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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 792689, 13 pages
http://dx.doi.org/10.1155/2011/792689
Research Article

Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

1Institute of Mathematics, National Academy of Science of Ukraine, 01601 Kyiv, Ukraine
2Department of Mathematics, University of Ε½ilina, 01026 Ε½ilina, Slovakia

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

Academic Editor: JosefΒ DiblΓ­k

Copyright Β© 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 Μ‡π‘₯=𝐴(𝑑)π‘₯+πœ€π΄1(𝑑)π‘₯+𝑓(𝑑),π‘‘βˆˆβ„,π‘‘βˆ‰π’―βˆΆ={πœπ‘–}β„€,Ξ”π‘₯|𝑑=πœπ‘–=𝛾𝑖+πœ€π΄1𝑖π‘₯(πœπ‘–βˆ’),πœπ‘–βˆˆπ’―βŠ‚β„,π›Ύπ‘–βˆˆβ„π‘›, and π‘–βˆˆβ„€ are considered. Under the assumption that the generating system (for π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 πœ€ with finitely many terms with negative powers of π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 [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 timeΜ‡π‘₯=𝐴(𝑑)π‘₯+𝑓(𝑑),π‘‘βˆˆβ„β§΅π’―,Ξ”π‘₯|𝑑=πœπ‘–=𝛾𝑖,πœπ‘–βˆˆπ’―,π‘–βˆˆβ„€,(2.1) where π΄βˆˆπ΅πΆπ’―(ℝ) is an 𝑛×𝑛 matrix of functions, π‘“βˆˆπ΅πΆπ’―(ℝ) is an 𝑛×1 vector function, 𝐡𝐢𝒯(ℝ) is the Banach space of real vector functions bounded on ℝ and left-continuous for π‘‘βˆˆβ„ with discontinuities of the first kind at π‘‘βˆˆπ’―βˆΆ={πœπ‘–}β„€ with the norm: β€–π‘₯‖𝐡𝐢𝒯(ℝ)∢=supπ‘‘βˆˆβ„β€–π‘₯(𝑑)β€–, 𝛾𝑖 are 𝑛-dimensional column constant vectors: π›Ύπ‘–βˆˆβ„π‘›; β‹―<πœβˆ’2<πœβˆ’1<𝜏0=0<𝜏1<𝜏2<β‹―, and Ξ”π‘₯|𝑑=πœπ‘–βˆΆ=π‘₯(πœπ‘–+)βˆ’π‘₯(πœπ‘–βˆ’).

The solution π‘₯(𝑑) of the system (2.1) is sought in the Banach space of 𝑛-dimensional bounded on ℝ and piecewise continuously differentiable vector functions with discontinuities of the first kind at π‘‘βˆˆπ’―βˆΆπ‘₯∈𝐡𝐢1𝒯(ℝ).

Parallel with the nonhomogeneous impulsive system (2.1), we consider the corresponding homogeneous systemΜ‡π‘₯=𝐴(𝑑)π‘₯,Ξ”π‘₯|𝑑=πœπ‘–=0,(2.2) which is the homogeneous system without impulses, and let 𝑋(𝑑) be the fundamental matrix of (2.2) such that 𝑋(0)=𝐼.

Assume that the homogeneous system (2.2) is exponentially dichotomous (e-dichotomous) [5, 10] on semiaxes β„βˆ’=(βˆ’βˆž,0] and ℝ+=[0,∞), that is, there exist projectors 𝑃 and 𝑄(𝑃2=𝑃,𝑄2=𝑄) and constants 𝐾𝑖β‰₯1, 𝛼𝑖>0(𝑖=1,2) such that the following inequalities are satisfied:‖‖𝑋(𝑑)π‘ƒπ‘‹βˆ’1β€–β€–(𝑠)≀𝐾1π‘’βˆ’π›Ό1(π‘‘βˆ’π‘ )‖‖𝑋,𝑑β‰₯𝑠,(𝑑)(πΌβˆ’π‘ƒ)π‘‹βˆ’1β€–β€–(𝑠)≀𝐾1π‘’βˆ’π›Ό1(π‘ βˆ’π‘‘),𝑠β‰₯𝑑,𝑑,π‘ βˆˆβ„+,‖‖𝑋(𝑑)π‘„π‘‹βˆ’1β€–β€–(𝑠)≀𝐾2π‘’βˆ’π›Ό2(π‘‘βˆ’π‘ )β€–β€–,𝑑β‰₯𝑠,𝑋(𝑑)(πΌβˆ’π‘„)π‘‹βˆ’1β€–β€–(𝑠)≀𝐾2π‘’βˆ’π›Ό2(π‘ βˆ’π‘‘),𝑠β‰₯𝑑,𝑑,π‘ βˆˆβ„βˆ’.(2.3)

