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 of n ordinary 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 conditions z(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 an explicit and analytical form 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 functional l) does not coincide with the number of unknowns in the differential system with a single delay.