TAILIEUCHUNG - Weak and strong convergence for nonexpansive nonself-mapping
Suppose C is a nonempty closed convex nonexpansive retract of real uniformly convex Banach space X with P a nonexpansive retraction. Let T : C → X be a nonexpansive nonself-mapping of C with F(T) := {x ∈ C : T x = x} 6= ∅. Suppose {xn} is generated iteratively by x1 ∈ C. | Weak and Strong Convergence for Nonexpansive Nonself-Mapping Nguyen Thanh Mai University of Science, Thainguyen University, Vietnam E-mail: thanhmai6759@ Abstract: Suppose C is a nonempty closed convex nonexpansive retract of real uniformly convex Banach space X with P a nonexpansive retraction. Let T : C → X be a nonexpansive nonself-mapping of C with F (T ) := {x ∈ C : T x = x} = 6 ∅. Suppose {xn } is generated iteratively by x1 ∈ C, yn = P ((1 − an − µn )xn + an T P ((1 − βn )xn + βn T xn ) + µn wn ), xn+1 = P ((1 − bn − δn )xn + bn T P ((1 − γn )yn + γn T yn ) + δn vn ), n ≥ 1, where {an }, {bn }, {µn }, {δn }, {βn } and{γn } are appropriate sequences in [0, 1] and {wn }, {vn } are bounded sequences in C. (1) If T is a completely continuous nonexpansive nonself-mapping, then strong convergence of {xn } to some x∗ ∈ F (T ) is obtained; (2) If T satisfies condition, then strong convergence of {xn } to some x∗ ∈ F (T ) is obtained; (3) If X is a uniformly convex Banach space which satisfies Opial’s condition, then weak convergence of {xn } to some x∗ ∈ F (T ) is proved. Keywords: Weak and strong convergence; Nonexpansive nonself-mapping. 2000 Mathematics Subject Classification: 47H10, 47H09, 46B20. 1 Introduction Fixed point iteration processes for approximating fixed points of nonexpansive mappings in Banach spaces have been studied by various authors (see [3, 4, 6, 9, 10, 15, 17, 19]) using the Mann iteration process (see [6]) or the Ishikawa iteration process (see [3, 4, 15, 19]). For nonexpansive nonself-mappings, some authors (see [19, 12, 14, 16]) have studied the strong and weak convergence theorems in Hilbert space or uniformly convex Banach spaces. In 2000, Noor [7] introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces. In 1998, Takahashi and Kim [14] proved strong convergence of approximants to fixed points of nonexpansive nonself-mappings in reflexive Banach spaces with a .
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