Computation of Fixed Points

Abstract

In this chapter we study the behavior of a class of iterative algorithms for computing fixed points of operators defined on subsets of a reflexiveBanach space B and subject to relatively unrestrictive conditions. In order to precise the framework within which we are working we introduce the following notions: Definition. Let ∫ : B → (−∞,+∞] be a lower semicontinuous convex function and let D := Dom(∫). The function ∫ is called a Bregman function on the set C ̸ Int(D) if, for each x ∈ C , the following conditions are satisfied:

• (i) ∫ is differentiable and totally convex at x;

• (ii) For any α ≥ 0, the set

  $$ R_{\alpha }^{f}(x;C) = \{ y \in C;{{D}_{f}}(x,y) \leqslant \alpha \} $$

  is bounded.