New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings |
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Authors: | S. Thianwan |
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Affiliation: | 1. Naresuan Phayao University, Phayao, Thailand
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Abstract: | Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T 1, T 2: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k n (i) } and {δ n (i) } ? [1, ∞),, i = 1, 2, respectively such that Σ n=1 ∞ (k n (i) ? 1) < ∞, Σ n=1 (i) δ n (i) < ∞, and F = F(T 1) ∩ F(T 2) ≠ ?. For an arbitrary x 1 ∈ C, let {x n } be the sequence in C defined by $$ begin{gathered} y_n = Pleft( {left( {1 - beta _n - gamma _n } right)x_n + beta _n T_2 left( {PT_2 } right)^{n - 1} x_n + gamma _n v_n } right), hfill x_{n + 1} = Pleft( {left( {1 - alpha _n - lambda _n } right)y_n + alpha _n T_1 left( {PT_1 } right)^{n - 1} x_n + lambda _n u_n } right), n geqslant 1, hfill end{gathered} $$ where {α n }, {β n }, {γ n } and {λ n } are appropriate real sequences in [0, 1) such that Σ n=1 ∞ ] γ n < ∞, Σ n=1 ∞ λ n < ∞, and {u n }, }v n } are bounded sequences in C. Then {x n } and {y n } converge strongly to a common fixed point of T 1 and T 2 under suitable conditions. |
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