Abstract
AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.
Highlights
Let H be a real Hilbert space and C be a nonempty, closed, and convex subset of H
The theory of equilibrium problems was first introduced by Fan [1] in 1972, the most significant contributions to this problem were made by Blum and Oettli [2] and Noor and Oettli [3] in 1994
It can be reformulated in the form of different mathematical problems such as an optimization problem, a convex feasibility problem, a variational inequality problem, a minimization problem, a minimax inequality problem, a fixed point problem, a complementarity problem, a saddle point problem, or a Nash equilibrium problem in noncooperative games
Summary
Let H be a real Hilbert space and C be a nonempty, closed, and convex subset of H. Ugwunnadi and Ali [20] established the following algorithm to solve the system of split equilibrium problems and showed that the sequence generated by their algorithm converges strongly to the common solution of considered problem and fixed point problem for a finite family of continuous pseudocontractive mappings. Onjai-uea and Phuengrattana [21] proposed another iterative algorithm to find a solution for the split mixed equilibrium problem for λ-hybrid multivalued mappings. They proved that the sequence generated by the following iterative algorithm converges weakly to a common solution of fixed point problem and split mixed equilibrium problem. We give some corollaries and numeric results to show that our results generalize and extend many results in the literature
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