## Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings.(English)Zbl 0865.47040

The demiclosedness principle of F. E. Browder [Bull. Am. Math. Soc. 74, 660-665 (1968; Zbl 0164.44801)] states that if $$X$$ is a uniformly convex Banach space, if $$C$$ is a nonempty closed convex subset of $$X$$, and if $$T:C\to X$$ is a nonexpansive map, then $$I-T$$ is demiclosed at each $$y$$ in $$X$$. In this paper, authors prove this principle at zero for asymptotically nonexpansive maps either in a Banach space with the locally uniform Opial condition or in a nearly uniformly convex Banach space with Opial’s condition. They also study the asymptotic behavior of the iterates for an asymptotically nonexpansive map. Finally, they prove that the uniform Opial condition in a Banach space $$Y$$ implies the fixed point property for asymptotically nonexpansive maps defined on weakly compact convex subsets of $$Y$$.

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46B20 Geometry and structure of normed linear spaces 47H10 Fixed-point theorems

Zbl 0164.44801
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### References:

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