The subgradient extragradient method for approximation of fixed-point problem and modification of equilibrium problem

ABSTRACT

In this paper, we consider the modification of equilibrium problem (MEP) and new subgradient extragradient algorithm by using the concept of the set of solutions of the modified variational inequality problem introduced by [Kangtunyakarn A. A new iterative scheme for fixed-point problems of infinite family of ${\kappa }_i$ pseudo contractive mappings, equilibrium problem, variational inequality problems. J Optim Theory Appl. 2013;56:1543–1562.]. Then, we establish and prove weak and strong convergence theorem of the new subgradient extragradient algorithm for finding a common element of the set of solutions of the MEP and two sets of the variational inequality problems under some suitable conditions on ${\alpha }_n$ and ${\text{β}}_n$ with ${\alpha }_n+{\text{β}}_n\le 1$. Moreover, we apply our main theorem to prove weak and strong convergence theorems to solve the generalized equilibrium problem, the system of equilibrium problem, the variational inequality problem and the general system of variational inequality problems. Finally, we give two numerical examples to support our main result.

KEYWORDS:

• Equilibrium problem
• subgradient extragradient algorithm
• variational inequality problem
• fixed-point problem

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by King Mongkut's Institute of Technology Ladkrabang.