The Chebyshev selections and fixed points of set-valued mappings in Banach spaces with some uniform convexity

The existence of a continuous Chebyshev selection for a Hausdorff continuous set-valued mapping is studied in a Banach space with some uniform convexity. As applications, some existence results of Chebyshev fixed point for condensing set-valued mappings are given, and the existence of Chebyshev solutions for an integral inclusion is proved.