Generalized Forward–Backward Methods and Splitting Operators for a Sum of Maximal Monotone Operators

Abstract

Suppose each of A1,…,An is a maximal monotone, and βB is firmly nonexpansive with β>0. In this paper, we have two purposes: the first is finding the zeros of ∑j=1nAj+B, and the second is finding the zeros of ∑j=1nAj. To address the first problem, we produce fixed-point equations on the original Hilbert space as well as on the product space and find that these equations associate with crucial operators which are called generalized forward–backward splitting operators. To tackle the second problem, we point out that it can be reduced to a special instance of n=2 by defining new operators on the product space. Iterative schemes are given, which produce convergent sequences and these sequences ultimately lead to solutions for the last two problems.