scholarly journals Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

2020 ◽  
Vol 53 (1) ◽  
pp. 38-43 ◽  
Author(s):  
Moosa Gabeleh ◽  
Hans-Peter A. Künzi
Keyword(s):  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Studia Math ◽  
Normal Structure ◽  
Projection Algorithm ◽  
Best Proximity Points ◽  
Iteration Sequence ◽  
Cyclic Contractions ◽  
Convergence Method ◽  
Strictly Convex Banach Spaces

AbstractIn this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

scholarly journals Existence and Convergence Theorems of Best Proximity Points

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Moosa Gabeleh ◽  
Naseer Shahzad
Keyword(s):  
Banach Spaces ◽  
Convergence Theorem ◽  
Best Proximity Point ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Best Proximity Points ◽  
Relatively Nonexpansive Mappings ◽  
Uniformly Convex Banach Spaces ◽  
Relatively Nonexpansive

The aim of this paper is to prove some best proximity point theorems for new classes of cyclic mappings, called pointwise cyclic orbital contractions and asymptotic pointwise cyclic orbital contractions. We also prove a convergence theorem of best proximity point for relatively nonexpansive mappings in uniformly convex Banach spaces.

scholarly journals Convergence of Ishikawa iterates of generalized nonexpansive mappings

1997 ◽  
Vol 20 (3) ◽  
pp. 517-520 ◽  
Author(s):  
M. K. Ghosh ◽  
L. Debnath
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Strictly Convex ◽  
Strictly Convex Banach Spaces

This paper is concerned with the convergence of Ishikawa iterates of generalized nonexpansive mappings in both uniformly convex and strictly convex Banach spaces. Several fixed point theorems are discussed.

scholarly journals Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1557-1569
Author(s):  
Nguyen Buong ◽  
Nguyen Anh ◽  
Khuat Binh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Uniformly Smooth ◽  
Differentiable Norm ◽  
Strictly Convex Banach Spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

scholarly journals Strong and weak convergence of Ishikawa iterations for best proximity pairs

2020 ◽  
Vol 18 (1) ◽  
pp. 10-21
Author(s):  
Moosa Gabeleh ◽  
S. I. Ezhil Manna ◽  
A. Anthony Eldred ◽  
Olivier Olela Otafudu
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Relatively Nonexpansive Mapping ◽  
Strictly Convex ◽  
Strong And Weak Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Relatively Nonexpansive ◽  
Strictly Convex Banach Spaces ◽  
Best Proximity Pair

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

Best proximity points for p-cyclic summing iterated contractions

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3275-3287 ◽  
Author(s):  
Mihaela Petric ◽  
Boyan Zlatanov
Keyword(s):  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Uniformly Convex ◽  
Best Proximity Points ◽  
Uniformly Convex Banach Spaces ◽  
Cyclic Contractions ◽  
New Type

We generalize the p - summing contractions maps. We found sufficient conditions for these new type of maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces. We apply the result for Kannan and Chatterjea type cyclic contractions and we obtain sufficient conditions for these maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces.

scholarly journals On Best Proximity Points in Metric and Banach Spaces

2011 ◽  
Vol 63 (3) ◽  
pp. 533-550 ◽  
Author(s):  
Rafa Espínola ◽  
Aurora Fernández-León
Keyword(s):  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Best Proximity Points ◽  
Geodesic Metric ◽  
Cyclic Contractions

Abstract In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.

scholarly journals A Shrinking Projection Algorithm with Errors for Costerro Bounded Linear Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Joseph Frank Gordon
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Projection Algorithm ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Finite Set ◽  
Shrinking Projection Algorithm

The purpose of this paper is to introduce and analyze the shrinking projection algorithm with errors for a finite set of costerro bounded linear mappings in the setting of uniformly convex smooth Banach spaces. Here, under finite dimensional or compactness restriction or the error term being zero, the strong limit point of the sequence stated in the iterative scheme for these mappings in uniformly convex smooth Banach spaces was studied. This paper extends Ezearn and Prempeh’s result for nonexpansive mappings in real Hilbert spaces.

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