bounded on the entire real axis of weakly perturbed linear impulsive differential systems.

2. Initial Problem

We consider the problem of existence and construction of solutions bounded on the entire real axis of linear systems of ordinary differential equations with impulsive action at fixed points of time where is an matrix of functions, is an vector function, is the Banach space of real vector functions bounded on and left-continuous for with discontinuities of the first kind at with the norm: , are -dimensional column constant vectors: ; , and .

The solution of the system (2.1) is sought in the Banach space of -dimensional bounded on and piecewise continuously differentiable vector functions with discontinuities of the first kind at .

Parallel with the nonhomogeneous impulsive system (2.1), we consider the corresponding homogeneous system which is the homogeneous system without impulses, and let be the fundamental matrix of (2.2) such that .

Assume that the homogeneous system (2.2) is exponentially dichotomous (e-dichotomous) [5, 10] on semiaxes and , that is, there exist projectors and and constants , such that the following inequalities are satisfied:

For getting the solution bounded on the entire axis, we assume that , that is, .

We use the following notation: ; is a Moore-Penrose pseudoinverse matrix to ; and are matrices (orthoprojectors) projecting onto and onto , respectively, that is, , , and , ; ;   and .

The existence conditions and the structure of solutions of system (2.1) bounded on the entire real axis was analyzed in [13]. Here the following theorem was formulated and proved.

Theorem 2.1. Assume that the linear nonhomogeneous impulsive differential system (2.1) has the corresponding homogeneous system (2.2) e-dichotomous on the semiaxes and with projectors and , respectively. Then the homogeneous system (2.2) has exactly linearly independent solutions bounded on the entire real axis. If nonhomogeneities and satisfy linearly independent conditions then the nonhomogeneous system (2.1) possesses an -parameter family of linearly independent solutions bounded on in the form

Here, is a matrix formed by a complete system of linearly independent rows of matrix ,

is an matrix formed by a complete system of linearly independent solutions bounded on of homogeneous system (2.2), and is the generalized Green operator of the problem of finding bounded solutions of the nonhomogeneous impulsive system (2.1), acting upon and , defined by the formula with the following property

These results are required to establish new conditions for the existence of solutions of weakly perturbed linear impulsive systems bounded on the entire real axis.

3. Perturbed Problems

Consider a weakly perturbed nonhomogeneous linear impulsive system in the form where is an matrix of functions, are constant matrices.

Assume that the condition of solvability (2.4) of the generating system (2.1) (obtained from system (3.1) for ) is not satisfied for all nonhomogeneities and , that is, system (2.1) does not have solutions bounded on the entire real axis. Therefore, we analyze whether the system (2.1) can be made solvable by introducing linear perturbations to the differential system and to the pulsed conditions. Also it is important to determine perturbations and required to make the problem (3.1) solvable in the space of functions bounded on the entire real axis, that is, it is necessary to specify pertubations for which the corresponding homogeneous system turns into a system e-trichotomous or e-dichotomous on the entire real axis [10].

We show that this problem can be solved using the matrix

constructed with the coefficients of the system (3.1). The Vishik-Lyusternik method developed in [14] enables us to establish conditions under which a solution of impulsive system (3.1) can be represented by a function bounded on the entire real axis in the form of a Laurent series in powers of the small fixed parameter with finitely many terms with negative powers of .

We use the following notation: is the unique matrix pseudoinverse to in the Moore-Penrose sense, is the matrix (orthoprojector) projecting the space to the null space of the matrix , that is, , and is the matrix (orthoprojector) projecting the space to the null space of the matrix (), that is, .

Now we formulate and prove a theorem that enables us to solve indicated problem.

Theorem 3.1. Suppose that the system (3.1) satisfies the conditions imposed above, and the homogeneous system (2.2) is e-dichotomous on and with projectors P and Q, respectively. Let nonhomogeneities and be given such that the condition (2.4) is not satisfied and the generating system (2.1) does not have solutions bounded on the entire real axis. If then the system (3.2) is e-trichotomous on and, for all nonhomogeneities and , the system (3.1) possesses at least one solution bounded on in the form of a series uniformly convergent for sufficiently small fixed .

