Abstract
Fredholm boundary-value problems for linear delay systems defined by pairwise permutable matrices
Highlights
The aim of the paper is to prove an existence result for the following boundary-value problem: z(t) = Az(t) + B1z(t − τ1) + · · · + Bnz(t − τn) + g(t), z(s) = ψ(s), if s ∈ [−τ, 0], lz(·) = α ∈ Rm, (1.1) (1.2)A
First of all we consider initial value problems for a system of linear differential equations with delays defined by pairwise permutable matrices: z(t) = Az(t) + B1z(t − τ1) + · · · + Bnz(t − τn) + g(t), t ∈ [0, b], (1.3)
A solution of differential system (1.6) is defined as a vector-function z(t) ∈ Dp[0, b] absolutely continuous on [0, b] with z(t) ∈ Lp[0, b], if it satisfies the system (1.6) almost everywhere on [0, b]. Such a treatment makes it possible to apply to the equation (1.6) with the linear and bounded operator L well developed methods of linear functional analysis
Summary
First of all we consider initial value problems for a system of linear differential equations with delays defined by pairwise permutable matrices: z(t) = Az(t) + B1z(t − τ1) + · · · + Bnz(t − τn) + g(t), t ∈ [0, b],. We will investigate the equation (1.6) assuming that the operator L maps a Banach space Dp[0, b] of absolutely continuous functions z : [0, b] → RN with the norm z(t) Dp = z(t) Lp + z(0) RN. A solution of differential system (1.6) is defined as a vector-function z(t) ∈ Dp[0, b] absolutely continuous on [0, b] with z(t) ∈ Lp[0, b], if it satisfies the system (1.6) almost everywhere on [0, b] Such a treatment makes it possible to apply to the equation (1.6) with the linear and bounded operator L well developed methods of linear functional analysis. In this case the problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayed matrix exponential defined in [6] and generalized to the case of several delays in [8]
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