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Two strong convergence theorems for a proximal method in reflexive Banach spaces. (English) Zbl 1200.47085

Let X be a reflexive Banach space and A i i=1 N a finite family of maximal monotone operators in X which have a common zero, i.e.,

Z:= i=1 N A i -1 (0 * )·

In order to approximate such a zero, the authors introduce two new proximal type algorithms with errors, for which they give corresponding strong convergence theorems. The obtained common zero, in both cases, is proj Z f (x 0 ), where x 0 is the initial approximation and proj Z f denotes the Bregman projection of X onto Z induced by the Legendre function f:X·

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties