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: $\dot{z}\left(t\right)=Az(t-\tau )+g\left(t\right)+ϵZ(z\left(h_i\right(t),t,ϵ)$, $t\in [a,b]$, assuming that these solutions satisfy the initial and boundary conditions $z\left(s\right):=\psi \left(s\right)$ if $s\notin [a,b]$, $lz(\cdot )=\alpha \in {ℝ}^m$. 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.