Proximal Point Method for Quasi-Equilibrium Problems in Banach Spaces. (3rd July 2020)

• Record Type: Journal Article
• Authors: Djafari Rouhani, Behzad; Mohebbi, Vahid
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 9(2020)
• Page Start: 1007
• Page End: 1026
• Publication Date: 2020-07-03
• Subjects: Bregman distance -- demiclosed -- proximal point method -- quasi-equilibrium problem -- quasi Dg-nonexpansive -- θ-undermonotone bifunction; 90C25 -- 90C30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1708392
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We study the proximal point method for solving quasi-equilibrium problems in Banach spaces, which generalizes the proximal point method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem without assuming neither pseudo-monotonicity nor any weak continuity assumption of the bifunction in its arguments that in many well-known methods have been used. Also, we show that the boundedness of the generated sequences implies that the solution set of the quasi-equilibrium problem is nonempty and prove the strong convergence of the generated sequence to a solution of the quasi-equilibrium problem when these conditions are satisfied.