Abstract and Applied Analysis

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Volume 2011 |Article ID 792689 | https://doi.org/10.1155/2011/792689

Alexander Boichuk, Martina LangerovĆ”, Jaroslava Å korĆ­kovĆ”, "Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems", Abstract and Applied Analysis, vol. 2011, Article ID 792689, 13 pages, 2011. https://doi.org/10.1155/2011/792689

Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

Academic Editor: Josef DiblĆ­k
Received31 Dec 2010
Revised01 Jul 2011
Accepted01 Jul 2011
Published15 Sep 2011

Abstract

The weakly perturbed linear nonhomogeneous impulsive systems in the form Ģ‡š‘„=š“(š‘”)š‘„+šœ€š“1(š‘”)š‘„+š‘“(š‘”),š‘”āˆˆā„,š‘”āˆ‰š’Æāˆ¶={šœš‘–}ā„¤,Ī”š‘„|š‘”=šœš‘–=š›¾š‘–+šœ€š“1š‘–š‘„(šœš‘–āˆ’),šœš‘–āˆˆš’ÆāŠ‚ā„,š›¾š‘–āˆˆā„š‘›, and š‘–āˆˆā„¤ are considered. Under the assumption that the generating system (for šœ€=0) 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 šœ€ with finitely many terms with negative powers of šœ€, 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 [1ā€“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 [6ā€“9], the theory of pseudoinverse matrices [10] and results obtained in analyzing boundary-value problems for ordinary differential equations (see [10ā€“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 timeĢ‡š‘„=š“(š‘”)š‘„+š‘“(š‘”),š‘”āˆˆā„ā§µš’Æ,Ī”š‘„|š‘”=šœš‘–=š›¾š‘–,šœš‘–āˆˆš’Æ,š‘–āˆˆā„¤,(2.1) where š“āˆˆšµš¶š’Æ(ā„) is an š‘›Ć—š‘› matrix of functions, š‘“āˆˆšµš¶š’Æ(ā„) is an š‘›Ć—1 vector function, šµš¶š’Æ(ā„) is the Banach space of real vector functions bounded on ā„ and left-continuous for š‘”āˆˆā„ with discontinuities of the first kind at š‘”āˆˆš’Æāˆ¶={šœš‘–}ā„¤ with the norm: ā€–š‘„ā€–šµš¶š’Æ(ā„)āˆ¶=supš‘”āˆˆā„ā€–š‘„(š‘”)ā€–, š›¾š‘– are š‘›-dimensional column constant vectors: š›¾š‘–āˆˆā„š‘›; ā‹Æ<šœāˆ’2<šœāˆ’1<šœ0=0<šœ1<šœ2<ā‹Æ, and Ī”š‘„|š‘”=šœš‘–āˆ¶=š‘„(šœš‘–+)āˆ’š‘„(šœš‘–āˆ’).

The solution š‘„(š‘”) of the system (2.1) is sought in the Banach space of š‘›-dimensional bounded on ā„ and piecewise continuously differentiable vector functions with discontinuities of the first kind at š‘”āˆˆš’Æāˆ¶š‘„āˆˆšµš¶1š’Æ(ā„).

Parallel with the nonhomogeneous impulsive system (2.1), we consider the corresponding homogeneous systemĢ‡š‘„=š“(š‘”)š‘„,Ī”š‘„|š‘”=šœš‘–=0,(2.2) which is the homogeneous system without impulses, and let š‘‹(š‘”) be the fundamental matrix of (2.2) such that š‘‹(0)=š¼.

