Generalized green operator of a boundary-value problem with degenerate pulse influence

Бойчук, А. А.; Chuiko, E. V.; Чуйко, С. М.

doi:10.1007/bf02384232

Cited by 6 publications

(13 citation statements)

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“…. , (I n + S p ) is singular; moreover, rankQ = n 1 < np, whence rank PQ * = np − n 1 > 0, and the problem with degenerate pulse action [11] corresponds to the critical case (PQ * = O) and is solvable not for all inhomogeneities but only for those, and only those, for which condition (16) is satisfied. The proved theorem also generalizes the corresponding statements for problems with boundary conditions of the "interface conditions" type [12] and more general impulsive conditions [1] to the case of problems with switchings.…”

Section: Impulsive Boundary-value Problem With Switchings In the Critmentioning

confidence: 97%

“…In particular, for S i = 0, this problem was studied in [10]. If, for some i, the matrices I n + S i are singular, then a degenerate pulse action takes place [11].…”

Section: Linear Homogeneous Boundary-value Problems With Pulse Actionmentioning

confidence: 99%

“…Indeed, for the systems with pushes at given moments of time investigated by Myshkis and Samoilenko, the inhomogeneous Cauchy problem z(a) = α for system (12) is solvable in the critical case not for all α ∈ R n (condition 3 in [15, p. 203]). For the problem with degenerate pulse action given in [11], the np × n(p + 1) matrix …”

Section: Impulsive Boundary-value Problem With Switchings In the Critmentioning

confidence: 99%

“…Among all fundamental systems of solutions of problem (6), it is natural to select a solution whose norm is equal to one for t ∈ [1, 2[. The structure of the fundamental system of solutions of problem (6) obtained in Example 2 is impossible both for problems with pulse action [11] and for problems with boundary conditions of the "interface conditions" type [12].…”

Section: Example 2 Solutions Of the Boundary-value Problemmentioning

confidence: 99%

“…The obtained fundamental matrix X(t) of problem (11) is normalized at the point τ i 0 +0 = +0, and, furthermore, X(1 + 0) = X(1 + 0) = 1. The rank of the normalized fundamental matrix of problem (11) is nonzero in the intervals 0; π 2 and π 2 ; 2π , which is impossible for the normal (X(a) = I n ) fundamental matrix X(t) of the problem of finding smooth solutions of system (1) that satisfy a linear boundary condition and the problem with nondegenerate pulse action [7].…”

Section: Lemma 3 Under the Condition P Q = O Problem (4) (5) Has Amentioning

confidence: 99%

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Generalized green operator for an impulsive boundary-value problem with switchings

Бойчук

Чуйко²

2007

Nonlinear Oscill

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We establish constructive existence conditions and construct a generalized Green operator for the construction of solutions of a Noetherian linear boundary-value problem for a system of ordinary differential equations with switchings and pulse action in critical and noncritical cases. Statement of the ProblemConsider the problem of finding a solutionof a linear homogeneous differential equation with switchingswhere A i (t) are n × n matrices continuous on the segments [a;Let W 0 (t) be the normal (W 0 (a) = I n ) fundamental matrix of system (1) on the segment [a; τ 1 ] and let W 1 (t) be a fundamental matrix of this system on the segment [τ 1 ; τ 2 ] that satisfies the condition W 0 (τ 1 ) = W 1 (τ 1 ). The existence of the normal (W 0 (τ 1 ) = W 1 (τ 1 )) fundamental matrix of system (1) on the segment [τ 1 ; τ 2 ] follows from the nonsingularity of the fundamental matrices of system (1) on the segments [a; τ 1 ] and [τ 1 ; τ 2 ]. Thus, the normal (X 0 (a) = I n ) fundamental matrix X 0 (t) of system (1) can be represented in the formThe matrix X 0 (t) is continuous on the segment [a; b] and satisfies system (1).

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Section: Impulsive Boundary-value Problem With Switchings In the Critmentioning

confidence: 97%

“…In particular, for S i = 0, this problem was studied in [10]. If, for some i, the matrices I n + S i are singular, then a degenerate pulse action takes place [11].…”

Section: Linear Homogeneous Boundary-value Problems With Pulse Actionmentioning

confidence: 99%

Section: Impulsive Boundary-value Problem With Switchings In the Critmentioning

confidence: 99%

Section: Example 2 Solutions Of the Boundary-value Problemmentioning

confidence: 99%

Section: Lemma 3 Under the Condition P Q = O Problem (4) (5) Has Amentioning

confidence: 99%

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Generalized green operator for an impulsive boundary-value problem with switchings

Бойчук

Чуйко²

2007

Nonlinear Oscill

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On the Regularization of a Linear Fredholm Boundary-Value Problem by a Degenerate Pulsed Action

Чуйко

2014

J Math Sci

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Boundary value problems for higher order linear impulsive differential equations

Uğur

Akhmet

2006

Journal of Mathematical Analysis and Applications

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In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.

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Generalized green operator of a boundary-value problem with degenerate pulse influence

Cited by 6 publications

References 1 publication

Generalized green operator for an impulsive boundary-value problem with switchings

Generalized green operator for an impulsive boundary-value problem with switchings

On the Regularization of a Linear Fredholm Boundary-Value Problem by a Degenerate Pulsed Action

Boundary value problems for higher order linear impulsive differential equations

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