For getting the solution π‘₯∈𝐡𝐢1𝒯(ℝ) bounded on the entire axis, we assume that 𝑑=0βˆ‰π’―, that is, π‘₯(0+)βˆ’π‘₯(0βˆ’)=𝛾0=0.

We use the following notation: 𝐷=π‘ƒβˆ’(πΌβˆ’π‘„); 𝐷+ is a Moore-Penrose pseudoinverse matrix to 𝐷; 𝑃𝐷 and π‘ƒπ·βˆ— are 𝑛×𝑛 matrices (orthoprojectors) projecting ℝ𝑛 onto 𝑁(𝐷)=ker𝐷 and onto 𝑁(π·βˆ—)=kerπ·βˆ—, respectively, that is, π‘ƒπ·βˆΆβ„π‘›β†’π‘(𝐷), 𝑃2𝐷=𝑃𝐷=π‘ƒβˆ—π·, and π‘ƒπ·βˆ—βˆΆβ„π‘›β†’π‘(π·βˆ—), 𝑃2π·βˆ—=π‘ƒπ·βˆ—=π‘ƒβˆ—π·βˆ—; 𝐻(𝑑)=[π‘ƒπ·βˆ—π‘„]π‘‹βˆ’1(𝑑);  𝑑=rank[π‘ƒπ·βˆ—π‘„]=rank[π‘ƒπ·βˆ—(πΌβˆ’π‘ƒ)] and π‘Ÿ=rank[𝑃𝑃𝐷]=rank[(πΌβˆ’π‘„)𝑃𝐷].

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 β„βˆ’=(βˆ’βˆž,0] and ℝ+=[0,∞) with projectors 𝑃 and 𝑄, respectively. Then the homogeneous system (2.2) has exactly π‘Ÿ linearly independent solutions bounded on the entire real axis. If nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘› satisfy 𝑑 linearly independent conditions ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ›Ύπ‘–=0,(2.4) then the nonhomogeneous system (2.1) possesses an π‘Ÿ-parameter family of linearly independent solutions bounded on ℝ in the form π‘₯𝑑,π‘π‘Ÿξ€Έ=π‘‹π‘Ÿ(𝑑)π‘π‘Ÿ+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π‘“π›Ύπ‘–βŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(𝑑),βˆ€π‘π‘Ÿβˆˆβ„π‘Ÿ.(2.5)

Here, 𝐻𝑑(𝑑)=[π‘ƒπ·βˆ—π‘„]π‘‘π‘‹βˆ’1(𝑑) is a 𝑑×𝑛 matrix formed by a complete system of 𝑑 linearly independent rows of matrix 𝐻(𝑑), π‘‹π‘Ÿξ€Ί(𝑑)∢=𝑋(𝑑)π‘ƒπ‘ƒπ·ξ€»π‘Ÿξ€Ί=𝑋(𝑑)(πΌβˆ’π‘„)π‘ƒπ·ξ€»π‘Ÿ(2.6)