Assume that the homogeneous system (2.2) is exponentially dichotomous (e-dichotomous) [5, 10] on semiaxes ā„āˆ’=(āˆ’āˆž,0] and ā„+=[0,āˆž), that is, there exist projectors š‘ƒ and š‘„(š‘ƒ2=š‘ƒ,š‘„2=š‘„) and constants š¾š‘–ā‰„1, š›¼š‘–>0(š‘–=1,2) such that the following inequalities are satisfied:ā€–ā€–š‘‹(š‘”)š‘ƒš‘‹āˆ’1ā€–ā€–(š‘ )ā‰¤š¾1š‘’āˆ’š›¼1(š‘”āˆ’š‘ )ā€–ā€–š‘‹,š‘”ā‰„š‘ ,(š‘”)(š¼āˆ’š‘ƒ)š‘‹āˆ’1ā€–ā€–(š‘ )ā‰¤š¾1š‘’āˆ’š›¼1(š‘ āˆ’š‘”),š‘ ā‰„š‘”,š‘”,š‘ āˆˆā„+,ā€–ā€–š‘‹(š‘”)š‘„š‘‹āˆ’1ā€–ā€–(š‘ )ā‰¤š¾2š‘’āˆ’š›¼2(š‘”āˆ’š‘ )ā€–ā€–,š‘”ā‰„š‘ ,š‘‹(š‘”)(š¼āˆ’š‘„)š‘‹āˆ’1ā€–ā€–(š‘ )ā‰¤š¾2š‘’āˆ’š›¼2(š‘ āˆ’š‘”),š‘ ā‰„š‘”,š‘”,š‘ āˆˆā„āˆ’.(2.3)

For getting the solution š‘„āˆˆšµš¶1š’Æ(ā„) bounded on the entire axis, we assume that š‘”=0āˆ‰š’Æ, that is, š‘„(0+)āˆ’š‘„(0āˆ’)=š›¾0=0.

We use the following notation: š·=š‘ƒāˆ’(š¼āˆ’š‘„); š·+ is a Moore-Penrose pseudoinverse matrix to š·; š‘ƒš· and š‘ƒš·āˆ— are š‘›Ć—š‘› matrices (orthoprojectors) projecting ā„š‘› onto š‘(š·)=kerš· and onto š‘(š·āˆ—)=kerš·āˆ—, respectively, that is, š‘ƒš·āˆ¶ā„š‘›ā†’š‘(š·), š‘ƒ2š·=š‘ƒš·=š‘ƒāˆ—š·, and š‘ƒš·āˆ—āˆ¶ā„š‘›ā†’š‘(š·āˆ—), š‘ƒ2š·āˆ—=š‘ƒš·āˆ—=š‘ƒāˆ—š·āˆ—; š»(š‘”)=[š‘ƒš·āˆ—š‘„]š‘‹āˆ’1(š‘”);ā€‰ā€‰š‘‘=rank[š‘ƒš·āˆ—š‘„]=rank[š‘ƒš·āˆ—(š¼āˆ’š‘ƒ)] and š‘Ÿ=rank[š‘ƒš‘ƒš·]=rank[(š¼āˆ’š‘„)š‘ƒš·].

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 ā„āˆ’=(āˆ’āˆž,0] and ā„+=[0,āˆž) with projectors š‘ƒ and š‘„, respectively. Then the homogeneous system (2.2) has exactly š‘Ÿ linearly independent solutions bounded on the entire real axis. If nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘› satisfy š‘‘ linearly independent conditions ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš›¾š‘–=0,(2.4) then the nonhomogeneous system (2.1) possesses an š‘Ÿ-parameter family of linearly independent solutions bounded on ā„ in the form š‘„ī€·š‘”,š‘š‘Ÿī€ø=š‘‹š‘Ÿ(š‘”)š‘š‘Ÿ+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š‘“š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (š‘”),āˆ€š‘š‘Ÿāˆˆā„š‘Ÿ.(2.5)

Here, š»š‘‘(š‘”)=[š‘ƒš·āˆ—š‘„]š‘‘š‘‹āˆ’1(š‘”) is a š‘‘Ć—š‘› matrix formed by a complete system of š‘‘ linearly independent rows of matrix š»(š‘”), š‘‹š‘Ÿī€ŗ(š‘”)āˆ¶=š‘‹(š‘”)š‘ƒš‘ƒš·ī€»š‘Ÿī€ŗ=š‘‹(š‘”)(š¼āˆ’š‘„)š‘ƒš·ī€»š‘Ÿ(2.6)

