Abstract
In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior knowledge of the Lipschitz constants. We also prove a strong convergence result for the sequence that is generated by our algorithm under mild conditions. We apply our result to solving variational inequality problems, and finally, we give some numerical examples to illustrate the efficiency and accuracy of the algorithm.
Highlights
In 1994, Blum and Oettli [1] revisited the Equilibrium Problem (EP) that was first introduced by Ky Fan which has become a fundamental concept and serves as an important mathematical tool for solving many concrete problems
Being motivated by the above results, we introduce a Halpern-subgradient extragradient method for solving pseudomonotone EP and finding common fixed point of countable family of quasi-Bregman nonexpansive mappings in real reflexive Banach spaces
In Algorithm 1, the stepsize is selected self-adaptively and does not involve any inner loop. (iii) Our algorithm extends the subgradient extragradient method of [32,33,34,35,36,37] to reflexive Banach spaces using Bregman distance
Summary
In 1994, Blum and Oettli [1] revisited the Equilibrium Problem (EP) that was first introduced by Ky Fan which has become a fundamental concept and serves as an important mathematical tool for solving many concrete problems. In 2013, Anh Pham Ngoc [24] presented a hybrid extragradient iteration method, where the extragradient method was extended to fixed point and equilibrium problem This was done for a pseudomonotone and Lipschitz-type continuous bifunction in a setting of real. Being motivated by the above results, we introduce a Halpern-subgradient extragradient method for solving pseudomonotone EP and finding common fixed point of countable family of quasi-Bregman nonexpansive mappings in real reflexive Banach spaces. We provide an application of our result to variational inequality problems and give some numerical experiments to show the numerical behaviour of our algorithm This improves the work of Eskandami et al [39] and extends the results of [32,33,34,35,36,37] to a reflexive Banach space while using Bregman distance techniques. Throughout this paper, E denotes a real Banach space with dual E∗; x∗, x denotes the duality pairing between x ∈ E and x∗ ∈ E∗; ∀ denotes the for all; min{A} is the minimum of a set A; max{B} is the maximum of a set B; xn → u implies the strong converges of a sequence {xn} ⊂ X to a point u ∈ E; xn u is the weak convergence of xn to u; · denotes the norm on E, while · ∗ denotes the norm on E∗; EP denotes the equilibrium problem, EP( f ) denotes the solution set of the equilibrium problem; F(T) is the set of fixed point of a mapping T, ∇ f is the gradient of a function f and R is the real number line
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