is an π‘›Γ—π‘Ÿ matrix formed by a complete system of π‘Ÿ linearly independent solutions bounded on ℝ of homogeneous system (2.2), and 𝐺[𝑓𝛾𝑖]ξ€Έ(𝑑) is the generalized Green operator of the problem of finding bounded solutions of the nonhomogeneous impulsive system (2.1), acting upon π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘›, defined by the formula βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π‘“π›Ύπ‘–βŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©ξ€œ(𝑑)=𝑋(𝑑)𝑑0π‘ƒπ‘‹βˆ’1(ξ€œπ‘ )𝑓(𝑠)π‘‘π‘ βˆ’βˆžπ‘‘(πΌβˆ’π‘ƒ)π‘‹βˆ’1(+𝑠)𝑓(𝑠)𝑑𝑠𝑗𝑖=1π‘ƒπ‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–βˆ’βˆžξ“π‘–=𝑗+1(πΌβˆ’π‘ƒ)π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–+𝑃𝐷+ξ‚»ξ€œ0βˆ’βˆžπ‘„π‘‹βˆ’1ξ€œ(𝑠)𝑓(𝑠)𝑑𝑠+∞0(πΌβˆ’π‘ƒ)π‘‹βˆ’1+(𝑠)𝑓(𝑠)π‘‘π‘ βˆ’1𝑖=βˆ’βˆžπ‘„π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–+βˆžξ“π‘–=1(πΌβˆ’π‘ƒ)π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–ξƒ°ξ€œ,𝑑β‰₯0;π‘‘βˆ’βˆžπ‘„π‘‹βˆ’1ξ€œ(𝑠)𝑓(𝑠)π‘‘π‘ βˆ’0𝑑(πΌβˆ’π‘„)π‘‹βˆ’1+(𝑠)𝑓(𝑠)π‘‘π‘ βˆ’(𝑗+1)𝑖=βˆ’βˆžπ‘„π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–βˆ’βˆ’1𝑖=βˆ’π‘—(πΌβˆ’π‘„)π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–+(πΌβˆ’π‘„)𝐷+ξ‚»ξ€œ0βˆ’βˆžπ‘„π‘‹βˆ’1ξ€œ(𝑠)𝑓(𝑠)𝑑𝑠+∞0(πΌβˆ’π‘ƒ)π‘‹βˆ’1+(𝑠)𝑓(𝑠)π‘‘π‘ βˆ’1𝑖=βˆ’βˆžπ‘„π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–+βˆžξ“π‘–=1(πΌβˆ’π‘ƒ)π‘‹βˆ’1ξ€·πœπ‘–ξ€Έπ›Ύπ‘–ξƒ°,𝑑≀0,(2.7) with the following property βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π‘“π›Ύπ‘–βŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π‘“π›Ύ(0βˆ’)βˆ’π‘–βŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€œ(0+)=βˆžβˆ’βˆžπ»(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»ξ€·πœπ‘–ξ€Έπ›Ύπ‘–.(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 formΜ‡π‘₯=𝐴(𝑑)π‘₯+πœ€π΄1(𝑑)π‘₯+𝑓(𝑑),π‘‘βˆˆβ„β§΅π’―,Ξ”π‘₯|𝑑=πœπ‘–=𝛾𝑖+πœ€π΄1𝑖π‘₯ξ€·πœπ‘–βˆ’ξ€Έ,πœπ‘–βˆˆπ’―,π›Ύπ‘–βˆˆβ„π‘›,π‘–βˆˆβ„€,(3.1) where 𝐴1βˆˆπ΅πΆπ’―(ℝ) is an 𝑛×𝑛 matrix of functions, 𝐴1𝑖 are 𝑛×𝑛 constant matrices.

Assume that the condition of solvability (2.4) of the generating system (2.1) (obtained from system (3.1) for πœ€=0) is not satisfied for all nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘›, 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 𝐴1(𝑑) and 𝐴1𝑖 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 systemΜ‡π‘₯=𝐴(𝑑)π‘₯+πœ€π΄1(𝑑)π‘₯,π‘‘βˆˆβ„β§΅π’―,Ξ”π‘₯|𝑑=πœπ‘–=πœ€π΄1𝑖π‘₯ξ€·πœπ‘–βˆ’ξ€Έ,πœπ‘–βˆˆπ’―,π‘–βˆˆβ„€,(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 π‘‘Γ—π‘Ÿ matrix 𝐡0=ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1(𝑑)π‘‹π‘Ÿ(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–π‘‹π‘Ÿξ€·πœπ‘–βˆ’ξ€Έ,(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 πœ€ with finitely many terms with negative powers of πœ€.

