We propose an approach to the investigation of even systems of functional differential equations with restrictions and control. According to this approach, the investigation of the consistency of the considered problem is reduced to the investigation of the solvability of a system of integral equations. We substantiate the application of the iteration and projection-iterative methods to this problem.
Object of InvestigationEven systems of equations are encountered in the mathematical simulation of processes in inhomogeneous media. We investigate a model according to which an evolution process running for t ∈ [a, c] in a certain medium and described by the system of functional differential equations with controlpasses at time t = c to a different medium, where it is described by the following system of functional differential equations for t ∈ [c, b]:We assume here that the elements of the given m × m matrices P (t) and Q(t) are square summable on the segments [a, c] and [c, b], respectively, and the given vector functions f : [a, c]×R m ×R m → R m and g : [c, b]× R m ×R m → R m are such that the Nemytskii operators generated by these functions map the spaces L 2 ([a, c], R m ) and L 2 ([c, b], R m ) , respectively, into themselves. Consider the case where the control is defined by the relationand the strictly monotonically increasing continuous functions ν(t) and τ (t) defined on [a, c] and [c, b], respectively, are such that ν(a) = a, ν(c) ≥ τ (c), τ(c) ≥ a, and τ (b) ≤ b; in particular, ν(t) = t and τ (t) = t − Δ, where Δ = const > 0 and Δ ≤ c − a.