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 ð ‘› 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: ̇ 𠑧 ( ð ‘¡ ) = ð ´ ð ‘§ ( ð ‘¡ − ð œ ) + ð ‘” ( ð ‘¡ ) + 𠜀 ð ‘ ( 𠑧 ( â„Ž ð ‘– ( ð ‘¡ ) , ð ‘¡ , 𠜀 ) , ð ‘¡ ∈ [ ð ‘Ž , ð ‘ ] , assuming that these solutions satisfy the initial and boundary conditions 𠑧 ( ð ‘ ) ∶ = 𠜓 ( ð ‘ ) i f ð ‘ âˆ‰ [ ð ‘Ž , ð ‘ ] , â„“ 𠑧 ( â‹… ) = ð ›¼ ∈ â„ ð ‘š . 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 â„“ ) does not coincide with the number of unknowns in the differential system with a single delay.