# American Institute of Mathematical Sciences

• Previous Article
$\theta$ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations
• NACO Home
• This Issue
• Next Article
Numerical solution of an obstacle problem with interval coefficients
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, 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.

show all references

##### 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.
 [1] Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024 [2] Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009 [3] Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129 [4] Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006 [5] Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 [6] Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252 [7] Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082 [8] Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159 [9] Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001 [10] Peter Benner, Ryan Lowe, Matthias Voigt. $\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 119-133. doi: 10.3934/naco.2018007 [11] Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059 [12] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [13] Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094 [14] Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086 [15] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 [16] Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003 [17] Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062 [18] Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007 [19] Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011 [20] Zalman Balanov, Yakov Krasnov. On good deformations of $A_m$-singularities. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1851-1866. doi: 10.3934/dcdss.2019122

Impact Factor: