# Smooth solutions to some differential-difference equations of neutral type

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## Abstract

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.

## Keywords

Linear System Smooth Solution Initial Function Exact Analytical Solution Neutral Type## Preview

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