Abstract
In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.
Highlights
The theory of nonlinear functional analysis, in particular fixed point theory, is a rich and thriving mathematical discipline
In 2010, Cianciaruso et al [13] extended the results presented in [34] to find a common solution of equilibrium problem and fixed point problem associated with the nonexpansive semigroup; see [24]
Inspired and motivated by the ongoing research in this direction, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of nonexpansive semigroup in real Hilbert spaces
Summary
The theory of nonlinear functional analysis, in particular fixed point theory, is a rich and thriving mathematical discipline. Moudafi [32] generalized the concept of SVIP to that of split monotone variational inclusions (SMVIP) which includes, as a special case, SVIP, split common fixed point problem, split equilibrium problem and split feasibility problem These problems have already been studied and successfully employed as a model in intensity-modulated radiation therapy treatment planning; see [9, 10]. In 2015, Thuy [41] proposed a hybrid extragradient method for equilibrium, variational inequality and fixed point problem of a nonexpansive semigroup in Hilbert spaces. In 2017, Dinh et al [18] proposed two new extragradientproximal point algorithms for solving split equilibrium problem and fixed point problem of nonexpansive mappings in real Hilbert spaces. Inspired and motivated by the ongoing research in this direction, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of nonexpansive semigroup in real Hilbert spaces.
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