Li, Gang and Sims, Brailey (2008) *$\tau$ AND WEAK DEMICLOSENESS PRINCIPLE AND ASYMPTOTIC BEHAVIOR FOR ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS.* In: Fixed point theory and its applications. Yokohama Publishers, Yokohama, pp. 103-108.

## Abstract

The purpose of this paper is to provide the new demicloseness principle{ $\tau$ (weakly) demicloseness principle. We prove that if $X$ is a Banach space with locally uniformly $\tau$-Opial condition, where $tau$ is a Hausdorff topology on $X$, $C$ is a nonempty $\tau$ compact subset of $X$, and $T : C \rightarrow C$ is a asymptotically nonexpansive type mapping. If $\{x_\alpha\}$ is a net in $C$ which converges to $x$ in the sense of $\tau$ topology and if the net $\{x_\alpha-T^mx_alpha\} converges to zero in the sense of $\tau$ topology for each $m \in N$, then $x-Tx = 0$. We also give the weakly demicloseness theorem in a Banach space with Opial property. This result is to be used to study convergence theorem for almost-orbits of asymptotically nonexpansive type mappings in a Banach space.

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