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: , assuming that these solutions satisfy the initial and boundary conditions z ( s ): = ψ ( s ) if s∉ [ a , b ], ℓ z (·) = α ∈ 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.