Geodesic
Fixed points of asymptotically regular mappings in spaces with uniformly normal structure
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 639-643 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that $$ \liminf _{n\rightarrow \infty } |\kern -0.8pt|\kern -0.8pt|T^n|\kern -0.8pt|\kern -0.8pt| k, $$ where $|\kern -0.8pt|\kern -0.8pt|T|\kern -0.8pt|\kern -0.8pt|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.
It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that $$ \liminf _{n\rightarrow \infty } |\kern -0.8pt|\kern -0.8pt|T^n|\kern -0.8pt|\kern -0.8pt| k, $$ where $|\kern -0.8pt|\kern -0.8pt|T|\kern -0.8pt|\kern -0.8pt|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.
Classification : 46B20, 47H10
Keywords: asymptotically regular mappings; uniformly normal structure; fixed points 
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     title = {Fixed points of asymptotically regular mappings in spaces with uniformly normal structure},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_4_a5/}
}
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Górnicki, Jarosław. Fixed points of asymptotically regular mappings in spaces with uniformly normal structure. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 639-643. http://geodesic.mathdoc.fr/item/CMUC_1991_32_4_a5/

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