Iterative Methods for Fixed Point Problems in Hilbert Spaces

Cegielski, Andrzej

doi:10.1007/978-3-642-30901-4

Cited by 222 publications

(333 citation statements)

References 105 publications

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“…We recall ( [1,5,6,29]) that (1) a mapping f : H → H is k-contractive if fx − fy k x − y for a constant k ∈ [0, 1) and ∀x, y ∈ H; (2) a mapping V : H → H is l-Lipschitzian if Vx − Vy l x − y for a constant l ∈ [0, ∞) and ∀x, y ∈ H; (3) a mapping T : H → H is nonexpansive if T x − T y x − y , ∀x, y ∈ H; (4) a mapping T : H → H is strongly nonexpansive if T is nonexpansive and…”

Section: Preliminaries and Lemmasmentioning

confidence: 99%

Hybrid iterative algorithms for the split common fixed point problems

Jung¹

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce two iterative algorithms (one implicit algorithm and one explicit algorithm) based on the hybrid steepest descent method for solving the split common fixed point problems. We establish the strong convergence of the sequences generated by the proposed algorithms to a solution of the split common fixed point problems, which is also a solution of a certain variational inequality. In particular, the minimum norm solution of the split common fixed point problems is obtained. As applications, variational problems and equilibrium problems are considered. c 2017 All rights reserved.Keywords: Split common fixed point problem, firmly nonexpansive mapping, nonexpansive mapping, variational inequality, minimum-norm, ρ-Lipschitzian and η-strongly monotone operator, bounded linear operator, variational problems, equilibrium problems, iterative algorithms. 2010 MSC: 47J05, 49J40, 49J52, 47J20, 47H10.

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Section: Preliminaries and Lemmasmentioning

confidence: 99%

Hybrid iterative algorithms for the split common fixed point problems

Jung¹

2017

J. Nonlinear Sci. Appl.

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“…But in other cases the subgradient projections are easier to compute than orthogonal projections since they do not call for the, computationally demanding, inner-loop of least Euclidean distance minimization, but rather employ the "subgradient projection" which is merely a step in the negative direction of a calculable subgradient of g j at the current iteration; see, e.g., [22,34,37,74]. For a general review on projection algorithms for the CFP see [8] and consult the recent work [29].…”

Section: Subgradient Projection Methodsmentioning

confidence: 99%

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

Censor

Reem

2014

Math. Program.

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The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and nonconvex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

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“…Math.Soc., 44 (1974), 147-150] upgraded Mann iterative scheme. Mann and Ishikawa iterative processes for nonexpansive and quasi-nonexpansive mappings have been extensively studied in uniformly convex Banach spaces and Hilbert spaces [1,4,5,8,10,11,16]. Das and Debata [6] studied strong convergence of Ishikawa iterates {x n } defined by…”

Section: Introductionmentioning

confidence: 99%