Covering Constants for Metric Projection Operator with Applications to Stochastic Fixed-Point Problems

TL;DR

This paper determines covering constants for the metric projection operator in Banach spaces using Mordukhovich derivatives, and applies these results to establish solvability of certain stochastic fixed-point problems with concrete examples.

Contribution

It introduces a method to compute covering constants for metric projections in Banach spaces and applies these to solve stochastic fixed-point problems.

Findings

Covering constants for metric projections are explicitly calculated.

Solvability of stochastic fixed-point problems is established.

Examples demonstrate specific solutions to these problems.

Abstract

In this paper, we use the Mordukhovich derivatives to precisely find the covering constants for the metric projection operator onto nonempty closed and convex subsets in uniformly convex and uniformly smooth Banach spaces. We consider three cases of the subsets: closed balls in uniformly convex and uniformly smooth Banach spaces, closed and convex cylinders in l, and the positive cone in L, for some p. By using Theorem 3.1 in [2] and as applications of covering constants obtained in this paper, we prove the solvability of some stochastic fixed-point problems. We also provide three examples with specific solutions of stochastic fixed-point problems.