In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex and reflexive Banach space E2 . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.
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In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.
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In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.
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