On the stability of asynchronous iterative processes

We consider an iterative process in which one out of a finite set of possible operators is applied at each iteration. We obtain necessary and sufficient conditions for convergence to a common fixed point of these operators, when the order at which different operators are applied is left completely free, except for the requirement that each operator is applied infinitely many times. The theory developed is similar in spirit to Lyapunov stability theory. We also derive some very different, qualitatively, results for partially asynchronous iterative processes, that is for the case where certain requirements are imposed on the order at which the different operators are applied.