On Kannan maps

Wong, Chi Song

doi:10.1090/S0002-9939-1975-0358468-0

On Kannan maps
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Proc. Amer. Math. Soc. 47 (1975), 105-111 Request permission

Abstract:

Let $K$ be a (nonempty) weakly compact convex subset of a Banach space $B$. Let $T$ be a self map on $K$ such that for all $x,y$ in $K,||T(x) - T(y)|| \leqslant (||x - T(x)|| + ||y - T(y)||)/2$. It is proved without the continuity of $T$ and Zorn’s lemma that $T$ has a fixed point if and only if $\inf \{ ||x - T(x)||:x \in K\} = 0$. A characterization of the existence of fixed points for such $T$ is obtained in terms of close-to-normal structure. As consequences, the following results are obtained: (i) $T$ has a unique fixed point if $B$ is locally uniformly convex or more generally if $B$ has the property ${\mathbf {A}}$: For any sequence $\{ {x_n}\}$ in $B,\{ {x_n}\}$ converges to a point $x$ in $B$ if it converges weakly to $x$ and $\{ ||x||\}$ converges to $||x||$; (ii) $T$ has a unique fixed point if $B$ is separable.

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Additional Information

Journal: Proc. Amer. Math. Soc. 47 (1975), 105-111
DOI: https://doi.org/10.1090/S0002-9939-1975-0358468-0
MathSciNet review: 0358468