Smooth solutions to some differential-difference equations of neutral type

Abstract

The paper is devoted to the scalar linear differential-difference equation of neutral type

$$dx(t)/dt + p(t)dx(t - 1)/dt = a(t)x(t - 1) + b(t)x(t) + f(t)$$

. We study the existence of and methods for finding solutions possessing required smoothness on intervals of length greater than 1.

The following two settings are considered

(1) To find an initial function g(t) defined on the initial set t ∈ [t ₀ − 1, t ₄] such that the continuous solution x(t), t > t ₀, generated by g(t) possesses the required smoothness at points divisible by the delay time. For the investigation, we apply the inverse initial-value problem method.

(2) Let a(t), b(t), p(t), and f(t) be polynomials and let the initial value x(0) = x ₀ be assigned at the initial point t = 0. Polynomials satisfying the initial-value condition are considered as quasi-solutions to the original equation. After substitution of a polynomial of degree N for x(t) in the original equation, there appears a residual Δ(t) = O(t ^N), for which sharp estimates are obtained by the method of polynomial quasi-solutions. Since polynomial quasi-solutions may contain free parameters, the problem of minimization of the residual on some interval can be considered on the basis of variational criteria.