is an š‘›Ć—š‘Ÿ matrix formed by a complete system of š‘Ÿ linearly independent solutions bounded on ā„ of homogeneous system (2.2), and ī€·šŗ[š‘“š›¾š‘–]ī€ø(š‘”) is the generalized Green operator of the problem of finding bounded solutions of the nonhomogeneous impulsive system (2.1), acting upon š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘›, defined by the formula āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š‘“š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ āŽ§āŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽØāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽŖāŽ©ī€œ(š‘”)=š‘‹(š‘”)š‘”0š‘ƒš‘‹āˆ’1(ī€œš‘ )š‘“(š‘ )š‘‘š‘ āˆ’āˆžš‘”(š¼āˆ’š‘ƒ)š‘‹āˆ’1(+š‘ )š‘“(š‘ )š‘‘š‘ š‘—ī“š‘–=1š‘ƒš‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–āˆ’āˆžī“š‘–=š‘—+1(š¼āˆ’š‘ƒ)š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–+š‘ƒš·+ī‚»ī€œ0āˆ’āˆžš‘„š‘‹āˆ’1ī€œ(š‘ )š‘“(š‘ )š‘‘š‘ +āˆž0(š¼āˆ’š‘ƒ)š‘‹āˆ’1+(š‘ )š‘“(š‘ )š‘‘š‘ āˆ’1ī“š‘–=āˆ’āˆžš‘„š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–+āˆžī“š‘–=1(š¼āˆ’š‘ƒ)š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–īƒ°ī€œ,š‘”ā‰„0;š‘”āˆ’āˆžš‘„š‘‹āˆ’1ī€œ(š‘ )š‘“(š‘ )š‘‘š‘ āˆ’0š‘”(š¼āˆ’š‘„)š‘‹āˆ’1+(š‘ )š‘“(š‘ )š‘‘š‘ āˆ’(š‘—+1)ī“š‘–=āˆ’āˆžš‘„š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–āˆ’āˆ’1ī“š‘–=āˆ’š‘—(š¼āˆ’š‘„)š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–+(š¼āˆ’š‘„)š·+ī‚»ī€œ0āˆ’āˆžš‘„š‘‹āˆ’1ī€œ(š‘ )š‘“(š‘ )š‘‘š‘ +āˆž0(š¼āˆ’š‘ƒ)š‘‹āˆ’1+(š‘ )š‘“(š‘ )š‘‘š‘ āˆ’1ī“š‘–=āˆ’āˆžš‘„š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–+āˆžī“š‘–=1(š¼āˆ’š‘ƒ)š‘‹āˆ’1ī€·šœš‘–ī€øš›¾š‘–īƒ°,š‘”ā‰¤0,(2.7) with the following property āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š‘“š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š‘“š›¾(0āˆ’)āˆ’š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€œ(0+)=āˆžāˆ’āˆžš»(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»ī€·šœš‘–ī€øš›¾š‘–.(2.8)

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 formĢ‡š‘„=š“(š‘”)š‘„+šœ€š“1(š‘”)š‘„+š‘“(š‘”),š‘”āˆˆā„ā§µš’Æ,Ī”š‘„|š‘”=šœš‘–=š›¾š‘–+šœ€š“1š‘–š‘„ī€·šœš‘–āˆ’ī€ø,šœš‘–āˆˆš’Æ,š›¾š‘–āˆˆā„š‘›,š‘–āˆˆā„¤,(3.1) where š“1āˆˆšµš¶š’Æ(ā„) is an š‘›Ć—š‘› matrix of functions, š“1š‘– are š‘›Ć—š‘› constant matrices.

Assume that the condition of solvability (2.4) of the generating system (2.1) (obtained from system (3.1) for šœ€=0) is not satisfied for all nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘›, 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 š“1(š‘”) and š“1š‘– 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 systemĢ‡š‘„=š“(š‘”)š‘„+šœ€š“1(š‘”)š‘„,š‘”āˆˆā„ā§µš’Æ,Ī”š‘„|š‘”=šœš‘–=šœ€š“1š‘–š‘„ī€·šœš‘–āˆ’ī€ø,šœš‘–āˆˆš’Æ,š‘–āˆˆā„¤,(3.2) 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 š‘‘Ć—š‘Ÿ matrix šµ0=ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1(š‘”)š‘‹š‘Ÿ(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–š‘‹š‘Ÿī€·šœš‘–āˆ’ī€ø,(3.3)

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 šœ€ with finitely many terms with negative powers of šœ€.

