Iterative approaches to convex feasibility problems in Banach spaces

O’Hara, John G.; Pillay, Paranjothi; Xu, Hong–Kun

doi:10.1016/j.na.2005.07.036

Cited by 85 publications

(25 citation statements)

References 24 publications

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“…The following result was given by Jung [8] (see also O'Hara et al [15] and Xu [23] for the case that f (x) = u a constant). We refer Jung and Sahu [10] for the case of non-LIpschizian mappings.…”

Section: Resultsmentioning

confidence: 89%

Strong Convergence of Composite Iterative Methods for Nonexpansive Mappings

Jung¹

2009

Journal of the Korean Mathematical Society

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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 ) = ∅. Let {xn} be generated by a new composite iterative scheme: yn = λnf (xn)+(1−λn)T xn, x n+1 = (1−βn)yn +βnT yn, (n ≥ 0). It is proved that {xn} converges strongly to a point in F (T ), which is a solution of certain variational inequality provided the sequence {λn} ⊂ (0, 1) satisfies limn→∞ λn = 0 and 2 ∞ n=0 λn = ∞, {βn} ⊂ [0, a) for some 0 < a < 1 and the sequence {xn} is asymptotically regular.

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Section: Resultsmentioning

confidence: 89%

Strong Convergence of Composite Iterative Methods for Nonexpansive Mappings

Jung¹

2009

Journal of the Korean Mathematical Society

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“…Assume that F := N i=1 F (T i ) is nonempty. Let {u n } be a sequence generated by 18) for each n ≥ 1, where α n ∈ (0, 1) such that lim n→∞ α n = 0, ∞ n=1 α n = ∞ and {β n,i } ⊂ [a, b] ⊂ (0, 1) for i = 1, 2, · · · , N satisfying β n,0 + β n,1 + β n,2 + · · · + β n,N = 1 for each n ≥ 1. Then the sequence {u n } converges strongly to a common minimum-norm point of F (T i ).…”

Section: Resultsmentioning

confidence: 99%

Approximation of a common minimum-norm fixed point of a finite family of σ -asymptotically quasi-nonexpansive mappings with applications

Pathak¹,

Sahu²,

Cho³

2016

J. Nonlinear Sci. Appl.

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In this paper, we use the iterative method proposed by Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages] which converges strongly to the common minimum-norm fixed point of a finite family of σ-asymptotically quasi-nonexpansive mappings. As consequence, convergence results to a common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings is proved. Our result generalize and improve a recent result of Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages]. In the sequel, we apply our main result to find solution of minimizer of a continuously Frechet-differentiable convex functional which has the minimum norm in Hilbert spaces.

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“…Remark 3.1. (i) We note that the authors in [2,6,7,9,10] have imposed some additional assumptions on parameters {λ n+1 } or mappings {T n } as follows:…”

Section: From Lemma 22 and Lemma 23 We Obtain Thatmentioning

confidence: 99%

“…See, for example [2,6,7,9,10] and the references therein. But we note that the authors have imposed some additional assumptions on parameters or mappings {T n }.…”

Section: Introductionmentioning

confidence: 99%