scholarly journals Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1557-1569
Author(s):  
Nguyen Buong ◽  
Nguyen Anh ◽  
Khuat Binh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Uniformly Smooth ◽  
Differentiable Norm ◽  
Strictly Convex Banach Spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

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 Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Haitao Che ◽  
Xintian Pan
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Common Element ◽  
Hybrid Steepest Descent Method ◽  
Pseudocontractive Mappings ◽  
Strictly Pseudocontractive Mappings

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

scholarly journals Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Projection Algorithm ◽  
Fixed Point Set

We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.

scholarly journals Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nguyen Buong
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Descent Method ◽  
Finite Family ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Point Problem ◽  
Lower Semi ◽  
The Family

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

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