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EXTRAGRADIENT METHODS FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES

Published online by Cambridge University Press:  01 October 2020

BEHZAD DJAFARI ROUHANI
Affiliation:
Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Avenue, El Paso, Texas TX 79968, USA e-mail: behzad@utep.edu
VAHID MOHEBBI
Affiliation:
Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Avenue, El Paso, Texas TX 79968, USA e-mail: vmohebbi@utep.edu

Abstract

We study the extragradient method for solving quasi-equilibrium problems in Banach spaces, which generalizes the extragradient 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 assuming neither any monotonicity assumption on the bifunction nor any weak continuity assumption of f in its arguments that in the many well-known methods have been used. Also, we give a necessary and sufficient condition for the solution set of the quasi-equilibrium problem to be nonempty and we show that, in this case, this iterative sequence converges strongly to a solution of the quasi-equilibrium problem. In other words, we prove strong convergence of the generated sequence to a solution of the quasi-equilibrium problem without assuming existence of a solution of the problem. Finally, we give an application of our main result to a generalized Nash equilibrium problem.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Vaithilingam Jeyakumar

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EXTRAGRADIENT METHODS FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES
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