scholarly journals Inertial Krasnoselskii–Mann Method in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 638
Author(s):  
Yekini Shehu ◽  
Aviv Gibali
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Splitting Method ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Mann Algorithm ◽  
Uniformly Smooth Banach Spaces

In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-backward splitting method for the inclusion. Our results extend many recent and related results obtained in real Hilbert spaces.

A CONVERGENCE THEOREM FOR COMMON ELEMENTS OF EQUILIBRIUM PROBLEMS AND MAPPINGS SATISFYING CONDITION (Φ-Eµ) IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH SPACES

2018 ◽  
Vol 1 (30) ◽  
pp. 67-77
Author(s):  
Hieu Trung Nguyen ◽  
Tien Cam Truong
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Satisfying Condition ◽  
Common Element ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Smooth ◽  
Point Set ◽  
Uniformly Smooth Banach Spaces

In this paper, we propose a new hybrid iteration for finding a common element of solution set of equilibrium problems and the fixed point set of mappings  satisfying condi- tion (Φ-Eµ), and establish the convergence of this iteration in uniformly convex and uniformly smooth Banach spaces. From this theorem, we geta corollary for the convergence for equilibrium problems and mappings satisfying condition (Eµ) in real Hilbert spaces. In addition, an example is provided to illustrate for the convergence of equilibrium problems and mappings satisfying condition (Φ-Eµ). These results are the generations and improvements of some  existing results in the literature

MODIFIED HALPERN ITERATIVE ALGORITHM FOR NONEXPANSIVE MAPPINGS II

2011 ◽  
Vol 04 (04) ◽  
pp. 683-694
Author(s):  
Mengistu Goa Sangago
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Convergence Conditions ◽  
Uniformly Smooth ◽  
Additional Control ◽  
Uniformly Smooth Banach Spaces

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Yao et al. [16] modification of Halpern iterative algorithm for nonexpansive mappings in uniformly smooth Banach spaces. Unfortunately, some deficiencies are found in the Yao et al. [16] control conditions imposed on the modified iteration to obtain strong convergence. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Yao et al. [16] are not sufficient and it is also proved that with some additional control conditions on the parameters strong convergence of the iteration is obtained.

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

scholarly journals Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Set ◽  
Variational Problems ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Nonconvex Sets ◽  
Generalized Projections ◽  
Uniformly Smooth Banach Spaces

The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

scholarly journals An iterative method for solving multiple-set split feasibility problems in Banach spaces

2020 ◽  
Vol 36 (1) ◽  
pp. 1-13
Author(s):  
SULIMAN AL-HOMIDAN ◽  
BASHIR ALI ◽  
YUSUF I. SULEIMAN
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Result ◽  
Uniformly Convex ◽  
Split Feasibility Problems ◽  
Uniformly Smooth ◽  
Convex Problem ◽  
Split Feasibility

"In this paper, we study generalized multiple-set split feasibility problems (in short, GMSSFP) in the frame workof p-uniformly convex real Banach spaces which are also uniformly smooth. We construct an iterative algo-rithm which is free from an operator norm and prove its strong convergence to a solution of GMSSFP, thatis, a solution of convex problem and a common fixed point of a countable family of Bregman asymptoticallyquasi-nonexpansive mappings without requirement for semi-compactness on the mappings. We illustrate ouralgorithm and convergence result by a numerical example. "

scholarly journals A New Hybrid Algorithm forλ-Strict Asymptotically Pseudocontractions in 2-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Xin-dong Liu ◽  
Shih-sen Chang
Keyword(s):  
Banach Spaces ◽  
Hybrid Algorithm ◽  
Metric Projection ◽  
Projection Algorithm ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Hybrid Projection Algorithm ◽  
Uniformly Smooth Banach Spaces ◽  
Asymptotically Pseudocontractive Mapping

A new hybrid projection algorithm is considered for aλ-strict asymptotically pseudocontractive mapping. Using the metric projection, a strong convergence theorem is obtained in a uniformly convex and 2-uniformly smooth Banach spaces. The result presented in this paper mainly improves and extends the corresponding results of Matsushita and Takahashi (2008), Dehghan (2011) Kang and Wang (2011), and many others.

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