Here, is a proper constant characterizing the range of convergence of the series (3.5) and the coefficients of the series (3.5) are determined from the corresponding impulsive systems as

Proof. We suppose that the problem (3.1) has a solution in the form of a Laurent series (3.5). We substitute this solution into the system (3.1) and equate the coefficients at the same powers of . The problem of determination of the coefficient of the term with in series (3.5) is reduced to the problem of finding solutions of homogeneous system without impulses bounded on the entire real axis. According to the Theorem 2.1, the homogeneous system (3.7) possesses -parameter family of solutions bounded on the entire real axis, where is an -dimensional vector column and is determined from the condition of solvability of the problem used for determining the coefficient of the series (3.5).For , the problem of determination of the coefficient of series (3.5) reduces to the problem of finding solutions of the following nonhomogeneous system: bounded on the entire real axis. According to the Theorem 2.1, the condition of solvability of this problem takes the form Using the matrix , we get the following algebraic system for : which is solvable if and only if the condition is satisfied, that is, if In this case, this algebraic system is solvable with respect to within an arbitrary vector constant from the null space of the matrix , and one of its solutions has the form Therefore, under condition (3.4), the nonhomogeneous system (3.9) possesses an -parameter set of solution bounded on in the form where is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.9), and is an -dimensional constant vector determined in the next step of the process from the condition of solvability of the impulsive problem for coefficient .We continue this process by problem of determination of the coefficient of the term with in the series (3.5). It reduces to the problem of finding solutions of the system bounded on the entire real axis. If the condition (3.4) is satisfied and by using the condition of solvability of this problem, that is, we determine the vector (within an arbitrary vector constant , ) as Thus, under the condition (3.4), system (3.16) possesses an -parameter set of solutions bounded on in the form where is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.16), and is an -dimensional constant vector determined in the next stage of the process from the condition of solvability of the problem for .If we continue this process, we prove (by induction) that the problem of determination of the coefficient in the series (3.5) is reduced to the problem of finding solutions of the system bounded on the entire real axis. If the condition (3.4) is satisfied, then a solution of this problem bounded on has the form where is the generalized Green operator of the problem of finding bounded solutions of impulsive system (3.20) and the constant vector is given by the formula (within an arbitrary vector constant , ).The fact that the series (3.5) is convergent can be proved by using the procedure of majorization.

In the case where the number of linear independent solutions of system (2.2) bounded on is equal to the number , Theorem 3.1 yields the following assertion.

Corollary 3.2. Suppose that the system (3.1) satisfies the conditions imposed above, and the homogeneous system (2.2) is e-dichotomous on and with projectors and , respectively. Let nonhomogeneities and be given such that the condition (2.4) is not satisfied, and the generating system (2.1) does not have solutions bounded on the entire real axis. If condition is satisfied, then the system (3.1) possesses a unique solution bounded on in the form of series (3.5) uniformly convergent for sufficiently small fixed .

Proof. If , then is a square matrix. Therefore, it follows from condition (3.4) that , which is equivalent to the condition (3.23). In this case, the constant vectors are uniquely determined from (3.22). The coefficients of the series (3.5) are also uniquely determined by (3.21), and, for all and , the system (3.1) possesses a unique solution bounded on , which means that system (3.2) is e-dichotomous.

We now illustrate the assertions proved above.

Example 3.3. Consider the impulsive system where The generating homogenous system (for ) has the form and is e-dichotomous (as shown in [6]) on the semiaxes and with projectors and . The normal fundamental matrix of this system is Thus, we have         In order that the generating impulsive system (2.1) with the matrix specified above has solutions bounded on the entire real axis, the nonhomogeneities and must satisfy condition (2.4). In this analyzed impulsive problem, this condition takes the form Let and be given such that the condition (3.30) is not satisfied and the corresponding generating system (2.1) does not have solutions bounded on the entire real axis. The system (3.24) will be an e-trichotomous on if the coefficients of the perturbing matrix and the coefficients of the perturbing matrix satisfy condition (3.4), that is, , where the matrix has the form Therefore, if and are such that at least one of the following inequalities is satisfied, then either the condition (3.4) or the equivalent condition from Theorem 3.1 is satisfied and the system (3.2) is e-trichotomous on . In this case, the coefficients , , ,  are arbitrary functions from the space , and , ,  are arbitrary constants from . Moreover, for any a solution of the system (3.24) bounded on is given by the series (3.5) (within a constant from the null space , ).

Another Perturbed ProblemIn this part, we show that the problem of finding bounded solutions of nonhomogeneous system (2.1), in the case if the condition (2.4) is not satisfied, can be made solvable by introducing linear perturbations only to the pulsed conditions.Therefore, we consider the weakly perturbed nonhomogeneous linear impulsive system in the form where are constant matrices. For , we obtain the generating system (2.1). We assume that this generating system does not have solutions bounded on the entire real axis, which means that the condition of solvability (2.4) is not satisfied (for some nonhomogeneities and ). Let us show that it is possible to make this problem solvable by adding linear perturbation only to the pulsed conditions. In the case, if this is possible, it is necessary to determine perturbations for which the corresponding homogeneous system turns into the system e-trichotomous or e-dichotomous on the entire real axis.This problem can be solved with help of the matrix constructed with the coefficients from the impulsive system (3.34).By using Theorem 3.1, we seek a solution in the form of the series (3.5). Thus, we have the following corollary.

Corollary 3.4. Suppose that the system (3.34) satisfies the conditions imposed above and the generating homogeneous system (2.2) is e-dichotomous on and with projectors and , respectively. Let nonhomogeneities and be given such that the condition (2.4) is not satisfied, and the generating system (2.1) does not have solutions bounded on the entire real axis. If the condition (3.4) is satisfied, then the system (3.35) is e-trichotomous on , and the system (3.34) possesses at least one solution bounded on in the form of series (3.5) uniformly convergent for sufficiently small fixed .

Acknowledgments

This research was supported by Grants 1/0771/08 of the Grant Agency of Slovak Republic (VEGA) and project APVV-0700-07 of Slovak Research and Development Agency.

References

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, Vyshcha Shkola, Kiev, Ukraine, 1974.

S. Schwabik, M. Tvrdý, and O. Vejvoda, Differential and Integral Equations, Academia, Prague, Czech Republic, 1979.

A. A. Boichuk, N. A. Perestyuk, and A. M. Samoilenko, “Periodic solutions of impulse differential systems in critical cases,” Differents. Uravn., vol. 27, no. 9, pp. 1516–1521, 1991.

R. J. Sacker, “The splitting index for linear differential systems,” Journal of Differential Equations, vol. 33, no. 3, pp. 368–405, 1979.

K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” Journal of Differential Equations, vol. 55, no. 2, pp. 225–256, 1984.

A. M. Samoilenko, A. A. Boichuk, and An. A. Boichuk, “Solutions, bounded on the whole axis, of linear weakly perturbed systems,” Ukrainian Mathematical Journal, vol. 54, no. 11, pp. 1517–1530, 2002.

A. A. Boichuk, “Solutions of weakly nonlinear differential equations bounded on the whole line,” Nonlinear Oscillations, vol. 2, no. 1, pp. 3–10, 1999.

A. Boichuk and A. Pokutnyi, “Bounded solutions of linear perturbed differential equations in a Banach space,” Tatra Mountains Mathematical Publications, vol. 39, pp. 1–12, 2007.

F. Battelli and M. Fečkan, “Chaos in singular impulsive O.D.E.,” Nonlinear Analysis, vol. 28, no. 4, pp. 655–671, 1997.

A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, Koninklijke Brill NV, Utrecht, Mass, USA, 2004.

A. A. Boichuk, J. Diblík, D. Khusainov, and M. Růžičková, “Fredholm's boundary-value problems for differential systems with a single delay,” Nonlinear Analysis, vol. 72, no. 5, pp. 2251–2258, 2010.

A. A. Boichuk, J. Diblík, D. Khusainov, and M. Růžičková, “Boundary-value problems for weakly nonlinear delay differential systems,” Abstract and Applied Analysis, vol. 2011, Article ID 631412, 19 pages, 2011.

A. Boichuk, M. Langerová, and J. Škoríková, “Solutions of linear impulsive differential systems bounded on the entire real axis,” Advances in Difference Equations, vol. 2010, Article ID 494379, 10 pages, 2010.

M. I. Vishik and L. A. Lyusternik, “The solution of some perturbation problems for matrices and selfadjoint differential equations,” Uspekhi Matematicheskikh Nauk, vol. 15, no. 3, pp. 3–80, 1960.