Abstract The theory of generalized differentiation in set-valued analysis is based on Mordukhovich derivative (Mordukhovich coderivative), which has been widely applied to optimization theory, equilibrium theory, variational analysis, with respect to set-valued mappings. In this paper, we use the Mordukhovich derivatives to precisely find the covering constants for metric projection onto nonempty closed and convex subsets in uniformly convex and uniformly smooth Banach spaces. This is considered as optimizing the metric projection with respect to covering values. We study three cases: closed balls in uniformly convex and uniformly smooth Banach spaces, closed and convex cylinders in l p and positive cones in L p , for some p with 1 < p <  $$\infty$$ ∞ . By Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem (Theorem 3.1 in (J Optim Theory Appl 196:177–198, 2023)), which is simply called Arutyunov Mordukhovich Zhukovskiy Theorem, and as applications of covering constants obtained in this paper, we prove solvability of some locally stochastic fixed-point problems. We also provide three examples with specific solutions of both locally and globally stochastic fixed-point problems.