Abstract
In this paper, we consider an iterative algorithm for finding the common element of the set of solutions for the generalized mixed equilibrium problems, the common fixed points set of two generalized nonexpansive type mappings, and the set of solutions of the variational inequality for an inverse-strongly skew-monotone operator in Banach spaces. Under mild conditions, the weak convergence theorem is established by using the sunny generalized nonexpansive retraction in Banach spaces. Our results refine, supplement, and extend the corresponding results in (Saewan et al. in Optim. Lett. 8:501-518, 2014), and other results announced by many other authors.
Highlights
Let E be a real Banach space with dual space E∗, whose inner product and norm are denoted by ·, · and ·, respectively
Many researchers have studied this algorithm in a Hilbert space and in a Banach space, for instance, [, ]
Motivated by [, ], and [ ], in this paper, using the projection algorithm method with the sunny generalized nonexpansive retraction RC, we introduce an iterative scheme to find a common element of the set of solutions for the generalized mixed equilibrium problem, the common fixed points for two generalized nonexpansive type mappings and the set of solutions of the variational inequality in Banach spaces
Summary
Let E be a real Banach space with dual space E∗, whose inner product and norm are denoted by ·, · and · , respectively. In [ ], Saewan et al introduced a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of fixed points for a closed φ-nonexpansive mapping by using the sunny generalized nonexpansive retraction in Banach spaces.
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