We use the following notation: 𝐡+0 is the unique matrix pseudoinverse to 𝐡0 in the Moore-Penrose sense, 𝑃𝐡0 is the π‘ŸΓ—π‘Ÿ matrix (orthoprojector) projecting the space π‘…π‘Ÿ to the null space 𝑁(𝐡0) of the π‘‘Γ—π‘Ÿ matrix 𝐡0, that is, 𝑃𝐡0:π‘…π‘Ÿβ†’π‘(𝐡0), and π‘ƒπ΅βˆ—0 is the 𝑑×𝑑 matrix (orthoprojector) projecting the space ℝ𝑑 to the null space 𝑁(π΅βˆ—0) of the π‘ŸΓ—π‘‘ matrix π΅βˆ—0 (π΅βˆ—0=𝐡𝑇), that is, π‘ƒπ΅βˆ—0βˆΆβ„π‘‘β†’π‘(π΅βˆ—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 ℝ+ and β„βˆ’ with projectors P and Q, respectively. Let nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘› 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 π‘ƒπ΅βˆ—0=0,(3.4) then the system (3.2) is e-trichotomous on ℝ and, for all nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘›, the system (3.1) possesses at least one solution bounded on ℝ in the form of a series π‘₯(𝑑,πœ€)=βˆžξ“π‘˜=βˆ’1πœ€π‘˜π‘₯π‘˜(𝑑),(3.5) uniformly convergent for sufficiently small fixed πœ€βˆˆ(0,πœ€βˆ—].

Here, πœ€βˆ— is a proper constant characterizing the range of convergence of the series (3.5) and the coefficients π‘₯π‘˜(𝑑) of the series (3.5) are determined from the corresponding impulsive systems as π‘₯π‘˜(𝑑)=π‘₯π‘˜ξ€·π‘‘,π‘π‘˜ξ€Έ=π‘‹π‘Ÿ(𝑑)π‘π‘˜+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1𝑖π‘₯ξ€·πœπ‘–βˆ’,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ π‘(𝑑)forπ‘˜=1,2,…,π‘˜=βˆ’π΅+0βŽ‘βŽ’βŽ’βŽ£ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1(βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄π‘‘)1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1𝑖π‘₯π‘˜βˆ’1ξ€·πœπ‘–βˆ’,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(+𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1𝑖π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€·πœπ‘–βˆ’ξ€ΈβŽ€βŽ₯βŽ₯⎦,π‘₯βˆ’1(𝑑)=π‘₯βˆ’1𝑑,π‘βˆ’1ξ€Έ=π‘‹π‘Ÿ(𝑑)π‘βˆ’1,π‘βˆ’1=𝐡+0ξƒ―ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–βˆ’ξ€Έπ›Ύπ‘–ξƒ°,π‘₯0(𝑑)=π‘₯0𝑑,𝑐0ξ€Έ=π‘‹π‘Ÿ(𝑑)𝑐0+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘‹π‘Ÿ(𝑑)π‘βˆ’1𝛾+𝑓(β‹…)𝑖+𝐴1π‘–π‘‹π‘Ÿξ€·πœπ‘–βˆ’ξ€Έπ‘βˆ’1⎀βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ π‘(𝑑),0=βˆ’π΅+0βŽ‘βŽ’βŽ’βŽ£ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1(βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄π‘‘)1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·πœπ‘–βˆ’,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(+𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€·πœπ‘–βˆ’ξ€ΈβŽ€βŽ₯βŽ₯⎦.(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 πœ€. The problem of determination of the coefficient π‘₯βˆ’1(𝑑) of the term with πœ€βˆ’1 in series (3.5) is reduced to the problem of finding solutions of homogeneous system without impulses Μ‡π‘₯βˆ’1=𝐴(𝑑)π‘₯βˆ’1,π‘‘βˆ‰π’―,Ξ”π‘₯βˆ’1|𝑑=πœπ‘–=0,π‘–βˆˆβ„€,(3.7) bounded on the entire real axis. According to the Theorem 2.1, the homogeneous system (3.7) possesses π‘Ÿ-parameter family of solutions π‘₯βˆ’1𝑑,π‘βˆ’1ξ€Έ=π‘‹π‘Ÿ(𝑑)π‘βˆ’1(3.8)bounded on the entire real axis, where π‘βˆ’1 is an π‘Ÿ-dimensional vector column π‘βˆ’1βˆˆβ„π‘Ÿ and is determined from the condition of solvability of the problem used for determining the coefficient π‘₯0 of the series (3.5).
For πœ€0, the problem of determination of the coefficient π‘₯0(𝑑) of series (3.5) reduces to the problem of finding solutions of the following nonhomogeneous system: Μ‡π‘₯0=𝐴(𝑑)π‘₯0+𝐴1(𝑑)π‘₯βˆ’1+𝑓(𝑑),π‘‘βˆ‰π’―,Ξ”π‘₯0|𝑑=πœπ‘–=𝐴1𝑖π‘₯βˆ’1ξ€·πœπ‘–βˆ’ξ€Έ+𝛾𝑖,π‘–βˆˆβ„€,(3.9) bounded on the entire real axis. According to the Theorem 2.1, the condition of solvability of this problem takes the form ξ€œβˆžβˆ’βˆžπ»π‘‘ξ€Ίπ΄(𝑑)1(𝑑)π‘‹π‘Ÿ(𝑑)π‘βˆ’1ξ€»++𝑓(𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–π΄ξ€Έξ€Ί1π‘–π‘‹π‘Ÿξ€·πœπ‘–βˆ’ξ€Έπ‘βˆ’1+𝛾𝑖=0.(3.10)Using the matrix 𝐡0, we get the following algebraic system for π‘βˆ’1βˆˆβ„π‘Ÿ: 𝐡0π‘βˆ’1ξ€œ=βˆ’βˆžβˆ’βˆžπ»π‘‘(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–βˆ’ξ€Έπ›Ύπ‘–,(3.11)which is solvable if and only if the condition π‘ƒπ΅βˆ—0ξƒ―ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–βˆ’ξ€Έπ›Ύπ‘–ξƒ°=0(3.12) is satisfied, that is, if π‘ƒπ΅βˆ—0=0.(3.13)In this case, this algebraic system is solvable with respect to π‘βˆ’1βˆˆβ„π‘Ÿ within an arbitrary vector constant 𝑃𝐡0𝑐(βˆ€π‘βˆˆβ„π‘Ÿ) from the null space of the matrix 𝐡0, and one of its solutions has the form π‘βˆ’1=𝐡+0ξƒ―ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝑓(𝑑)𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–βˆ’ξ€Έπ›Ύπ‘–ξƒ°.(3.14) Therefore, under condition (3.4), the nonhomogeneous system (3.9) possesses an π‘Ÿ-parameter set of solution bounded on ℝ in the form π‘₯0𝑑,𝑐0ξ€Έ=π‘‹π‘Ÿ(𝑑)𝑐0+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝛾+𝑓(β‹…)𝑖+𝐴1𝑖π‘₯βˆ’1ξ€·πœπ‘–βˆ’,π‘βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(𝑑),(3.15)where [(πΊβˆ—βˆ—])(𝑑) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.9), and 𝑐0 is an π‘Ÿ-dimensional constant vector determined in the next step of the process from the condition of solvability of the impulsive problem for coefficient π‘₯1(𝑑).
