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

In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive self-mapping 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⊂Xz∈E⊂X such thatd(z,y)=d(y,E)andz=t(z)∈T(z).