Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces

2020 ◽  
pp. 1-15
Author(s):  
Rais Ahmad ◽  
Imran Ali ◽  
Mijanur Rahaman ◽  
Mohd. Ishtyak ◽  
J. C. Yao
Keyword(s):  
Banach Spaces ◽  
Inclusion Problem ◽  
Resolvent Equation ◽  
Generalized Resolvent ◽  
Equation Problem ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

scholarly journals Algorithms for General System of Generalized Resolvent Equations with Corresponding System of Variational Inclusions

2012 ◽  
Vol 2012 ◽  
pp. 1-24
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Banach Spaces ◽  
General System ◽  
Approximate Solutions ◽  
Variational Inclusions ◽  
Convergence Criteria ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces ◽  
Generalized Resolvent Equations

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.

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 EQUILATERAL SETS IN UNIFORMLY SMOOTH BANACH SPACES

Mathematika ◽  
2014 ◽  
Vol 60 (1) ◽  
pp. 219-231 ◽  
Author(s):  
D. Freeman ◽  
E. Odell ◽  
B. Sari ◽  
Th. Schlumprecht
Keyword(s):  
Banach Spaces ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces ◽  
Equilateral Sets

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.

ON ASYMPTOTIC UNIFORM SMOOTHNESS AND NONLINEAR GEOMETRY OF BANACH SPACES

2019 ◽  
pp. 1-38
Author(s):  
B. M. Braga
Keyword(s):  
Banach Spaces ◽  
Open Problem ◽  
Concentration Inequalities ◽  
Uniformly Smooth ◽  
Uniform Smoothness ◽  
Weakly Sequentially Continuous ◽  
Nonlinear Geometry ◽  
Lipschitz Embeddings ◽  
Uniformly Smooth Banach Spaces ◽  
Descriptive Set

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach–Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

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