An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-ϕ-nonexpansive mappings in Banach spaces

In this paper, we study the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasi-ϕ-nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex, and reflexive Banach space E2. An iterative algorithm with Armijo linesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergence result is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally, numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm in comparison with other existing algorithms in the literature. Our results extend and improve many recent results in this direction.