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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 and Optimization, 2020, 10 (1) : 75-92. doi: 10.3934/naco.2019034
##### References:
 [1] A. S. Antipin, The fixed points of extremal maps: Computation by gradient methods, Zh. Vychisl. Mat. Mat. Fiz., 37 (1997), 42-53. [2] M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244. [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Mathematics Student-India, 63 (1994), 123-145. [4] O. Chadli, H. Mahdioui and J. C. Yao, Bilevel mixed equilibrium problems in Banach spaces: Existence and algorithmic aspects, Numerical Algebra, Control and Optimization, 1 (2011), 549-561.  doi: 10.1155/2012/843486. [5] Y. Q. Chen, On the semimonotone operator theory and applications, J. Math. Anal. Appl., 231 (1999), 177-192.  doi: 10.1006/jmaa.1998.6245. [6] X. P. Ding and K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math, 63 (1992), 233-247.  doi: 10.4064/cm-63-2-233-247. [7] X. P. Ding, Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces, Appl. Math. Mech. -Engl. Ed., 32 (2011), 231-240.  doi: 10.1007/s10483-011-1409-9. [8] Ky Fan, A minimax inequality and applications, in Inequalities III (eds. O. Shisha), Academic Press, (1972), 103–113. [9] Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545. [10] Y. P. Fang and N. J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118 (2003), 327-337.  doi: 10.1023/A:1025499305742. [11] S. D. Flåm and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1996), 29-41. [12] J. L. Joly and U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Anal., 34 (1979), 107-137.  doi: 10.1016/0022-1236(79)90028-4. [13] S. M. Kang, S. Y. Cho and Z. Liu, Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings, J. Inequal. Appl., 2010 (2010), Article ID 827082. doi: 10.1155/2010/827082. [14] N. K. Mahato and C. Nahak, Equilibrium problems with generalized relaxed monotonicities in Banach spaces, Opsearch, 51 (2014), 257-269.  doi: 10.1007/s12597-013-0142-5. [15] H. Mahdioui and O. Chadli, On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: Existence and algorithmic aspects, Advances in Operations Research, 2012 (2012), Article ID 843486. doi: 10.1155/2012/843486. [16] G. Mastroeni, On auxiliary principle for equilibrium problems, in Equilibrium Problems and Variational Models (eds. P. Daniele, F. Giannessi and A. Maugeri), Springer, (2003), 289–298. doi: 10.1007/978-1-4613-0239-1_15. [17] A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, in Ill-Posed Variational Problems and Regularization Techniques (eds. M. Théra and R. Tichatschke), Springer, (1999), 187–201. doi: 10.1007/978-3-642-45780-7_12. [18] U. Mosco, Implicit variational problems and quasi-variational inequalities, in Nonlinear operators and the calculus of variations, Proceedings of Summer School (Bruxelles 1975) (eds. J.P. Gossez, E.J. Lami Dozo, J. Mawhin, et al.), Lecture notes in mathematics, Springer-Verlag, 543 (1976), 83–156. [19] H. Nikaido and K. Isoda, Note on noncooperative convex games, Pacific J. Math., 5 (1955), 807-815. [20] M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl., 122 (2004), 371-386.  doi: 10.1023/B:JOTA.0000042526.24671.b2. [21] M. A. Noor, K. Inayat Noor and V. Gupta, On equilibrium-like problems, Appl. Anal., 86 (2007), 807-818.  doi: 10.1080/00036810701450454. [22] M. A. Noor and K. I. Noor, General equilibrium bifunction variational inequalities, Comput. Math. Appl., 64 (2012), 3522-3526.  doi: 10.1016/j.camwa.2012.09.001. [23] G. Pany and S. Pani, Nonlinear mixed variational-like inequality with respect to weakly relaxed η- α monotone mapping in Banach spaces, in Mathematical Analysis and its Applications: Roorkee, India, December 2014 (eds. P. N. Agrawal, R. N. Mohapatra, U. Singh and H. M. Srivastava), Springer, (2015), 185–196. doi: 10.1007/978-81-322-2485-3_14. [24] V. Preda, M. Beldiman and A. Bătătorescu, On Variational-like Inequalities with generalized monotone mappings, in Generalized Convexity and Related Topics (eds. I. Konnov, D.T. Luc and A. Rubinov), Lecture Notes in Economics and Mathematical Systems, Springer, 583 (2006), 415–431. doi: 10.1007/978-3-540-37007-9_25. [25] H. A. Rizvi, A. Kılıçman and R. Ahmad, Generalized equilibrium problem with mixed relaxed monotonicity, The Scientific World Journal, 2014 (2014). [26] R. Tremolieres, J. L. Lions and R. Glowinski, Numerical Analysis of Variational Inequalities, Elsevier, 2011. [27] R. Wangkeeree and U. Kamraksa, An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems, Nonlinear Analysis: Hybrid Systems, 3 (2009), 615-630.  doi: 10.1016/j.nahs.2009.05.005.

