Let
E be a reflexive Banach space with a uniformly Gâteaux 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∈
C and
t∈(0, 1), let
xt be a unique fixed point of a contraction
Gt C→
E, defined by
Gtx=
tTx+(1−
t)
u,
x∈
C. It is proved that if {
xt} is bounded, then the strong lim
t→1xt 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 Gâteaux differentiable norm.
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