Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

Abstract

Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems ofnordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity:ż(t)=Az(t-τ)+g(t)+εZ(z(hi(t),t,ε), t∈[a,b], assuming that these solutions satisfy the initial and boundary conditionsz(s):=ψ(s) if s∉[a,b], lz(⋅)=α∈Rm. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to anexplicitandanalyticalform of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functionall) does not coincide with the number of unknowns in the differential system with a single delay.

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…”

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

“…The convergence of the series (3.4) can be proved by traditional methods of majorization [5]. The proof of this statement is analogous to the proof for differential [10,11], [5, page 271] and difference [12] equations in a finite-dimensional case and for systems of differential equations in an infinite-dimensional space [13,14]. In detail, to complete the proof of convergence of the series (3.4), we now show that for sufficiently small fixed " 2 .0, " , the series (3.4) converges for t 2 OEa, b T .…”

Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays

“…The properties of delayed matrix exponential functions for their continuous and discrete variants and their applications are the topic of recent papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. We note that the definition of the delayed matrix exponential was first defined for the continuous case in [4] and, for the discrete case, in [1,2].…”