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Fixed points of asymptotically regular mappings in spaces with uniformly normal structure. (English) Zbl 0768.47027
Summary: 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 $$T: A\to A$$ is an asymptotically regular mapping such that $$\liminf_{n\to\infty}||| T^ n|||<k$$, where $$||| T|||$$ is the Lipschitz constant (norm) of $$T$$, then $$T$$ has a fixed point in $$A$$.

##### MSC:
 47H10 Fixed-point theorems 46B20 Geometry and structure of normed linear spaces
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