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

Abstract

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).

A periodic boundary value problem with switchingsunder nonlinear perturbations

“…and investigate the solvability of the boundary value problem (15) in the class of functions z(t, ε)…”

“…In this case The contraction condition (14) holds for ε ∈ [0, ε * ]; the value of ε * > 0 for which the iterative scheme ( 13) is applicable can be found numerically. For the considered problem, practical convergence is preserved up to ε * ≈ 1.09: The computed approximations of the solution to the periodic boundary value problem (15) are characterized by the discrepancies…”

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

“…We study the problem of regularization [11,13] of the boundaryvalue problem (1) with the help of the pulsed action [5,9,10,12,[14][15][16] z. / D S z.…”

Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action

“…In this case, a solution of problem (1), (2) can be expressed in terms of the generalized Green operator [4] G f s t…”

“…Under the condition P Q ≠ 0 , the homogenous part of problem (1),(2) has a family of solutions[4] z t c ( , ) = X t c ( ) , c R n ∈ , that can be represented by the normalized X i…”