STRONG CONVERGENCE OF COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS
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초록
Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {T_i}_{i=1}^N be N nonexpansive self-mappings of K with F = ∩^N_{i=1}F(Ti) not= ∅ (here F(Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}^N_{i=1} is semi-compact. Let {a_n} ⊂ [d; 1 − d] for some d ∈ (0; 1) and {b_n} ⊂ [t; 1] for some t ∈ (0; 1]. For arbitrary x_0 ∈ K, let the sequence {x_n} be defined iteratively by x_n = a_n x_{n−1} + (1 − a_n)T_n y_n; y_n = b_n x_{n−1} + (1 − b_n)T_n x_n; ∀n ≥ 1; where T_n = T_n(mod N). Then {x_n} convergence strongly to a common fixed point of the mappings family {T_i}^N_{i=1}. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of b_n ≡ 1 or N = 1 are also new.
Abstract
Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {T_i}_{i=1}^N be N nonexpansive self-mappings of K with F = ∩^N_{i=1}F(Ti) not= ∅ (here F(Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}^N_{i=1} is semi-compact. Let {a_n} ⊂ [d; 1 − d] for some d ∈ (0; 1) and {b_n} ⊂ [t; 1] for some t ∈ (0; 1]. For arbitrary x_0 ∈ K, let the sequence {x_n} be defined iteratively by x_n = a_n x_{n−1} + (1 − a_n)T_n y_n; y_n = b_n x_{n−1} + (1 − b_n)T_n x_n; ∀n ≥ 1; where T_n = T_n(mod N). Then {x_n} convergence strongly to a common fixed point of the mappings family {T_i}^N_{i=1}. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of b_n ≡ 1 or N = 1 are also new.
참고문헌 (7)
- C. E. Chidume, "Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear Anal. and Appl", pp.383~429 (2003), Kluwer Academic Publishers
- C. E. Chidume, "Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings", Nonlinear Anal, 62, pp.1149~1156 (2005)
- N. Shahzad, "Approximating fixed points of non-self nonexpansive mappings in Banach spaces", Nonlinear Anal, 6, pp.1031~1039 (2005)
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- Y. Zhou, "Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces", Numer. Funct. Anal. Appl, 23, pp.911~921 (2002)