ANALYSIS OF THE DIFFERENTIAL-DIFFERENCE EQUATION y(x+12)−y(x−12)=y′(x)

Abstract

In this paper we study some solution techniques of solving the differential-difference equation

$$y^{\prime }(x)=y\left(x+\frac{1}{2}\right)-y\left(x-\frac{1}{2}\right),$$

first without an initial condition and then with some initial function $h$ defined on the unit interval $\left[-\frac{1}{2},\frac{1}{2}\right]$. 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 $0$.