Strong convergence of composite iterative methods for nonexpansive mappings
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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.
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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