On the Stability of Asynchronous [ terative 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 constraints are imposed on the order at which the different operators are applied. The problem investigated in this paper is the following: we are given a set T,.... ,TK of operators on a common space X with a unique common fixed point x*. These operators are to be applied successively, starting from an arbitrary initial element of X. We derive necessary and sufficient conditions under which the outcome of such a sequence of operations converges to the desired common fixed point, when the order at which the operators are applied is left free; we only impose the requirement that each operator is applied an infinite number of times. (A process of this type will be called a "totally asynchronous" iterative process.) Our main results may be expressed in the following general form: convergence is obtained if and only if there exists a Lyapunov function (suitably defined)

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