Nonlinear Noetherian Second-Order Boundary-Value Problems in the Critical Case

Чуйко, С. М.; Бойчук, А. А.; Boichuk, I. A.

doi:10.1007/s10958-014-1795-1

Cited by 4 publications

(9 citation statements)

References 4 publications

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“…In [7], we show that the periodic problem for the Duffing-type equation represents a critical case of the second order. Since the procedure of determination of periodic solutions for the Duffing-type equation according to the iterative scheme [7] is fairly cumbersome, we use the established theorem and the fact that the equation for generating functions…”

Section: Sufficient Conditionmentioning

confidence: 95%

“…"// specified by the operator system (7). Moreover, in a small neighborhood of the generating solution z 0 .t; c 0 .…”

Section: Sufficient Conditionmentioning

confidence: 99%

“…According to the iterative scheme [8], the third approximation to the solution of the 2 -periodic problem for the Duffing-type equation (8) In [7], we have obtained the following approximation to the solution of the 2 -periodic problem for the Duffing-type equation (8):…”

Section: Sufficient Conditionmentioning

confidence: 99%

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Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

2015

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UDC 517.9We establish necessary and sufficient conditions for the existence of solutions and propose an iterative scheme for the determination of solutions of the Fredholm weakly nonlinear boundary-value problem for a system of ordinary differential equations in the case of multiple roots of the equation for generating constants. Statement of the ProblemWe study the problem of construction of the solution z.t; "/ W z.;We seek the solution of problem (1) in a small neighborhood of the solution of the generating problemHere, A.t/ is an .n n/ matrix and f .t / is an n -dimensional column vector whose elements are real functions continuous on the segment OEa; b and`z./ is a linear bounded vector functional`z./ W C OEa; b ! R m : Assume that the nonlinearities Z.z; t; "/ and J.z.; "/; "/ of the Fredholm .m ¤ n/ problem (1) 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 " in a small positive neighborhood of zero. In addition, we also assume that the vector function Z.z; t; "/ is continuous in the independent variable t on the segment OEa; b: We investigate the critical case .P Q 6 D 0/ under the assumption thatIn this case, the generating problem has an .r D n n 1 / -parameter family of solutionsHere, X.t/ is a normal .X.a/ D I n / fundamental matrix of the homogeneous part of the generating system, Q D`X./ is an .m n/ matrix, rank Q D n 1 ; X r .t / D X.t /P Q r ; P Q r is an .n r/ matrix formed by r

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Section: Sufficient Conditionmentioning

confidence: 95%

“…"// specified by the operator system (7). Moreover, in a small neighborhood of the generating solution z 0 .t; c 0 .…”

Section: Sufficient Conditionmentioning

confidence: 99%

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Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

2015

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“…for the solution of the boundary-value problem (1), ( 2), one can use the procedure of linearization and apply either the technique of reduction to the critical case of the first order [23] or the correction of expansions of nonlinearities of the boundary-value problem (1), (2). Moreover, the critical cases of the second and higher orders are also possible [1,7,[24][25][26].…”

Section: Critical Case Of the First Ordermentioning

confidence: 99%

On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

Бойчук

Чуйко

2021

J Math Sci

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We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton-Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Liénard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Liénard equations, we use discrepancies in the original equations. ⇤.�/ π D 0:(3)

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“…The matrix B 0 ; which plays the key role in the critical case, is obtained in explicit form and coincides with the derivative of the equations for generating constants (2.7) for matrix boundary value problem (1.1), (1.2). We differentiate solvability condition The method for construction of solvability conditions and construction of the generalized Green operator for Noetherian linear boundary value problem for the matrix differential equations and solvability conditions and the scheme for constructing solutions of nonlinear Noetherian boundary value problem for matrix differential equation can be generalized to boundary value problem for the matrix differential equations in various critical and noncritical cases [6,16], in particular, to autonomous boundary value problems [2,12] and nonlinear Noetherian second-order boundaryvalue problems in the critical case [11]. The method for construction of solvability conditions and construction of the generalized Green operator for Noetherian linear boundary value problem for the matrix differential equations and solvability conditions and the scheme for constructing solutions of nonlinear Noetherian boundary value problem for matrix differential equation can be generalized to boundary value problem for the matrix differential-algebraic equations [10].…”

Section: Matrix Equations Of Duffingmentioning

confidence: 99%

Weakly nonlinear boundary value problem for a matrix differential equation

Чуйко¹

2016

MMN

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We set forth solvability conditions and construction of the generalized Green operator for Noetherian linear boundary value problem for the matrix differential equations and solvability conditions and the constructive scheme for constructing solutions of nonlinear Noetherian boundary value problem for matrix differential equation. We show that the principal results in the theory of weakly nonlinear periodic oscillations remain valid for nonlinear Noetherian boundary value problem for matrix differential equation. The study is illustrated by the periodic problem for a Duffing-type matrix differential equation.

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Nonlinear Noetherian Second-Order Boundary-Value Problems in the Critical Case

Cited by 4 publications

References 4 publications

Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

Reduction of the Fredholm Boundary-Value Problem to the Critical Case of the First Order

On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

Weakly nonlinear boundary value problem for a matrix differential equation

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