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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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