The Best Approximation Theorems and Fixed Point Theorems for Discontinuous Increasing Mappings in Banach Spaces

Abstract EN

We prove that Fan’s theorem is true for discontinuous increasing mappings f in a real partially ordered reflexive, strictly convex, and smooth Banach space X.

The main tools of analysis are the variational characterizations of the generalized projection operator and order-theoretic fixed point theory.

Moreover, we get some properties of the generalized projection operator in Banach spaces.

As applications of our best approximation theorems, the fixed point theorems for non-self-maps are established and proved under some conditions.

Our results are generalizations and improvements of the recent results obtained by many authors.