Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization

A generalized "measure of distance" defined by D_f(x, y): = f(x) - f(y) - 〈∇f(y), x - y〉, is generated from any member f of the class of Bregman functions. Although it is not, technically speaking, a distance function, it has been used in the past to define and study projection operators. In this paper we give new definitions of paracontractions, convex combinations, and firmly nonexpansive operators, based on D_f(x, y), and study sequential and simultaneous iterative algorithms employing them for the solution of the problem of finding a common asymptotic fixed point of a family of operators. Applications to the convex feasibility problem, to optimization and to monotone operator theory are also included.