A theorem for solving Banach generalized system of variational inequality problems and fixed point problem in uniformly convex and 2-uniformly smooth Banach space

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
Bunyawee Chaloemyotphong ◽  
Atid Kangtunyakarn
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Fixed Point Problem ◽  
Variational Inequality Problems ◽  
Point Problem ◽  
Uniformly Convex ◽  
Uniformly Smooth Banach Space ◽  
Generalized System ◽  
Uniformly Smooth

A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

2018 ◽  
Vol 62 (1) ◽  
pp. 241-257 ◽  
Author(s):  
C. E. Chidume ◽  
M. O. Uba ◽  
M. I. Uzochukwu ◽  
E. E. Otubo ◽  
K. O. Idu
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Maximal Monotone ◽  
Variational Inequality Problems ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Monotone Maps

AbstractLetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

Some comments on the paper of Khuangsatung and Kangtunyakarn

2018 ◽  
Vol 13 (03) ◽  
pp. 2050051
Author(s):  
Kanokwan Wongchan
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Convergence Theorem ◽  
Fixed Point Problem ◽  
Variational Inequality Problems ◽  
Point Problem ◽  
Nonlinear Mappings

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

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