Autonomous noetherian boundary-value problem in the critical case

Чуйко, С. М.; Boichuk, I. A.

doi:10.1007/s11072-010-0085-1

Cited by 18 publications

(9 citation statements)

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“…Thus, the following statement is proved [10]: (4) is satisfied and the boundary-value problem (1), (2) corresponds to the critical case…”

Section: Statement Of the Problemmentioning

confidence: 90%

“…J(z 0 (·, c ) + x(·, ε), ε) = J(z 0 (·, c r ), 0) + 1 x(·, ε) + ε 3 (z 0 (·, c r ), 0) + εJ 3 (z 0 (·, c r ) + x(·, ε), ε); (10) this expansion can be represented in terms of Fréchet derivatives as follows:…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Reasoning by analogy, we arrive at the following statement [10]: (1), (2) corresponds to the special critical case P Q * = 0, F 0 (c r ) ≡ 0. Then, for every root c * r ∈ R r of Eq.…”

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confidence: 91%

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Weakly nonlinear boundary-value problem in a special critical case

Чуйко

2009

Ukr Math J

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We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case. Statement of the ProblemWe investigate the problem of the determination of conditions for the existence of a solution z(t, ε)that satisfy the boundary condition z(·, ε) = α + εJ(z(·, ε), ε)and the construction of this solution. We seek a solution of the Noetherian (m = n) problem (1), (2) in a small neighborhood of the generating problemwhere A(t) is an n × n matrix, f (t) is an n-dimensional column vector whose elements are real functions continuous on the segment [a, b], and z(·) is a linear bounded vector functional. Assume that the nonlinearitiesof problem (1), (2) are twice continuously differentiable with respect to the unknown z in a small neighborhood of the generating solution and with respect to the small parameter ε in a small positive neighborhood of zero. In

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“…Thus, the following statement is proved [10]: (4) is satisfied and the boundary-value problem (1), (2) corresponds to the critical case…”

Section: Statement Of the Problemmentioning

confidence: 90%

Section: Statement Of the Problemmentioning

confidence: 99%

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Weakly nonlinear boundary-value problem in a special critical case

Чуйко

2009

Ukr Math J

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“…Óìîâè iñíóâàííÿ, à òàêîae iòåðàöiéíà ñõåìà äëÿ çíàõîäaeåííÿ ðîçâ'ÿçêiâ íåâèðîäaeåíî¨ñëàáêîíåëiíiéíî¨íåòåðîâî¨äèôåðåíöiàëüíî-àëãåáðà¨÷íî¨êðàéîâî¨çàäà÷i (9) â íåêðèòè÷íîìó âèïàäêó, à ñàìå: çà óìîâèP Q * = 0, çíàéäåíi â ñòàòòi[5]. Ìåòîþ äàíî¨ñòàòòi ¹ çíàõîäaeåííÿ óìîâ çâiäíîñòi íåëiíiéíî¨íåâèðîäaeåíî¨íåòåðîâî¨äèôåðåíöiàëüíî-àëãåáðà¨÷íî¨êðàéîâîç àäà÷i (9) â êðèòè÷íîìó âèïàäêó äî íåêðèòè÷íîãî âèïàäêó àíàëîãi÷íî[5,6].…”

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On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

2019

MAMM

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The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works 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 are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of nonlinear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the nonlinear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the differential algebraic equations, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear differential algebraic boundary value problems. In this article we found the conditions of the existence and constructed the iterative scheme for finding the solutions of the weakly nonlinear Noetherian differential-algebraic boundary value problem. The proposed scheme of the research of the nonlinear differential-algebraic boundary value problems in the article can be transferred to the nonlinear matrix differential-algebraic boundary value problems. On the other hand, the proposed scheme of the research of the nonlinear Noetherian differential-algebraic boundary value problems in the critical case in this article can be transferred to the autonomous seminonlinear differential-algebraic boundary value problems.

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“…В заключение заметим, что частным случаем неавтономной периодической задачи в случае параметрического резонанса, в том числе и неавтономной периодической задачи для уравнения типа Хилла, является автономная периодическая задача, которая после замены независимой переменной приведена к неавтономной периодической задаче, содержащей дополнительную неизвестную, отвечающую за поправку на период искомого решения [Малкин, 1956;Бойчук, Чуйко, 1992;Chujko, Boichuk, 2009].…”

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Periodic boudary-value problem for Hill's equation in the case of parametric resonance

Кулиш¹,

Starkova²,

Chujko³

2014

CRM

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Получено 27 октября 2013 г.Найдены необходимые и достаточные условия существования решений нелинейной неавтономной периодической задачи для уравнения типа Хилла в случае параметрического резонанса. Характерной особенностью поставленной задачи является необходимость нахождения как искомого решения, так и соответствующей собственной функции, обеспечивающей разрешимость периодической задачи для уравнения типа Хилла в случае параметрического резонанса. Для построения решений периодической задачи для уравнения типа Хилла и соответствующей собственной функции в случае параметрического резонанса предложены итерационные схемы, построенные методу простых итераций, а также с использованием техники наименьших квадратов.Ключевые слова: нелинейная неавтономная периодическая краевая задача, уравнение типа Хилла, случай параметрического резонанса, метод простых итераций Abstract. -Necessary and sufficient conditions for the existence of solutions of nonlinear nonautonomous periodic problem for Hill's equation in the case of parametric resonance. A characteristic feature of the task is the need of finding, as desired solution, and the corresponding eigenfunction, which ensures solvability of the periodic problem for Hill's equation in the case of parametric resonance. To construct solutions of the periodic problem for Hill's equation and the corresponding eigenfunction in the case of parametric resonance proposed iterative scheme, based on the method of simple iterations with used list-square technics. Keywords: nonlinear nonautonomous periodic boudary-value problem, Hill's equation, the case of parametric resonance, he method of simple iterations

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Autonomous noetherian boundary-value problem in the critical case

Cited by 18 publications

References 2 publications

Weakly nonlinear boundary-value problem in a special critical case

Weakly nonlinear boundary-value problem in a special critical case

On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

Periodic boudary-value problem for Hill's equation in the case of parametric resonance

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