On solving the split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings

Abstract

We consider the split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings in real Hilbert spaces. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a common solution of the considered problems. A numerical example is presented to illustrate the convergence result. Our results improve and extend the corresponding results in the literature.

On Split Generalized Equilibrium and Fixed Point Problems of Bregman W-Mappings With Multiple Output Sets in Reflexive Banach Spaces

“…To solve GEP (1.1), we need the following assumptions: Let G : D × D → R. Assumption 1.3: In 2018, Phuengrattana and Lerkchayaphum [32] introduced a shrinking projection method for solving the common solution of split generalized equilibrium problem and fixed point problem of multivalued nonexpansive mappings in real Hilbert spaces. They proved that the sequence {x n } converges strongly to proj g ∆ x 0 , where ∆ := Sol(GEP (1.1) ∩ F (T ) is nonempty.…”

“…They proved that the sequence {x n } converges strongly to proj g ∆ x 0 , where ∆ := Sol(GEP (1.1) ∩ F (T ) is nonempty. Our proposed method is endowed with the following characteristics: (1) We extend the results of [1,2,32] from real Hilbert spaces to a more general space which is convex, continuous and strongly coercive Bregman function, which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.…”

On the existence and approximation of solutions of generalized equilibrium problem on Hadamard manifolds

“…In particular, the GEP is applicable in sensor networks, data compression, robustness to marginal changes and equilibrium stability, etc. (see [2,3,15,18,30,37]).…”

“…Also, for methods of approximating a solution of the EP in both the linear and nonlinear spaces, see [1,20,28,40]. We refer the readers to see the following [18,30] for methods of solving the GEP in the Hilbert and Banach spaces.…”

An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings

“…Many problems arising in different areas of mathematics such as optimization, variational analysis, differential equations, mathematical economics, and game theory can be modeled as fixed point equations of the form x ∈ T x, where T is a multivalued nonexpansive mapping. There are many effective algorithms for solving the fixed point problem [5,7,20,22]. One of the most efficient methods for approximating fixed points of single-valued nonexpansive mappings dates back to 1953 and is the Mann's method.…”