A note on the Cauchy problem for differential difference equations

Abstract

has a unique continuous solution on the interval [0, T] if functions x,, and g are continuous (see [ 1, 2,4]). However, the function y(t) which coincides with x(t) on [0, T] and with -u,(t) on ( -3, 0) will in general be discontinuous or have discontinuous derivatives at t = 0, unless x,(t) is carefully chosen. We endeavoured to investigate whether this situation could be improved by using the classical Cauchy type initial condition, x(0, S) = c instead of (2), and imposing some continuity conditions. This leads, however, to a non-uniqueness of the resulting solution even if we demand analyticity in t. For example, the equation,