Menger Convexity and Hausdorff Metric

Shimizu and Takahashi have shown 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 $(\alpha, \beta)-$generalized hybrid mapping, and using the obtained nonempty intersection property we derive the fixed point results for this generalized mapping defined on Menger convex spaces.