Smooth solutions to some differential-difference equations of neutral type

V. B. Cherepennikov ; P. G. Ermolaeva

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 78-87

Résumé

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 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 $t\in[t_0-1,t_4]$ such that the continuous solution $x(t)$, $t>t_0$, generated by $g(t)$ possesses required smoothness at the 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 $\Delta(t)=O(t^N)$, 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.