Fixed point and selection theorems in hyperconvex spaces

Khamsi, M.; Kirk, W.; Martinez Yañez, Carlos

doi:10.1090/S0002-9939-00-05777-4

Fixed point and selection theorems in hyperconvex spaces
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by M. A. Khamsi, W. A. Kirk and Carlos Martinez Yañez PDF

Proc. Amer. Math. Soc. 128 (2000), 3275-3283 Request permission

Abstract:

It is shown that a set-valued mapping $T^{\ast }$ of a hyperconvex metric space $M$ which takes values in the space of nonempty externally hyperconvex subsets of $M$ always has a lipschitzian single valued selection $T$ which satisfies $d(T(x),T(y))\leq d_{H}(T^{\ast }(x),T^{\ast }(y))$ for all $x,y\in M$. (Here $d_{H}$ denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded $\lambda$-lipschitzian self-mappings of $M$ is itself hyperconvex. Several related results are also obtained.

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