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##### References:
 [1] A. S. Antipin, The fixed points of extremal maps: Computation by gradient methods, Zh. Vychisl. Mat. Mat. Fiz., 37 (1997), 42-53. [2] M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244. [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Mathematics Student-India, 63 (1994), 123-145. [4] O. Chadli, H. Mahdioui and J. C. Yao, Bilevel mixed equilibrium problems in Banach spaces: Existence and algorithmic aspects, Numerical Algebra, Control and Optimization, 1 (2011), 549-561.  doi: 10.1155/2012/843486. [5] Y. Q. Chen, On the semimonotone operator theory and applications, J. Math. Anal. Appl., 231 (1999), 177-192.  doi: 10.1006/jmaa.1998.6245. [6] X. P. Ding and K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math, 63 (1992), 233-247.  doi: 10.4064/cm-63-2-233-247. [7] X. P. Ding, Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces, Appl. Math. Mech. -Engl. Ed., 32 (2011), 231-240.  doi: 10.1007/s10483-011-1409-9. [8] Ky Fan, A minimax inequality and applications, in Inequalities III (eds. O. Shisha), Academic Press, (1972), 103–113. [9] Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545. [10] Y. P. Fang and N. J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118 (2003), 327-337.  doi: 10.1023/A:1025499305742. [11] S. D. Flåm and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1996), 29-41. [12] J. L. Joly and U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Anal., 34 (1979), 107-137.  doi: 10.1016/0022-1236(79)90028-4. [13] S. M. Kang, S. Y. Cho and Z. Liu, Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings, J. Inequal. Appl., 2010 (2010), Article ID 827082. doi: 10.1155/2010/827082. [14] N. K. Mahato and C. Nahak, Equilibrium problems with generalized relaxed monotonicities in Banach spaces, Opsearch, 51 (2014), 257-269.  doi: 10.1007/s12597-013-0142-5. [15] H. Mahdioui and O. Chadli, On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: Existence and algorithmic aspects, Advances in Operations Research, 2012 (2012), Article ID 843486. doi: 10.1155/2012/843486. [16] G. Mastroeni, On auxiliary principle for equilibrium problems, in Equilibrium Problems and Variational Models (eds. P. Daniele, F. Giannessi and A. Maugeri), Springer, (2003), 289–298. doi: 10.1007/978-1-4613-0239-1_15. [17] A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, in Ill-Posed Variational Problems and Regularization Techniques (eds. M. Théra and R. Tichatschke), Springer, (1999), 187–201. doi: 10.1007/978-3-642-45780-7_12. [18] U. Mosco, Implicit variational problems and quasi-variational inequalities, in Nonlinear operators and the calculus of variations, Proceedings of Summer School (Bruxelles 1975) (eds. J.P. Gossez, E.J. Lami Dozo, J. Mawhin, et al.), Lecture notes in mathematics, Springer-Verlag, 543 (1976), 83–156. [19] H. Nikaido and K. Isoda, Note on noncooperative convex games, Pacific J. Math., 5 (1955), 807-815. [20] M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl., 122 (2004), 371-386.  doi: 10.1023/B:JOTA.0000042526.24671.b2. [21] M. A. Noor, K. Inayat Noor and V. Gupta, On equilibrium-like problems, Appl. Anal., 86 (2007), 807-818.  doi: 10.1080/00036810701450454. [22] M. A. Noor and K. I. Noor, General equilibrium bifunction variational inequalities, Comput. Math. Appl., 64 (2012), 3522-3526.  doi: 10.1016/j.camwa.2012.09.001. [23] G. Pany and S. Pani, Nonlinear mixed variational-like inequality with respect to weakly relaxed η- α monotone mapping in Banach spaces, in Mathematical Analysis and its Applications: Roorkee, India, December 2014 (eds. P. N. Agrawal, R. N. Mohapatra, U. Singh and H. M. Srivastava), Springer, (2015), 185–196. doi: 10.1007/978-81-322-2485-3_14. [24] V. Preda, M. Beldiman and A. Bătătorescu, On Variational-like Inequalities with generalized monotone mappings, in Generalized Convexity and Related Topics (eds. I. Konnov, D.T. Luc and A. Rubinov), Lecture Notes in Economics and Mathematical Systems, Springer, 583 (2006), 415–431. doi: 10.1007/978-3-540-37007-9_25. [25] H. A. Rizvi, A. Kılıçman and R. Ahmad, Generalized equilibrium problem with mixed relaxed monotonicity, The Scientific World Journal, 2014 (2014). [26] R. Tremolieres, J. L. Lions and R. Glowinski, Numerical Analysis of Variational Inequalities, Elsevier, 2011. [27] R. Wangkeeree and U. Kamraksa, An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems, Nonlinear Analysis: Hybrid Systems, 3 (2009), 615-630.  doi: 10.1016/j.nahs.2009.05.005.
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