Convergence for a family of discrete advection–reaction operators

We define a family of discrete Advection–reaction operators, denoted by AaλAaλ, which associate to a given scalar sequence s={sn}s={sn} the sequence given by Aaλ(s)≡{bn}Aaλ(s)≡{bn}, where bn=an−2sn−1+λnsnbn=an−2sn−1+λnsn for n=1,2,…n=1,2,…. For AaλAaλ 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.