We use the following notation: šµ+0 is the unique matrix pseudoinverse to šµ0 in the Moore-Penrose sense, š‘ƒšµ0 is the š‘ŸĆ—š‘Ÿ matrix (orthoprojector) projecting the space š‘…š‘Ÿ to the null space š‘(šµ0) of the š‘‘Ć—š‘Ÿ matrix šµ0, that is, š‘ƒšµ0:š‘…š‘Ÿā†’š‘(šµ0), and š‘ƒšµāˆ—0 is the š‘‘Ć—š‘‘ matrix (orthoprojector) projecting the space ā„š‘‘ to the null space š‘(šµāˆ—0) of the š‘ŸĆ—š‘‘ matrix šµāˆ—0 (šµāˆ—0=šµš‘‡), that is, š‘ƒšµāˆ—0āˆ¶ā„š‘‘ā†’š‘(šµāˆ—0).

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 ā„+ and ā„āˆ’ with projectors P and Q, respectively. Let nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘› 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 š‘ƒšµāˆ—0=0,(3.4) then the system (3.2) is e-trichotomous on ā„ and, for all nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘›, the system (3.1) possesses at least one solution bounded on ā„ in the form of a series š‘„(š‘”,šœ€)=āˆžī“š‘˜=āˆ’1šœ€š‘˜š‘„š‘˜(š‘”),(3.5) uniformly convergent for sufficiently small fixed šœ€āˆˆ(0,šœ€āˆ—].

Here, šœ€āˆ— is a proper constant characterizing the range of convergence of the series (3.5) and the coefficients š‘„š‘˜(š‘”) of the series (3.5) are determined from the corresponding impulsive systems as š‘„š‘˜(š‘”)=š‘„š‘˜ī€·š‘”,š‘š‘˜ī€ø=š‘‹š‘Ÿ(š‘”)š‘š‘˜+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘–š‘„ī€·šœš‘–āˆ’,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ š‘(š‘”)forš‘˜=1,2,ā€¦,š‘˜=āˆ’šµ+0āŽ”āŽ¢āŽ¢āŽ£ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1(āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“š‘”)1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘–š‘„š‘˜āˆ’1ī€·šœš‘–āˆ’,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (+š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘–š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€·šœš‘–āˆ’ī€øāŽ¤āŽ„āŽ„āŽ¦,š‘„āˆ’1(š‘”)=š‘„āˆ’1ī€·š‘”,š‘āˆ’1ī€ø=š‘‹š‘Ÿ(š‘”)š‘āˆ’1,š‘āˆ’1=šµ+0īƒÆī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–āˆ’ī€øš›¾š‘–īƒ°,š‘„0(š‘”)=š‘„0ī€·š‘”,š‘0ī€ø=š‘‹š‘Ÿ(š‘”)š‘0+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘‹š‘Ÿ(š‘”)š‘āˆ’1š›¾+š‘“(ā‹…)š‘–+š“1š‘–š‘‹š‘Ÿī€·šœš‘–āˆ’ī€øš‘āˆ’1āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ š‘(š‘”),0=āˆ’šµ+0āŽ”āŽ¢āŽ¢āŽ£ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1(āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“š‘”)1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·šœš‘–āˆ’,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (+š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€·šœš‘–āˆ’ī€øāŽ¤āŽ„āŽ„āŽ¦.(3.6)

