Analysis of the Differential-Difference Equation $y(x+1\mathrm{/}2)-y(x-1\mathrm{/}2)=y^{\mathrm{\prime }}(x)$

In this paper we study some solution techniques of differential-difference equation $$y^{\mathrm{\prime }}(x)=y(x+1\mathrm{/}2)-y(x-1\mathrm{/}2),$$ first without an initial condition and then with some initial function $h$ defined on the unit interval $[-1\mathrm{/}2,1\mathrm{/}2]$. We show some sufficient conditions that an initial function $h$ is admissible, i.e., it yields a unique continuous solution on some symmetric interval about $0$.