Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Chidume, C.; Zegeye, H.

doi:10.1090/S0002-9939-03-07101-6

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps
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by C. E. Chidume and H. Zegeye PDF

Proc. Amer. Math. Soc. 132 (2004), 831-840 Request permission

Abstract:

Let $K$ be a nonempty closed convex subset of a real Banach space $E$ and $T$ be a Lipschitz pseudocontractive self-map of $K$ with $F(T):=\{x\in K:Tx=x\}\neq \emptyset$. An iterative sequence $\{x_n\}$ is constructed for which $||x_n-Tx_n||\rightarrow 0$ as $n\rightarrow \infty$. If, in addition, $K$ is assumed to be bounded, this conclusion still holds without the requirement that $F(T)\neq \emptyset .$ Moreover, if, in addition, $E$ has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of $K$ has the fixed point property for nonexpansive self-mappings, then the sequence $\{x_n\}$ converges strongly to a fixed point of $T$. Our iteration method is of independent interest.

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