Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces

“…Since by commutativity of the mappings t and T , t (y) ∈ T (t(x)) ∩ E = T (x) ∩ E for each x ∈ A and y ∈ T (x) ∩ E, it follows that T (x) ∩ E is invariant under t for each x ∈ A. By [22,Theorem 4.4] the invariance of T (x) ∩ E under t and the nonexpansiveness of t imply T (x) ∩ A = ∅ for x ∈ A. By [22,Corollary 3.7] the mapping T (·) ∩ A of A into the closed convex subsets of A is nonexpansive and has a fixed point, that is, there is z ∈ A such z ∈ T (z).…”

Metric fixed point theory on hyperconvex spaces: recent progress

“…The next theorem is a version of this result under milder topological conditions, see [54] for its proof. …”

“…In contrast to this, best proximity pair theorems provide approximate solutions that are optimal. A few advances have been obtained for hyperconvex spaces about existence of best proximity pairs in the last ten years [6,20,38,47,52,54]. In this section we collect some of them.…”

Coupled best approximation theorems for discontinuous operators in partially ordered Banach spaces

“…A remarkable application of fixed-point theorems is to prove the existence of best approximation point in best approximation theory [8,10,11,15,18,19,21], which was introduced by Fan in [3] for normed linear spaces. Various aspects of best approximation theorem have been studied by many authors under different assumptions and many interesting results are obtained in the last decades [7,9,14,16,17,[23][24][25].…”