Representation of solutions of neutral differential equations with delay and linear parts defined by pairwise permutable matrices

Abstract

This paper is devoted to the study of linear systems of neutral differential equations with delay. Assuming the linear parts to be given by pairwise permutable matrices, representation of a solution of a nonhomogeneous initial value problem using matrix polynomial of a degree depending on time is derived. Examples illustrating the obtained results are given.

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

“…Notice that the fractional analogue of the same problem was considered by Li and Wang 2 in the case A = Θ. For more recent contributions on oscillating system with pure delay, relative controllability of system with pure delay, asymptotic stability of nonlinear multidelay differential equations, finite time stability of differential equations, one can refer to previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and reference therein.…”

“…Under the assumptions that A and B are permutation matrices, Khusainov & Shuklin give a representation of a solution of a linear homogeneous system with delay by introducing the concept of delayed matrix exponential corresponding to delay h and matrix B : They proved that fundamental matrix of linear delay system (delayed perturbation of exponential matrix e A t ) can be given by Notice that the fractional analogue of the same problem was considered by Li and Wang in the case A = Θ. For more recent contributions on oscillating system with pure delay, relative controllability of system with pure delay, asymptotic stability of nonlinear multidelay differential equations, finite time stability of differential equations, one can refer to previous studies and reference therein.…”

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

“…Motivated by delayed exponential representing a solution of a system of differential or difference equations with one or multiple fixed or variable delays [1][2][3][4][5][6], which has many applications in theory of controllability, asymptotic properties, boundary-value problems, and so forth [3][4][5][7][8][9][10][11][12][13][14][15], we extended representation of a solution of a system of differential equations of second order with delay [1] ( ) = − 2 ( − ) (1) to the case of two delays…”

“…Indeed, if is × matrix, ≥ 3 is odd, and has a simple real nonzero eigenvalue , then there exists a regular matrix such that −1 = = ( 0 0̃) wherẽ is ( − 1) × ( − 1) matrix. On letting = , one gets = − 2 ( − ) (4) or rewrites as the system…”

“…We note that the delayed matrix exponential from [1][2][3][4][5] as well as the representation of a solution of second-order differential equations derived in [1,16] and in this paper can lead to new results in nonlinear boundary value problems for impulsive functional differential equations considered in [17] or stochastic delayed differential equations from [18].…”