Convergence of string-averaging method for a class of operators

Abstract

We analyse a fixed-point iterative method with a finite pool of operators which are subfamily of strictly quasi-nonexpansive operators. These operators, which are not necessarily continuous, may be employed in iterative methods used in convex feasibility problems. Furthermore, members of this subfamily are able to handle convex constraints. The current iterate of the fixed-point iterative method is made by averaging of strings' endpoints and each string consists of a composition of operators which lie in the pool. To examine the study, we deal with two important pools of operators. The first one is a class of operators which define the algebraic iterative methods, as block iterative projection methods, for solving linear systems of equations (inequalities). The second class consists of the parallel subgradient projection operators for solving nonlinear convex feasibility problems. In both classes, we use optimal relaxation or optimal weight parameters which may break the continuity of the operators used in the classes. The advantages and disadvantages of using these parameters are illustrated using some numerical examples.