  The one-dimensional Dunkl operator $D_k$ with a non-negative parameter $k$,
is considered under an arbitrary nonlocal boundary value condition. The right
inverse operator of $D_k$, satisfying this condition is studied. An operational
calculus of Mikusinski type is developed. In the frames of this operational
calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy
boundary value problems for Dunkl functional-differential equations $P(D_k)u=f$
with a given polynomial $P$ is proposed. The solution of these equations in
mean-periodic functions reduces to such problems. Necessary and sufficient
condition for existence of unique solution in mean-periodic functions is found.