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 šœ€. The problem of determination of the coefficient š‘„āˆ’1(š‘”) of the term with šœ€āˆ’1 in series (3.5) is reduced to the problem of finding solutions of homogeneous system without impulses Ģ‡š‘„āˆ’1=š“(š‘”)š‘„āˆ’1,š‘”āˆ‰š’Æ,Ī”š‘„āˆ’1|š‘”=šœš‘–=0,š‘–āˆˆā„¤,(3.7) bounded on the entire real axis. According to the Theorem 2.1, the homogeneous system (3.7) possesses š‘Ÿ-parameter family of solutions š‘„āˆ’1ī€·š‘”,š‘āˆ’1ī€ø=š‘‹š‘Ÿ(š‘”)š‘āˆ’1(3.8)bounded on the entire real axis, where š‘āˆ’1 is an š‘Ÿ-dimensional vector column š‘āˆ’1āˆˆā„š‘Ÿ and is determined from the condition of solvability of the problem used for determining the coefficient š‘„0 of the series (3.5).
For šœ€0, the problem of determination of the coefficient š‘„0(š‘”) of series (3.5) reduces to the problem of finding solutions of the following nonhomogeneous system: Ģ‡š‘„0=š“(š‘”)š‘„0+š“1(š‘”)š‘„āˆ’1+š‘“(š‘”),š‘”āˆ‰š’Æ,Ī”š‘„0|š‘”=šœš‘–=š“1š‘–š‘„āˆ’1ī€·šœš‘–āˆ’ī€ø+š›¾š‘–,š‘–āˆˆā„¤,(3.9) bounded on the entire real axis. According to the Theorem 2.1, the condition of solvability of this problem takes the form ī€œāˆžāˆ’āˆžš»š‘‘ī€ŗš“(š‘”)1(š‘”)š‘‹š‘Ÿ(š‘”)š‘āˆ’1ī€»++š‘“(š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–š“ī€øī€ŗ1š‘–š‘‹š‘Ÿī€·šœš‘–āˆ’ī€øš‘āˆ’1+š›¾š‘–ī€»=0.(3.10)Using the matrix šµ0, we get the following algebraic system for š‘āˆ’1āˆˆā„š‘Ÿ: šµ0š‘āˆ’1ī€œ=āˆ’āˆžāˆ’āˆžš»š‘‘(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–āˆ’ī€øš›¾š‘–,(3.11)which is solvable if and only if the condition š‘ƒšµāˆ—0īƒÆī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–āˆ’ī€øš›¾š‘–īƒ°=0(3.12) is satisfied, that is, if š‘ƒšµāˆ—0=0.(3.13)In this case, this algebraic system is solvable with respect to š‘āˆ’1āˆˆā„š‘Ÿ within an arbitrary vector constant š‘ƒšµ0š‘(āˆ€š‘āˆˆā„š‘Ÿ) from the null space of the matrix šµ0, and one of its solutions has the form š‘āˆ’1=šµ+0īƒÆī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š‘“(š‘”)š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–āˆ’ī€øš›¾š‘–īƒ°.(3.14) Therefore, under condition (3.4), the nonhomogeneous system (3.9) possesses an š‘Ÿ-parameter set of solution bounded on ā„ in the form š‘„0ī€·š‘”,š‘0ī€ø=š‘‹š‘Ÿ(š‘”)š‘0+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš›¾+š‘“(ā‹…)š‘–+š“1š‘–š‘„āˆ’1ī€·šœš‘–āˆ’,š‘āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (š‘”),(3.15)where [(šŗāˆ—āˆ—])(š‘”) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.9), and š‘0 is an š‘Ÿ-dimensional constant vector determined in the next step of the process from the condition of solvability of the impulsive problem for coefficient š‘„1(š‘”).
We continue this process by problem of determination of the coefficient š‘„1(š‘”) of the term with šœ€1 in the series (3.5). It reduces to the problem of finding solutions of the systemĢ‡š‘„1=š“(š‘”)š‘„1+š“1(š‘”)š‘„0,š‘”āˆ‰š’Æ,Ī”š‘„1|š‘”=šœš‘–=š“1š‘–š‘„0ī€·šœš‘–āˆ’ī€ø,š‘–āˆˆā„¤,(3.16) 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, ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1āŽ”āŽ¢āŽ¢āŽ£š‘‹(š‘”)š‘Ÿ(š‘”)š‘0+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·šœš‘–āˆ’,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ āŽ¤āŽ„āŽ„āŽ¦+(š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–āˆ’ī€øš“1š‘–āŽ”āŽ¢āŽ¢āŽ£š‘‹š‘Ÿī€·šœš‘–āˆ’ī€øš‘0+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€·šœš‘–āˆ’ī€øāŽ¤āŽ„āŽ„āŽ¦=0,(3.17)we determine the vector š‘0āˆˆā„š‘Ÿ (within an arbitrary vector constant š‘ƒšµ0š‘, āˆ€š‘āˆˆā„š‘Ÿ) as š‘0=āˆ’šµ+0āŽ”āŽ¢āŽ¢āŽ£ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“(š‘”)1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·šœš‘–āˆ’,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ +(š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€øš“+š‘“(ā‹…)1š‘–š‘„āˆ’1ī€·ā‹…,š‘āˆ’1ī€ø+š›¾š‘–āŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€·šœš‘–āˆ’ī€øāŽ¤āŽ„āŽ„āŽ¦.(3.18)Thus, under the condition (3.4), system (3.16) possesses an š‘Ÿ-parameter set of solutions bounded on ā„ in the form š‘„1ī€·š‘”,š‘1ī€ø=š‘‹š‘Ÿ(š‘”)š‘1+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„0ī€·ā‹…,š‘0ī€øš“1š‘–š‘„ī€·šœš‘–āˆ’,š‘0ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (š‘”),(3.19)where [(šŗāˆ—āˆ—])(š‘”) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.16), and š‘1 is an š‘Ÿ-dimensional constant vector determined in the next stage of the process from the condition of solvability of the problem for š‘„2(š‘”).
If we continue this process, we prove (by induction) that the problem of determination of the coefficient š‘„š‘˜(š‘”) in the series (3.5) is reduced to the problem of finding solutions of the system Ģ‡š‘„š‘˜=š“(š‘”)š‘„š‘˜+š“1(š‘”)š‘„š‘˜āˆ’1,š‘”āˆ‰š’Æ,Ī”š‘„š‘˜|š‘”=šœš‘–=š“1š‘–š‘„š‘˜āˆ’1ī€·šœš‘–āˆ’ī€ø,š‘–āˆˆā„¤,š‘˜=1,2,ā€¦,(3.20) bounded on the entire real axis. If the condition (3.4) is satisfied, then a solution of this problem bounded on ā„ has the form š‘„š‘˜(š‘”)=š‘„š‘˜ī€·š‘”,š‘š‘˜ī€ø=š‘‹š‘Ÿ(š‘”)š‘š‘˜+āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘˜š‘„š‘˜āˆ’1ī€·šœš‘–āˆ’,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ (š‘”),(3.21) where [(šŗāˆ—āˆ—])(š‘”) is the generalized Green operator of the problem of finding bounded solutions of impulsive system (3.20) and the constant vector š‘š‘˜āˆˆš‘…š‘Ÿ is given by the formula š‘š‘˜=āˆ’šµ+0āŽ”āŽ¢āŽ¢āŽ£ī€œāˆžāˆ’āˆžš»š‘‘(š‘”)š“1āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“(š‘”)1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘–š‘„š‘˜āˆ’1ī€·šœš‘–āˆ’,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ +(š‘”)š‘‘š‘”āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–āŽ›āŽœāŽœāŽšŗāŽ”āŽ¢āŽ¢āŽ£š“1(ā‹…)š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øš“1š‘–š‘„š‘˜āˆ’1ī€·ā‹…,š‘š‘˜āˆ’1ī€øāŽ¤āŽ„āŽ„āŽ¦āŽžāŽŸāŽŸāŽ ī€·šœš‘–āˆ’ī€øāŽ¤āŽ„āŽ„āŽ¦(3.22) (within an arbitrary vector constant š‘ƒšµ0š‘, š‘āˆˆš‘…š‘Ÿ).
The fact that the series (3.5) is convergent can be proved by using the procedure of majorization.

