Proximal normal structure and relatively nonexpansive mappings
The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy k T x − T yk ≤ k x − yk for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A, B) has proximal normal structure, then a relatively nonexpansive mapping T : A ∪ B → A ∪ B satisfying (i) T(A) ⊆ B and T(B) ⊆ A, has a proximal point in the sense that there exists x0 ∈ A ∪ B such that k x0 − T x0k = dist(A, B). If in addition the norm of X is strictly convex, and if (i) is replaced with (i) ' T(A) ⊆ A and T(B) ⊆ B, then the conclusion is that there exist x0 ∈ A and y0 ∈ B such that x0 and y0 are fixed points of T and k x0 − y0k = dist(A, B). Because every bounded closed convex pair in a uniformly convex Banach space has proximal normal structure, these results hold in all uniformly convex spaces. A Krasnosel'skiuo type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.