  In this article, we establish that any uniformly local Mizoguchi-Takahashi
contraction is actually a set-valued contraction due to Feng and Liu on a
metrically convex complete metric space. Through an example, we demonstrate
that this result need not hold on any arbitrary metric space. Furthermore, when
the metric space is compact, we derive that any Mizoguchi-Takahashi local
contraction and Nadler local contraction are equivalent. Moreover, a result
related to invariant best approximation is established.

Functional Analysis