In the case where the number š‘Ÿ=rankš‘ƒš‘ƒš·=rank(š¼āˆ’š‘„)š‘ƒš· of linear independent solutions of system (2.2) bounded on ā„ is equal to the number š‘‘=rank[š‘ƒš·āˆ—š‘„]=rank[š‘ƒš·āˆ—(š¼āˆ’š‘ƒ)], 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 ā„+ and ā„āˆ’ with projectors š‘ƒ and š‘„, respectively. Let nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘› 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 detšµ0ā‰ 0(š‘Ÿ=š‘‘),(3.23) is satisfied, then the system (3.1) possesses a unique solution bounded on ā„ in the form of series (3.5) uniformly convergent for sufficiently small fixed šœ€āˆˆ(0,šœ€āˆ—].

Proof. If š‘Ÿ=š‘‘, then šµ0 is a square matrix. Therefore, it follows from condition (3.4) that š‘ƒšµ0=š‘ƒšµāˆ—0=0, which is equivalent to the condition (3.23). In this case, the constant vectors š‘š‘˜āˆˆā„š‘Ÿ are uniquely determined from (3.22). The coefficients of the series (3.5) are also uniquely determined by (3.21), and, for all š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘›, the system (3.1) possesses a unique solution bounded on ā„, which means that system (3.2) is e-dichotomous.

We now illustrate the assertions proved above.

Example 3.3. Consider the impulsive system Ģ‡š‘„=š“(š‘”)š‘„+šœ€š“1(š‘”)š‘„+š‘“(š‘”),š‘”āˆˆā„ā§µš’Æ,Ī”š‘„|š‘”=šœš‘–=š›¾š‘–+šœ€š“1š‘–š‘„ī€·šœš‘–āˆ’ī€ø,š›¾š‘–=āŽ§āŽŖāŽŖāŽØāŽŖāŽŖāŽ©š›¾š‘–(1)š›¾š‘–(2)š›¾š‘–(3)āŽ«āŽŖāŽŖāŽ¬āŽŖāŽŖāŽ­āˆˆā„3,š‘–āˆˆā„¤,(3.24) where ī€½š‘“š“(š‘”)=diag{āˆ’tanhš‘”,āˆ’tanhš‘”,tanhš‘”},š‘“(š‘”)=col1(š‘”),š‘“2(š‘”),š‘“3ī€¾(š‘”)āˆˆšµš¶š’Æš“(ā„),1ī€½š‘Ž(š‘”)=š‘–š‘—ī€¾(š‘”)3š‘–,š‘—=1āˆˆšµš¶š’Æ(ā„),š“1š‘–=ī€½Ģƒš‘Žš‘–š‘—ī€¾3š‘–,š‘—=1.(3.25) The generating homogenous system (for šœ€=0) has the form Ģ‡š‘„=š“(š‘”)š‘„,Ī”š‘„|š‘”=šœš‘–=0(3.26)and is e-dichotomous (as shown in [6]) on the semiaxes ā„+ and ā„āˆ’ with projectors š‘ƒ=diag{1,1,0} and š‘„=diag{0,0,1}. The normal fundamental matrix of this system is ī‚»2š‘‹(š‘”)=diagš‘’š‘”+š‘’āˆ’š‘”,2š‘’š‘”+š‘’āˆ’š‘”,š‘’š‘”+š‘’āˆ’š‘”2ī‚¼.(3.27)Thus, we have š·=0,š·+=0,š‘ƒš·=š‘ƒš·āˆ—=š¼3,š‘Ÿ=rankš‘ƒš‘ƒš·=2,š‘‘=rankš‘ƒš·āˆ—š‘‹š‘„=1,š‘ŸāŽ›āŽœāŽœāŽœāŽœāŽœāŽ2(š‘”)=š‘’š‘”+š‘’āˆ’š‘”002š‘’š‘”+š‘’āˆ’š‘”āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ ,š»00(3.28)š‘‘ī‚€2(š‘”)=0,0,š‘’š‘”+š‘’āˆ’š‘”ī‚.(3.29) ā€ƒā€ƒIn order that the generating impulsive system (2.1) with the matrix š“(š‘”) specified above has solutions bounded on the entire real axis, the nonhomogeneities š‘“(š‘”)=col{š‘“1(š‘”),š‘“2(š‘”),š‘“3(š‘”)}āˆˆšµš¶š’Æ(ā„) and š›¾š‘–=col{š›¾š‘–(1),š›¾š‘–(2),š›¾š‘–(3)}āˆˆā„3 must satisfy condition (2.4). In this analyzed impulsive problem, this condition takes the form ī€œāˆžāˆ’āˆž2š‘“3(š‘”)š‘’š‘”+š‘’āˆ’š‘”š‘‘š‘”+āˆžī“š‘–=āˆ’āˆž2š‘’šœš‘–+š‘’āˆ’šœš‘–š›¾š‘–(3)=0,āˆ€š‘“1(š‘”),š‘“2(š‘”)āˆˆšµš¶š’Æ(ā„),āˆ€š›¾š‘–(1),š›¾š‘–(2)āˆˆā„.(3.30) Let š‘“3 and š›¾š‘–(3) 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 ā„ if the coefficients š‘Ž31(š‘”),š‘Ž32(š‘”)āˆˆšµš¶š’Æ(ā„) of the perturbing matrix š“1(š‘”) and the coefficients Ģƒš‘Ž31,Ģƒš‘Ž32āˆˆā„ of the perturbing matrix š“1š‘– satisfy condition (3.4), that is, š‘ƒšµāˆ—0=0, where the matrix šµ0 has the form šµ0=ī€œāˆžāˆ’āˆžī‚øš‘Ž31(š‘”)(š‘’š‘”+š‘’āˆ’š‘”)2,š‘Ž32(š‘”)(š‘’š‘”+š‘’āˆ’š‘”)2ī‚¹š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžī‚øĢƒš‘Ž31(š‘’šœš‘–āˆ’+š‘’āˆ’šœš‘–āˆ’)2,Ģƒš‘Ž32(š‘’šœš‘–āˆ’+š‘’āˆ’šœš‘–āˆ’)2ī‚¹.(3.31) Therefore, if š‘Ž31(š‘”),š‘Ž32(š‘”)āˆˆšµš¶š’Æ(ā„) and Ģƒš‘Ž31,Ģƒš‘Ž32āˆˆā„ are such that at least one of the following inequalities ī€œāˆžāˆ’āˆžš‘Ž31(š‘”)(š‘’š‘”+š‘’āˆ’š‘”)2š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžĢƒš‘Ž31(š‘’šœš‘–āˆ’+š‘’āˆ’šœš‘–āˆ’)2ī€œā‰ 0,āˆžāˆ’āˆžš‘Ž32(š‘”)(š‘’š‘”+š‘’āˆ’š‘”)2š‘‘š‘”+āˆžī“š‘–=āˆ’āˆžĢƒš‘Ž32(š‘’šœiāˆ’+š‘’āˆ’šœš‘–āˆ’)2ā‰ 0(3.32) is satisfied, then either the condition (3.4) or the equivalent condition rankšµ0=š‘‘=1 from Theorem 3.1 is satisfied and the system (3.2) is e-trichotomous on ā„. In this case, the coefficients š‘Ž11(š‘”),ā€‰š‘Ž12(š‘”),ā€‰š‘Ž13(š‘”),ā€‰š‘Ž21(š‘”),š‘Ž22(š‘”),š‘Ž23(š‘”),š‘Ž33(š‘”) are arbitrary functions from the space šµš¶š’Æ(ā„), and Ģƒš‘Ž11,ā€‰Ģƒš‘Ž12,ā€‰Ģƒš‘Ž13,Ģƒš‘Ž21,Ģƒš‘Ž22,Ģƒš‘Ž23,Ģƒš‘Ž33 are arbitrary constants from ā„. Moreover, for any š‘“ī€½š‘“(š‘”)=col1(š‘”),š‘“2(š‘”),š‘“3ī€¾(š‘”)āˆˆšµš¶š’Æ(ā„)(3.33)a solution of the system (3.24) bounded on ā„ is given by the series (3.5) (within a constant from the null space š‘(šµ0), dimš‘(šµ0)=š‘Ÿāˆ’rankšµ0=1).

