Weak and strong convergence of a scheme with errors for two nonexpansive mappings
30 Jun 2005-Nonlinear Analysis-theory Methods & Applications (Pergamon)-Vol. 61, Iss: 8, pp 1295-1301
TL;DR: In this article, a weak and strong convergence of an iterative scheme in a uniformly convex Banach space under a condition weaker than compactness was studied. But the convergence of the scheme was not considered.
Abstract: In this paper, we are concerned with the study of an iterative scheme with errors involving two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme in a uniformly convex Banach space under a condition weaker than compactness.
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TL;DR: In this article, a generalized iterative process with errors is considered to approximate the common fixed points of two asymptotically quasi-none-expansive mappings, and a convergence theorem has been obtained which generalizes a known result.
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TL;DR: In this article, an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces is proposed and analyzed, and results concerning Δ-convergence and strong convergence are proved.
Abstract: In this article, we propose and analyze an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Results concerning Δ-convergence as well as strong convergence of the proposed algorithm are proved. Our results are refinement and generalization of several recent results in CAT(0) spaces and uniformly convex Banach spaces. Mathematics Subject Classification (2010): Primary: 47H09; 47H10; Secondary: 49M05.
105 citations
01 Jan 2009
97 citations
TL;DR: In this paper, it was proved that the modified Mann iteration process converges weakly to a fixed point of an asymptotically κ -strict pseudocontractive mapping T in the intermediate sense which is not necessarily Lipschitzian.
Abstract: It is proved that the modified Mann iteration process: x n + 1 = ( 1 − α n ) x n + α n T n x n , n ∈ N , where { α n } is a sequence in (0, 1) with δ ≤ α n ≤ 1 − κ − δ for some δ ∈ ( 0 , 1 ) , converges weakly to a fixed point of an asymptotically κ -strict pseudocontractive mapping T in the intermediate sense which is not necessarily Lipschitzian. We also develop CQ method for this modified Mann iteration process which generates a strongly convergent sequence.
88 citations
TL;DR: This work gets some results on strong and @?-convergence in CAT(0) spaces for an iterative scheme which is both faster than and independent of the Ishikawa scheme.
Abstract: In this paper, we get some results on strong and @?-convergence in CAT(0) spaces for an iterative scheme which is both faster than and independent of the Ishikawa scheme. We also obtain some results for two mappings using the Ishikawa-type iteration scheme. The motivation of the present work comes from that of Dhompongsa and Panyanak (2008) [3].
83 citations
Cites background from "Weak and strong convergence of a sc..."
...Following Senter andDotson [17], Khan and Fukhar-ud-din [20] introduced the so-called condition (A′) for twomappings and gave an improved version of it in [21] as follows: Two mappings S, T : C → C are said to satisfy the condition (A′) if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that either d(x, Tx) ≥ f (d(x, F)) for all x ∈ C or d(x, Sx) ≥ f (d(x, F)) for all x ∈ C ....
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...Following Senter andDotson [17], Khan and Fukhar-ud-din [20] introduced the so-called condition (A) for twomappings and gave an improved version of it in [21] as follows: Two mappings S, T : C → C are said to satisfy the condition (A) if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that either d(x, Tx) ≥ f (d(x, F)) for all x ∈ C or d(x, Sx) ≥ f (d(x, F)) for all x ∈ C ....
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2,091 citations
845 citations
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01 Jan 1976
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TL;DR: In this article, it was shown that under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces, the convergence of xn to a fixed point is shown to be strong.
Abstract: Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.
677 citations
01 Feb 1974
TL;DR: In this article, it was shown that if T satisfies one additional condition, which is weaker than the requirement that T be demicompact, then an iterative process of the type introduced by W. R. Mann [8] converges to a fixed point of T.
Abstract: A condition is given for nonexpansive mappings which assures convergence of certain iterates to a fixed point of the mapping in a uniformly convex Banach space. A relationship between the given condition and the requirement of demicompactness is established. Introduction. Browder [1] and Kirk [7] have shown that a nonexpansive mapping T which maps a closed, bounded, convex subset C of a uniformly convex Banach space into itself has a nonempty fixed point set in C. In general, however, for arbitrary x E C the Picard iterates T'x do not converge to a fixed point of T. It will be shown that if T satisfies one additional condition, then an iterative process of the type introduced by W. R. Mann [8] converges to a fixed point of T. For nonexpansive mappings T which have fixed points, this additional condition is weaker than the requirement that T be demicompact. Convergence to a fixed point. Let X be a Banach space with norm I and C a convex subset of X. A self-mapping T of C is said to be nonexpansive provided 1 Tx-Ty1 0 for r E (0, oo), such that Ix-Txl?f(d(x, F)) for all x E C, where d(x, F)=inf{lx-zl :z E F}. Let P denote the set of positive integers. For x1 E C, M(x1, tn, T) is the sequence {x,} defined by x?+1=(I -tn)x +t Tx, where tn c [a, b] for all n e P and 0< a < b <1. This iterative process has been previously investigated by Dotson in [4]. Our main result for nonexpansive mappings is the following: THEOREM 1. Suppose X is a uniformly convex Banach space, C is a closed, bounided, convex, nonempty subset of X, and T is a nonexpansive mapping of C into C. Let F denote thefixed point set of T in C, and suppose Received by the editors February 26, 1973. AMS (MOS) subject classifications (1970). Primary 47H10.
510 citations