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Approximation of fixed points of weakly contractive nonself maps in Banach spaces. (English) Zbl 1005.47053

The authors present extensions and generalisations of theorems related with approximating fixed points due to Ya. Alber and S. Guerre-Delabriere [Analysis, München 21, No. 1, 17-39 (2001; Zbl 0985.47044)] from real Hilbert spaces to more general real uniformly smooth Banach spaces. We cite two results briefly stated in the abstract:

Let $K$ be a closed convex subset of a real uniformly smooth Banach space $E·$ Suppose $K$ is a nonexpansive retract of $E$ with $P$ as the nonexpansive retraction. Let $T:K\to E$ be a d-weakly contractive map such that a fixed point ${x}^{*}\in int\left(K\right)$ of $T$ exists. It is proved that a descent-like approximation sequence converges strongly to ${x}^{*}·$ Furthermore, if $K$ is a nonempty closed convex subset of an arbitrary real Banach space and $T:K\to E$ is a uniformly continuous d-weakly contractive map with $F\left(T\right):=\left\{x\in K:Tx=x\right\}\ne \varnothing$, it is proved that a descent-like approximation sequence converges strongly to ${x}^{*}\in F\left(T\right)·$

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties