Nonlinear common fixed point properties of semitopological semigroups in uniformly convex spaces

2016 ◽  
Vol 19 (2) ◽  
pp. 1041-1057 ◽  
Author(s):  
Khadime Salame
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Uniformly Convex ◽  
Fixed Point Properties ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

scholarly journals Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Florian Lauster ◽  
D. Russell Luke
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Uniformly Convex ◽  
Metric Subregularity ◽  
Splitting Algorithms ◽  
Projected Gradients ◽  
Fixed Point Iterations ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

AbstractIn the setting of $\operatorname{CAT}(\kappa)$ CAT ( κ ) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.

scholarly journals $$\alpha $$-Firmly nonexpansive operators on metric spaces

2022 ◽  
Vol 24 (1) ◽  
Author(s):  
Arian Bërdëllima ◽  
Florian Lauster ◽  
D. Russell Luke
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Convergence Analysis ◽  
Metric Spaces ◽  
Uniformly Convex ◽  
Point Mapping ◽  
Splitting Algorithms ◽  
Uniformly Convex Spaces ◽  
Cluster Points ◽  
Convex Spaces

AbstractWe extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.

scholarly journals Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

2021 ◽  
Vol 10 (1) ◽  
pp. 1061-1070
Author(s):  
Rahul Shukla ◽  
Andrzej Wiśnicki
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Lipschitz Mapping ◽  
Uniformly Convex ◽  
Halpern Iteration ◽  
Lipschitz Mappings ◽  
Nonlinear Ergodic Theorem ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

Abstract We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

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