Convergence and Stability of a Perturbed Mann Iterative Algorithm with Errors for a System of Generalized Variational-Like Inclusion Problems in q-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.

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Inertial Krasnoselskii–Mann Method in Banach Spaces

Mathematics ◽

10.3390/math8040638 ◽

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.

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An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces

Optimization Letters ◽

10.1007/s11590-011-0415-y ◽

2011 ◽

Vol 7 (2) ◽

pp. 267-287 ◽

Cited By ~ 9

Author(s):

Gang Cai ◽

Shangquan Bu

Keyword(s):

Fixed Point ◽

Variational Inequalities ◽

Banach Spaces ◽

Iterative Algorithm ◽

General System ◽

Fixed Point Problems ◽

System Of Variational Inequalities ◽

Uniformly Smooth ◽

Uniformly Smooth Banach Spaces

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Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach spaces

Filomat ◽

10.2298/fil1205897a ◽

2012 ◽

Vol 26 (5) ◽

pp. 897-908

Author(s):

Rais Ahmad ◽

Mohammad Dilshad ◽

Mohammad Akram

Keyword(s):

Banach Spaces ◽

Iterative Algorithm ◽

Resolvent Operator ◽

Existence Of Solutions ◽

Operator Technique ◽

Uniformly Smooth ◽

Resolvent Operator Technique ◽

Uniformly Smooth Banach Spaces ◽

Matlab Programming ◽

Cocoercive Operator

In this paper, we apply H(?,?)-?-cocoercive operator introduced in [2] for solving a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. By using the approach of resolvent operator associated with H(?,?)-?-cocoercive operator, an iterative algorithm for solving a system of generalized variational-like inclusions is constructed. We prove the existence of solutions of system of generalized variational-like inclusions and convergence of iterative sequences generated by the algorithm. An example through Matlab programming is constructed.

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