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

Abstract

In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive selfmapping T of a hyperconvex metric space (or a CAT(0) space) X, we look for a solution to the problem of existence of z ∈ E ⊂ X such that d(z, y) = d(y, E) and z = t (z) ∈ T (z).

Common fixed points for some generalized multivalued nonexpansive mappings in uniformly convex metric spaces

“…He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see e.g., [4][5][6][7][8][9][10][11][12][13][14] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in ℝ -trees) can be applied to graph theory, biology, and computer science (see e.g., [15][16][17][18][19][20]).…”

“…Shahzad and Markin [14] studied an invariant approximation problem and provided sufficient conditions for the existence of z K ⊆ X such that d(z, y) = dist(y, K) and z = t(z) T(z) where y X, t and T are commuting nonexpansive mappings on K. Shahzad [21] also obtained common fixed point and invariant approximation results for t and T, which are weakly commuting.…”

A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued

“…Shahzad and Markin [2] improved Theorem 1.1 by removing the assumption that the nonexpansive multivalued mapping T in that theorem has a convex-valued contractive approximation. They also noted that Theorem 1.1 needs the additional assumption that T(·) ∩ E ≠ ∅ for that result to be valid.…”

“…[ [2], Theorem 4.2] Let X be a complete CAT(0) space, and E a bounded closed and convex subset of X. Assume t : E E and T : E 2 X are nonexpansive mappings with T(x) a compact convex subset of X and T(x) ∩ E ≠ ∅ for each x E. If the mappings t and T commute, then Fix(t) ∩ Fix(T) ≠ ∅.…”

Strong andΔ-Convergence Theorems for Common Fixed Points of a Finite Family of Multivalued Demicontractive Mappings in CAT(0)Spaces

“…He showed that every nonexpansive mapping defined on a nonempty closed convex and bounded subset of a CAT(0) space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings has received much attention (see, e.g., [6][7][8][9][10][11][12][13]). In 1976, Lim [14] introduced a notion of convergence in a general metric space which he called Δ-convergence (see Definition 8).…”