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1
Content available remote Corrigendum to the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings”
Gabeleh Moosa
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
9--10
EN
The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings, ”Demonstr. Math. 53 (2020), 38-43.
2
Content available remote Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings
Gabeleh Moosa, Künzi Hans-Peter A.
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
38--43
EN
In 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.
3
Content available remote Noncyclic Meir-Keeler contractions and best proximity pair theorems
Gabeleh M., Markin J.
Demonstratio Mathematica
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2018
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Vol. 51, nr 1
171--181
EN
In this article, we consider the class of noncyclic Meir-Keeler contractions and study the existence and convergence of best proximity pairs for such mappings in the setting of complete CAT(0) spaces. We also discuss asymptotic pointwise noncyclic Meir-Keeler contractions in the framework of uniformly convex Banach spaces and generalize a main result of Chen [Chen C. M., A note on asymptotic pointwise weaker Meir-Keeler type contractions, Appl. Math. Lett., 2012, 25, 1267-1269]. Examples are given to support our main results.
4
Content available A strong convergence result for systems of nonlinear operator equations involving total asymptotically nonexpansive mappings in uniformly convex Banach spaces
Abdelhakim A. A.
Silesian Journal of Pure and Applied Mathematics
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2016
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Vol. 6, iss. 1
27--48
EN
We prove the strong convergence of an implicit iterative procedure to a solution of a system of nonlinear operator equations involving total asymptotically nonexpansive operators in uniformly convex Banach spaces.
5
Content available remote Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces
Kozlowski W. M.
Commentationes Mathematicae
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2014
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Vol 54, No. 2
203--208
EN
Let C be a convex compact subset of a uniformly convex Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
6
Content available remote On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces
Kozlowski W. M.
Commentationes Mathematicae
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2012
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Vol. 52, [Z] 2
113--136
EN
Let C be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space X. We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings Tt: C → C. Each of Tt: is assumed to be pointwise Lipschitzian, that is, there exists a family of functions αt: C → [0, ∞) such that ||Tt(x) — Tt(y)\\ ≤ αt:(x) || - y|| for x,y € C. The paper demonstrates how the weak compactness of C plays an essential role in proving the weak convergence of these processes to common fixed points.
7
Content available remote Convergence theorems of a family of uniformly (L, (alfa))-Lipschitz asymptotically quasi-nonexpansive type mappings
Saluja G. S.
Demonstratio Mathematica
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2010
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Vol. 43, nr 4
877-886
EN
In this paper, we study the convergence of the Ishikawa iterative sequence of rank-r to common fixed points of a finite family of asymptotically quasi-nonexpansive type mappings in uniformly convex Banach spaces. Our results extend and improve some known recent results given in the literature
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