Convergence for a family of discrete advection-reaction operators

We define a family of discrete Advection-reaction operators, denoted by A"a"@l, which associate to a given scalar sequence s={s"n} the sequence given by A"a"@l(s)={b"n}, where b"n=a"n"-"2s"n"-"[email protected]"ns"n for n=1,2,.... For A"a"@l we explicitly find their iterates and study their convergence properties. Finally, we show the relationship between the family of discrete operators with the continuous one dimensional advection-reaction equation.