Approximately intertwining mappings

Let AA be a Banach algebra, and let E   be a weak Banach AA-bimodule. An approximately intertwining mapping corresponding to a functional equation E(f)=0E(f)=0 is a mapping f:A→E with f(0)=0f(0)=0 such that‖E(f)‖⩽ε,‖E(f)‖⩽ε, and for each a∈Aa∈A the mappingsfa(x)=f(ax)−af(x),fa(x)=f(ax)−af(x),fa(x)=f(xa)−f(x)a, are continuous at a point. In this paper, we show that every approximately intertwining mapping corresponding to Cauchy, generalized Jensen or Trif functional equation can be estimated by an intertwining mapping.