TY - JOUR
T1 - Strong and weak convergence theorems for asymptotically nonexpansive mappings
AU - Chidume, C. E.
AU - Ofoedu, E. U.
AU - Zegeye, Habz
PY - 2003/4/15
Y1 - 2003/4/15
N2 - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.
AB - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.
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U2 - 10.1016/S0022-247X(03)00061-1
DO - 10.1016/S0022-247X(03)00061-1
M3 - Article
AN - SCOPUS:0038730516
SN - 0022-247X
VL - 280
SP - 364
EP - 374
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -