On the proximal point method for an infinite family of equilibrium problems in Banach spaces
Bull. Korean Math. Soc. 2019 Vol. 56, No. 3, 757-777
https://doi.org/10.4134/BKMS.b180547
Published online May 31, 2019
Hadi Khatibzadeh, Vahid Mohebbi
University of Zanjan; Instituto de Matem\'atica Pura e Aplicada
Abstract : In this paper, we study the convergence analysis of the sequences generated by the proximal point method for an infinite family of pseudo-monotone equilibrium problems in Banach spaces. We first prove the weak convergence of the generated sequence to a common solution of the infinite family of equilibrium problems with summable errors. Then, we show the strong convergence of the generated sequence to a common equilibrium point by some various additional assumptions. We also consider two variants for which we establish the 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 the 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, 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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