On the proximal point method for an infinite family of equilibrium problems in Banach spaces
Bull. Korean Math. Soc.
Published online 2019 Mar 18
Hadi Khatibzadeh and Vahid Mohebbi
University of Zanjan, IMPA
Abstract : In this paper, we study convergence analysis of the generated sequences for an infinite family of pseudo-monotone equilibrium problems in Banach spaces. We first prove weak convergence of the sequence generated by the proximal point method to a common solution of the infinite family of equilibrium problems with summable errors. Then we show strong convergence of the generated sequence to a common equilibrium point by some various additional assumptions. We also consider two variants for which we stablish strong convergence without any additional assumption. For both of them, each iteration consists of a proximal step followed by a computationally inexpensive step which ensures strong convergence of the generated sequence. Also, for this two variants we are able to characterize the strong limit of the sequence: for the first variant it is the solution lying closest to an arbitrarily selected point, and for the second one it is the solution of the problem which lies closest to the initial iterate.
Finally in order to illustrate an application of our main results, we give a concrete example where the main results can be applied.
Keywords : Equilibrium problem, Halpern regularization, hybrid projection method, proximal point method, pseudo-monotone bifunction.
MSC numbers : 90C25, 90C30.
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