Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces
Jong Soo Jung, Tae Hwa Kim · Kodai Mathematical Journal · 1998
Let E be a reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of E, and T C -• E a nonexpansive mapping satisfying the inwardness condition.Assume that every weakly compact convex subset of E has the fixed point property.For u e C and / e (0,1), let x t be a unique fixed point of a contraction G t C -> E, defined by G t x = tTx + (1t)u, x e C. It is proved that if {x t } is bounded, then the strong lim^ix, exists and belongs to the fixed point set of T Furthermore, the strong convergence of other two schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm.