A Generalized Green Operator for a Linear Noetherian
Differential-Algebraic Boundary Value Problem

Чуйко, С. М.

doi:10.3103/s1055134420030037

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…where KOEf .j /; ⌫ 0 .j /ç.k/ WD KOEF 0 .j; ⌫ 0 .j //ç.k/ is the Green operator of the Cauchy problem for system (3). By analogy with the classification of differentialalgebraic equations [13][14][15] under condition (4), we can say that the system of linear difference-algebraic equations (3) is nondegenerate. Note that, unlike the ordinary system of linear difference equations, the solution of the system of linear difference-algebraic equations (3) under condition (4) depends on an arbitrary bounded vector function…”

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confidence: 99%

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

2022

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We determine necessary and sufficient conditions for the existence of solutions of a nonlinear boundaryvalue problem for a system of difference-algebraic equations in the case of parametric resonance. We construct a convergent iterative algorithm for finding approximations to the solutions of nonlinear boundaryvalue problem for the system of difference-algebraic equations in the case of parametric resonance. As an example of application of the constructed iterative scheme, we obtain approximations to the solutions of a two-point boundary-value problem for a system of Mathieu-type difference-algebraic equations with parametric perturbation. To check the accuracy of the obtained approximations to the solutions of the two-point boundary-value problem for the system of Mathieu-type difference-algebraic equations with parametric perturbation, we use the discrepancies in the original equation.

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Section: Statement Of the Problemmentioning

confidence: 99%

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

2022

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“…Достатнi умови звiдностi диференцiально-алгебраїчної лiнiйної системи до центральної канонiчної форми були отриманi А. М. Самойленком i В. П. Яковцем [6]. У статтях [7,8] запропоновано достатнi умови розв'язностi, а також конструкцiю узагальненого оператора Грiна для лiнiйної диференцiально-алгебраїчної крайової задачi (1), (3) без використання центральної канонiчної форми i досконалих пар i трiйок матриць.…”

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“…За умови A(t) ≡ I n задача про знаходження умов iснування та побудову розв'язкiв системи диференцiальних рiвнянь (1) з невиродженим iмпульсним впливом була розв'язана А. М. Самойленком та М. О. Перестюком у монографiї [1] та узагальнювала задачу з крайовими умовами типу "shock conditions" [9]. За умови [7,8] P A * (t) ≡ 0 (4) система (1) розв'язна вiдносно похiдної…”

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“…Позначимо X 0 (t) нормальну фундаментальну матрицю X 0 (t) = A + (t)B(t)X 0 (t), X 0 (a) = I n отриманої традицiйної системи звичайних диференцiальних рiвнянь (5). За умови (4) для довiльної неперервної вектор-функцiї ν 0 (t) система (5), а вiдповiдно i система (1), має розв'язок вигляду [7,8]…”

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“…Бойчуком у статтi [11]. Таким чином, метою даної статтi є перенесення результатiв [3,4,5,6,7,8] на диференцiально-алгебраїчну крайову задачу (1) (3) з виродженим iмпульсним впливом. На вiдмiну вiд статтi [11], отримана нормальна фундаментальна матриця X(t) системи (1) з виродженим iмпульсним впливом (2) вироджена [12], причому її ранг вздовж промiжку [a, b] не зростає.…”

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Linear differential-algebraic boundary value problem with singular pulse influence

Чуйко

Chuiko

Shevtsova

2021

MAMM

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The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

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Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Чуйко

2021

J Math Sci

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A Generalized Green Operator for a Linear Noetherian Differential-Algebraic Boundary Value Problem

Cited by 10 publications

References 13 publications

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

Linear differential-algebraic boundary value problem with singular pulse influence

Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

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