We continue this process by problem of determination of the coefficient π‘₯1(𝑑) of the term with πœ€1 in the series (3.5). It reduces to the problem of finding solutions of the systemΜ‡π‘₯1=𝐴(𝑑)π‘₯1+𝐴1(𝑑)π‘₯0,π‘‘βˆ‰π’―,Ξ”π‘₯1|𝑑=πœπ‘–=𝐴1𝑖π‘₯0ξ€·πœπ‘–βˆ’ξ€Έ,π‘–βˆˆβ„€,(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, ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1βŽ‘βŽ’βŽ’βŽ£π‘‹(𝑑)π‘Ÿ(𝑑)𝑐0+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·πœπ‘–βˆ’,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦+(𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–βˆ’ξ€Έπ΄1π‘–βŽ‘βŽ’βŽ’βŽ£π‘‹π‘Ÿξ€·πœπ‘–βˆ’ξ€Έπ‘0+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€·πœπ‘–βˆ’ξ€ΈβŽ€βŽ₯βŽ₯⎦=0,(3.17)we determine the vector 𝑐0βˆˆβ„π‘Ÿ (within an arbitrary vector constant 𝑃𝐡0𝑐, βˆ€π‘βˆˆβ„π‘Ÿ) as 𝑐0=βˆ’π΅+0βŽ‘βŽ’βŽ’βŽ£ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄(𝑑)1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·πœπ‘–βˆ’,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠+(𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1𝐴+𝑓(β‹…)1𝑖π‘₯βˆ’1ξ€·β‹…,π‘βˆ’1ξ€Έ+π›Ύπ‘–βŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€·πœπ‘–βˆ’ξ€ΈβŽ€βŽ₯βŽ₯⎦.(3.18)Thus, under the condition (3.4), system (3.16) possesses an π‘Ÿ-parameter set of solutions bounded on ℝ in the form π‘₯1𝑑,𝑐1ξ€Έ=π‘‹π‘Ÿ(𝑑)𝑐1+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯0ξ€·β‹…,𝑐0𝐴1𝑖π‘₯ξ€·πœπ‘–βˆ’,𝑐0ξ€ΈβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(𝑑),(3.19)where [(πΊβˆ—βˆ—])(𝑑) is the generalized Green operator (2.7) of the problem of finding bounded solutions of system (3.16), and 𝑐1 is an π‘Ÿ-dimensional constant vector determined in the next stage of the process from the condition of solvability of the problem for π‘₯2(𝑑).
If we continue this process, we prove (by induction) that the problem of determination of the coefficient π‘₯π‘˜(𝑑) in the series (3.5) is reduced to the problem of finding solutions of the system Μ‡π‘₯π‘˜=𝐴(𝑑)π‘₯π‘˜+𝐴1(𝑑)π‘₯π‘˜βˆ’1,π‘‘βˆ‰π’―,Ξ”π‘₯π‘˜|𝑑=πœπ‘–=𝐴1𝑖π‘₯π‘˜βˆ’1ξ€·πœπ‘–βˆ’ξ€Έ,π‘–βˆˆβ„€,π‘˜=1,2,…,(3.20) bounded on the entire real axis. If the condition (3.4) is satisfied, then a solution of this problem bounded on ℝ has the form π‘₯π‘˜(𝑑)=π‘₯π‘˜ξ€·π‘‘,π‘π‘˜ξ€Έ=π‘‹π‘Ÿ(𝑑)π‘π‘˜+βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1π‘˜π‘₯π‘˜βˆ’1ξ€·πœπ‘–βˆ’,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠(𝑑),(3.21) where [(πΊβˆ—βˆ—])(𝑑) is the generalized Green operator of the problem of finding bounded solutions of impulsive system (3.20) and the constant vector π‘π‘˜βˆˆπ‘…π‘Ÿ is given by the formula π‘π‘˜=βˆ’π΅+0βŽ‘βŽ’βŽ’βŽ£ξ€œβˆžβˆ’βˆžπ»π‘‘(𝑑)𝐴1βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄(𝑑)1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1𝑖π‘₯π‘˜βˆ’1ξ€·πœπ‘–βˆ’,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠+(𝑑)π‘‘π‘‘βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–βŽ›βŽœβŽœβŽπΊβŽ‘βŽ’βŽ’βŽ£π΄1(β‹…)π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1𝐴1𝑖π‘₯π‘˜βˆ’1ξ€·β‹…,π‘π‘˜βˆ’1ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦βŽžβŽŸβŽŸβŽ ξ€·πœπ‘–βˆ’ξ€ΈβŽ€βŽ₯βŽ₯⎦(3.22) (within an arbitrary vector constant 𝑃𝐡0𝑐, π‘βˆˆπ‘…π‘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 π‘Ÿ=rank𝑃𝑃𝐷=rank(πΌβˆ’π‘„)𝑃𝐷 of linear independent solutions of system (2.2) bounded on ℝ is equal to the number 𝑑=rank[π‘ƒπ·βˆ—π‘„]=rank[π‘ƒπ·βˆ—(πΌβˆ’π‘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 ℝ+ and β„βˆ’ with projectors 𝑃 and 𝑄, respectively. Let nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘› 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 det𝐡0β‰ 0(π‘Ÿ=𝑑),(3.23) is satisfied, then the system (3.1) possesses a unique solution bounded on ℝ in the form of series (3.5) uniformly convergent for sufficiently small fixed πœ€βˆˆ(0,πœ€βˆ—].

