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Iterative approximations for multivalued nonexpansive mappings in reflexive banach spaces. (English) Zbl 1183.47073

Let \(K\) be a nonempty closed convex subset of a Banach space \(E\) and let \(T:K\rightarrow C(E)\) be a nonexpansive weakly inward mapping on \(K\), where \(C(E)\) denotes the family of nonempty closed and bounded subsets of \(E\).
In order to approximate the fixed points of \(T\), the authors consider two methods, namely, an implicit method, given by
\[ x_t\in (1-t) Tx_t+tu,\,t\in (0,1),\,u\in K, \]
and an explicit method of Halpern type, defined by
\[ x_{n+1}\in (1-\alpha_n) Tx_n+\alpha_n u,\,\alpha_n\in (0,1),\,u\in K. \]
Some strong convergence theorems for \(\{x_t\}\) and \(\{x_n\}\) in particular Banach spaces are obtained, which improve and extend several results in literature.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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