门格凸性与不动点结果
Menger Convexity and Fixed Point Results
AI总结：
本文推广了广义混合映射，基于门格凸度量空间的非空交集性质，推导了该广义映射的不动点结果，拓展了凸度量空间的相关理论。
摘要：
Shimizu和Takahashi证明了完备一致Takahashi凸度量空间中任意非空、有界、闭、凸子集的递减序列具有非空交集。已知门格（Menger）凸性是Takahashi凸性的推广。本文中，我们针对门格凸度量空间，基于豪斯多夫（Hausdorff）度量，得到了非空交集性质，该性质也提供了一类自反门格凸空间。我们引入了$(\boldsymbol{\beta})$-广义混合映射的推广形式，并利用所得非空交集性质，推导了定义在门格凸空间上的该广义映射的不动点结果。
英文摘要：
Shimizu and Takahashi proved that every decreasing sequence of nonempty, bounded, closed, convex subsets of a complete, uniformly Takahashi convex metric space has nonempty intersection. It is well known that the Menger convexity is a generalization of the Takahashi convexity. In this article, we acquire a nonempty intersection property, in terms of the Hausdorff metric, for Menger convex metric spaces, that also provides a class of reflexive Menger convex spaces. We introduce a generalization of $(α, β)-$generalized hybrid mappings, and using the obtained nonempty intersection property we derive the fixed point results for this generalized mapping defined on Menger convex spaces.
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