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Zbl 1005.47053
Chidume, C.E.; Zegeye, H.; Aneke, S.J.
Approximation of fixed points of weakly contractive nonself maps in Banach spaces. (English)
[J] J. Math. Anal. Appl. 270, No.1, 189-199 (2002). ISSN 0022-247X

The authors present extensions and generalisations of theorems related with approximating fixed points due to {\it Ya. Alber} and {\it 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:\par 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 $T: K \rightarrow E$ be a d-weakly contractive map such that a fixed point $x^* \in int(K)$ 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 $T : K \rightarrow E$ is a uniformly continuous d-weakly contractive map with $F(T) :=\{x \in K: T x = x\} \ne\emptyset $, it is proved that a descent-like approximation sequence converges strongly to $x^* \in F(T).$
[S.L.Singh (Rishikesh)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: uniformly smooth Banach space; descent-like approximation sequence

Citations: Zbl 0985.47044

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Zentralblatt MATH Berlin [Germany]