Another Perturbed Problem
In 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 Ģ‡š‘„=š“(š‘”)š‘„+š‘“(š‘”),š‘”āˆˆā„ā§µš’Æ,š“,š‘“āˆˆšµš¶š’Æ(ā„),Ī”š‘„|š‘”=šœš‘–=š›¾š‘–+šœ€š“1š‘–š‘„ī€·šœš‘–āˆ’ī€ø,š›¾š‘–āˆˆā„š‘›,š‘–āˆˆā„¤,(3.34) where š“1š‘– are š‘›Ć—š‘› constant matrices. For šœ€=0, 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 š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘›). 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 š“1š‘– for which the corresponding homogeneous system Ģ‡š‘„=š“(š‘”)š‘„,š‘”āˆˆā„ā§µš’Æ,Ī”š‘„|š‘”=šœš‘–=šœ€š“1š‘–š‘„ī€·šœš‘–āˆ’ī€ø,š‘–āˆˆā„¤,(3.35) turns into the system e-trichotomous or e-dichotomous on the entire real axis.
This problem can be solved with help of the š‘‘Ć—š‘Ÿ matrix šµ0=āˆžī“š‘–=āˆ’āˆžš»š‘‘ī€·šœš‘–ī€øš“1š‘–š‘‹š‘Ÿī€·šœš‘–āˆ’ī€ø(3.36)
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 ā„+ and ā„āˆ’ with projectors š‘ƒ and š‘„, respectively. Let nonhomogeneities š‘“āˆˆšµš¶š’Æ(ā„) and š›¾š‘–āˆˆā„š‘› 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 ā„, and the system (3.34) possesses at least one solution bounded on ā„ in the form of series (3.5) uniformly convergent for sufficiently small fixed šœ€āˆˆ(0,šœ€āˆ—].

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.

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