Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

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: (z)over dot(t) = Az(t - tau) + g(t) + epsilon Z(z(h(i)(t), t, epsilon), t is an element of [a, b], assuming that these solutions satisfy the initial and boundary conditions z(s) := psi(s) if s is an element of [a, b], lz(.) = alpha is an element of R(m). The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical formof 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.