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March  2020, 10(1): 75-92. doi: 10.3934/naco.2019034

## Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces

 1 Department of Economics, Faculty of Economics and Social Sciences, Ibn Zohr University, B.P. 8658 Poste Dakhla, Agadir, Morocco 2 National Institute of Science Education and Research Bhubaneswar, Pin-752050, India 3 Department of Mathematics, University of Central Florida, USA

* Corresponding author

Received  September 2018 Revised  March 2019 Published  May 2019

We study a new class of mixed equilibrium problem, in short MEP, under weakly relaxed $\alpha$-monotonicity in Banach spaces. This class of problems extends and generalizes some related fundamental results such as mixed variational-like inequalities, variational inequalities, and classical equilibrium problems as special cases. Existence and uniqueness of the solution to the problem is established. Auxiliary principle technique is used to obtain an iterative algorithm. Solvability of the auxiliary problem is established in the paper and finally the convergence of the iterates to the exact solution is proved. As applications of the approach developed in this paper, we study the existence and algorithmic approach for a general class of nonlinear mixed variational-like inequalities. The results obtained in this paper are interesting and improve considerably many existing results in literature.

Citation: Ouayl Chadli, Gayatri Pany, Ram N. Mohapatra. Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 75-92. doi: 10.3934/naco.2019034
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