Proof. If π‘Ÿ=𝑑, then 𝐡0 is a square matrix. Therefore, it follows from condition (3.4) that 𝑃𝐡0=π‘ƒπ΅βˆ—0=0, which is equivalent to the condition (3.23). In this case, the constant vectors π‘π‘˜βˆˆβ„π‘Ÿ are uniquely determined from (3.22). The coefficients of the series (3.5) are also uniquely determined by (3.21), and, for all π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘›, the system (3.1) possesses a unique solution bounded on ℝ, which means that system (3.2) is e-dichotomous.

We now illustrate the assertions proved above.

Example 3.3. Consider the impulsive system Μ‡π‘₯=𝐴(𝑑)π‘₯+πœ€π΄1(𝑑)π‘₯+𝑓(𝑑),π‘‘βˆˆβ„β§΅π’―,Ξ”π‘₯|𝑑=πœπ‘–=𝛾𝑖+πœ€π΄1𝑖π‘₯ξ€·πœπ‘–βˆ’ξ€Έ,𝛾𝑖=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›Ύπ‘–(1)𝛾𝑖(2)𝛾𝑖(3)⎫βŽͺβŽͺ⎬βŽͺβŽͺβŽ­βˆˆβ„3,π‘–βˆˆβ„€,(3.24) where 𝑓𝐴(𝑑)=diag{βˆ’tanh𝑑,βˆ’tanh𝑑,tanh𝑑},𝑓(𝑑)=col1(𝑑),𝑓2(𝑑),𝑓3ξ€Ύ(𝑑)βˆˆπ΅πΆπ’―π΄(ℝ),1ξ€½π‘Ž(𝑑)=𝑖𝑗(𝑑)3𝑖,𝑗=1βˆˆπ΅πΆπ’―(ℝ),𝐴1𝑖=ξ€½Μƒπ‘Žπ‘–π‘—ξ€Ύ3𝑖,𝑗=1.(3.25) The generating homogenous system (for πœ€=0) has the form Μ‡π‘₯=𝐴(𝑑)π‘₯,Ξ”π‘₯|𝑑=πœπ‘–=0(3.26)and is e-dichotomous (as shown in [6]) on the semiaxes ℝ+ and β„βˆ’ with projectors 𝑃=diag{1,1,0} and 𝑄=diag{0,0,1}. The normal fundamental matrix of this system is ξ‚»2𝑋(𝑑)=diag𝑒𝑑+π‘’βˆ’π‘‘,2𝑒𝑑+π‘’βˆ’π‘‘,𝑒𝑑+π‘’βˆ’π‘‘2ξ‚Ό.(3.27)Thus, we have 𝐷=0,𝐷+=0,𝑃𝐷=π‘ƒπ·βˆ—=𝐼3,π‘Ÿ=rank𝑃𝑃𝐷=2,𝑑=rankπ‘ƒπ·βˆ—π‘‹π‘„=1,π‘ŸβŽ›βŽœβŽœβŽœβŽœβŽœβŽ2(𝑑)=𝑒𝑑+π‘’βˆ’π‘‘002𝑒𝑑+π‘’βˆ’π‘‘βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,𝐻00(3.28)𝑑2(𝑑)=0,0,𝑒𝑑+π‘’βˆ’π‘‘ξ‚.(3.29)   In order that the generating impulsive system (2.1) with the matrix 𝐴(𝑑) specified above has solutions bounded on the entire real axis, the nonhomogeneities 𝑓(𝑑)=col{𝑓1(𝑑),𝑓2(𝑑),𝑓3(𝑑)}βˆˆπ΅πΆπ’―(ℝ) and 𝛾𝑖=col{𝛾𝑖(1),𝛾𝑖(2),𝛾𝑖(3)}βˆˆβ„3 must satisfy condition (2.4). In this analyzed impulsive problem, this condition takes the form ξ€œβˆžβˆ’βˆž2𝑓3(𝑑)𝑒𝑑+π‘’βˆ’π‘‘π‘‘π‘‘+βˆžξ“π‘–=βˆ’βˆž2π‘’πœπ‘–+π‘’βˆ’πœπ‘–π›Ύπ‘–(3)=0,βˆ€π‘“1(𝑑),𝑓2(𝑑)βˆˆπ΅πΆπ’―(ℝ),βˆ€π›Ύπ‘–(1),𝛾𝑖(2)βˆˆβ„.(3.30) Let 𝑓3 and 𝛾𝑖(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 ℝ if the coefficients π‘Ž31(𝑑),π‘Ž32(𝑑)βˆˆπ΅πΆπ’―(ℝ) of the perturbing matrix 𝐴1(𝑑) and the coefficients Μƒπ‘Ž31,Μƒπ‘Ž32βˆˆβ„ of the perturbing matrix 𝐴1𝑖 satisfy condition (3.4), that is, π‘ƒπ΅βˆ—0=0, where the matrix 𝐡0 has the form 𝐡0=ξ€œβˆžβˆ’βˆžξ‚Έπ‘Ž31(𝑑)(𝑒𝑑+π‘’βˆ’π‘‘)2,π‘Ž32(𝑑)(𝑒𝑑+π‘’βˆ’π‘‘)2𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžξ‚ΈΜƒπ‘Ž31(π‘’πœπ‘–βˆ’+π‘’βˆ’πœπ‘–βˆ’)2,Μƒπ‘Ž32(π‘’πœπ‘–βˆ’+π‘’βˆ’πœπ‘–βˆ’)2ξ‚Ή.(3.31) Therefore, if π‘Ž31(𝑑),π‘Ž32(𝑑)βˆˆπ΅πΆπ’―(ℝ) and Μƒπ‘Ž31,Μƒπ‘Ž32βˆˆβ„ are such that at least one of the following inequalities ξ€œβˆžβˆ’βˆžπ‘Ž31(𝑑)(𝑒𝑑+π‘’βˆ’π‘‘)2𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžΜƒπ‘Ž31(π‘’πœπ‘–βˆ’+π‘’βˆ’πœπ‘–βˆ’)2ξ€œβ‰ 0,βˆžβˆ’βˆžπ‘Ž32(𝑑)(𝑒𝑑+π‘’βˆ’π‘‘)2𝑑𝑑+βˆžξ“π‘–=βˆ’βˆžΜƒπ‘Ž32(π‘’πœiβˆ’+π‘’βˆ’πœπ‘–βˆ’)2β‰ 0(3.32) is satisfied, then either the condition (3.4) or the equivalent condition rank𝐡0=𝑑=1 from Theorem 3.1 is satisfied and the system (3.2) is e-trichotomous on ℝ. In this case, the coefficients π‘Ž11(𝑑),β€‰π‘Ž12(𝑑),β€‰π‘Ž13(𝑑),β€‰π‘Ž21(𝑑),π‘Ž22(𝑑),π‘Ž23(𝑑),π‘Ž33(𝑑) are arbitrary functions from the space 𝐡𝐢𝒯(ℝ), and Μƒπ‘Ž11,β€‰Μƒπ‘Ž12,β€‰Μƒπ‘Ž13,Μƒπ‘Ž21,Μƒπ‘Ž22,Μƒπ‘Ž23,Μƒπ‘Ž33 are arbitrary constants from ℝ. Moreover, for any 𝑓𝑓(𝑑)=col1(𝑑),𝑓2(𝑑),𝑓3ξ€Ύ(𝑑)βˆˆπ΅πΆπ’―(ℝ)(3.33)a solution of the system (3.24) bounded on ℝ is given by the series (3.5) (within a constant from the null space 𝑁(𝐡0), dim𝑁(𝐡0)=π‘Ÿβˆ’rank𝐡0=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 Μ‡π‘₯=𝐴(𝑑)π‘₯+𝑓(𝑑),π‘‘βˆˆβ„β§΅π’―,𝐴,π‘“βˆˆπ΅πΆπ’―(ℝ),Ξ”π‘₯|𝑑=πœπ‘–=𝛾𝑖+πœ€π΄1𝑖π‘₯ξ€·πœπ‘–βˆ’ξ€Έ,π›Ύπ‘–βˆˆβ„π‘›,π‘–βˆˆβ„€,(3.34) where 𝐴1𝑖 are 𝑛×𝑛 constant matrices. For π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 π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘›). 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 𝐴1𝑖 for which the corresponding homogeneous system Μ‡π‘₯=𝐴(𝑑)π‘₯,π‘‘βˆˆβ„β§΅π’―,Ξ”π‘₯|𝑑=πœπ‘–=πœ€π΄1𝑖π‘₯ξ€·πœπ‘–βˆ’ξ€Έ,π‘–βˆˆβ„€,(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 π‘‘Γ—π‘Ÿ matrix 𝐡0=βˆžξ“π‘–=βˆ’βˆžπ»π‘‘ξ€·πœπ‘–ξ€Έπ΄1π‘–π‘‹π‘Ÿξ€·πœπ‘–βˆ’ξ€Έ(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 ℝ+ and β„βˆ’ with projectors 𝑃 and 𝑄, respectively. Let nonhomogeneities π‘“βˆˆπ΅πΆπ’―(ℝ) and π›Ύπ‘–βˆˆβ„π‘› 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 ℝ, and the system (3.34) possesses at least one solution bounded on ℝ in the form of series (3.5) uniformly convergent for sufficiently small fixed πœ€βˆˆ(0,πœ€βˆ—].

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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