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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 $\left\{A_i\right\}_{i=1}^N$ a finite family of maximal monotone operators in $X$ which have a common zero, i.e., $$ Z:=\bigcap_{i=1}^{N} A_i^{-1}(0^*)\neq \emptyset. $$ 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$^{f}_{Z}(x_0)$, where $x_0$ is the initial approximation and proj$^{f}_{Z}$ denotes the Bregman projection of $X$ onto $Z$ induced by the Legendre function $f:X\rightarrow \mathbb{R}.$

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