Strong convergence of composite iterative methods for nonexpansive mappings

다국어 초록 (Multilingual Abstract)

Let E be a reflexive Banach space with a weakly sequentially
continuous duality mapping, C be a nonempty closed convex subset
of E, f : C → C a contractive mapping (or a weakly contractive
mapping), and T : C → C a nonexpansive mapping with the fixed
point set F(T) ≠Φ. Let {x_n} be generated by a new
composite iterative scheme: y_n = λ_nf(x_n) + (1 - λ_n)Tx_n, x_{n+1} = (1 - β_n)y_n + β_nTy_n,(n≥0). It is proved that {x_n} converges strongly to a point in
F(T), which is a solution of certain variational inequality
provided the sequence {λ_n} ⊂ (0,1) satisfies
lim_{n→∞}λ_n = 0 and ∑{n =0}^{∞}λ_n=∞ ,{β_n}⊂[0,a) for some 0 < a < 1 and the
sequence {x_n} is asymptotically regular.

다국어 초록 (Multilingual Abstract)

Let E be a reflexive Banach space with a weakly sequentially
continuous duality mapping, C be a nonempty closed convex subset
of E, f : C → C a contractive mapping (or a weakly contractive
mapping), and T : C → C a nonexpansive mapping with the fixed
point set F(T) ≠Φ. Let {x_n} be generated by a new
composite iterative scheme: y_n = λ_nf(x_n) + (1 - λ_n)Tx_n, x_{n+1} = (1 - β_n)y_n + β_nTy_n,(n≥0). It is proved that {x_n} converges strongly to a point in
F(T), which is a solution of certain variational inequality
provided the sequence {λ_n} ⊂ (0,1) satisfies
lim_{n→∞}λ_n = 0 and ∑{n =0}^{∞}λ_n=∞ ,{β_n}⊂[0,a) for some 0 < a < 1 and the
sequence {x_n} is asymptotically regular.

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