STRONG CONVERGENCE OF A HYBRID ITERATION FOR GENERALIZED MIXED EQUILIBRIUM PROBLEM AND BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

Tóm tắt

The purpose of this paper is to combine the Bregman distance with the shrinking projection method to introduce a new hybrid iteration process for a generalized mixed equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping. After that, under some suitable conditions, we prove that the proposed iteration strongly converges to the Bregman projection of the initial point onto common element set of the solution set of a generalized mixed equilibrium problem and the fixed point set of a Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. This theorem extends and improves the results in (Alizadeh & Moradlou, 2016) from a generalized hybrid mapping and an equilibrium problem in Hilbert spaces to a Bregman totally quasi-asymptotically nonexpansive mapping and a generalized mixed equilibrium problem in reflexive Banach spaces. The obtained result is applied to a generalized mixed equilibrium problem and a Bregman quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. In addition, an example is provided to illustrate for the proposed iteration process.