The best approximation theorems and fixed point theorems for discontinuous increasing mappings in Banach spaces (Q1668903)

Summary: 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.