Ayed E. Hashoosh, Mohsen Ali mohammady & Amneh Almusawi

Department of mathematics, University of Thi- Qar, Iraq.

Department of Mathematics, University of Mazandaran, Babolsar, Iran.

DOI: https://doi.org/10.17656/jzs.10598

The main goal of this paper is to establish some well-posedness results for a nonstandard equilibrium problem (for short ) and for optimization problem involving α-monotone bifunction, whose solution is sought in a subset K of a real reflexive Banach space X. Moreover, we establish some metric characterizations of well-posedness for a nonstandard equilibrium problems and for an optimization problem.

Equilibrium problem; Well-posed optimizat -ion problems; bifunction monotone; Metric char- acterizeations.

[1] L. Q. Anh, P. Q. Khanh, and D. T. M. Van, "Well-posedness under relaxed semicontinuouty for bilevel equilibrium and optimization problems with equilibrium constraint", Journal of Optimization Theory and Application, Vol. 153, pp. 42-59. (2012).

[2] E. Bednarczuk and J. P. Penot, "Metrically well-set minimization problems", Appl. Math. Optim. Vol. 26, pp. 273-285. (1992).

[3] M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems, Math. Methods Oper. Res. Vol. 70, pp. 171-182. (2009).

[4] M. Bianchi and R. Pini, "Sensitivity for parametric vector equilbria", Optimization Vol. 55, pp. 221-130. (2006).

[5] J.W. Chen, Y. J. Cho and X.Q. Qu, "Levitin-polyak well-posedness for set-valued optimization problems with constraints", Filomat, Vol. 28, pp. 1345-1352. (2014).

[6] Y. P. Fang, N. J. Huang and J.C. Yao, "Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems", Journal of Global Optimization, Vol. 41, pp. 117-133. (2008).

[7] Y. P. Fang, R. Hu and N. J. Huang, "Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints", Comput. Math. Appl. Vol. 55, pp. 89-100. (2008).

[8] A.E. Hashoosh, and M. Alimohammady, "On well-posedness of generalized equilibrium problems involving α- monotone bifunction", Journal of Hyperstructures (2016) accepted.

[9] A. E. Hashoosh, M. Alimohammady and H. M. Buite, "Existence results for nonstandard equilibrium problems with generalized monotone bifunctions", Journal of Applied Mathematics, (2016) accepted.

[10] A. E. Hashoosh, M. Alimohammady and M. K. Kalleji, "Existence Results for Some Equilibrium Problems involving α-Monotone Bifunction", International Journal of Mathematics and Mathematical Sciences, 2016, pp. 1–5. (2016).

[11] X. X. Huang, "Extended and strongly extended well-posedness of set-valued optimization problems", Math. Methods. Oper. Res. Vol. 53, pp. 101–116. (2001).

[12] K. Kimura, Y.C. Liou, S. Y. Wu and J. C. Yao, "Well-posedness for parametric vector equilibrium with applications, J. Ind, Manag. Optim. Vol. 4, pp. 313–327. (2008).

[13] K. Kuratowski, "Topology", vols, Academic Press, New York, NY, 1–2, (1989).

[14] E. S. Levitin and B. T. Polyak, "Convergence of minimizing sequences in conditional extremum problems Soviet" Math. Dokl. Vol. 7, pp. 764–767. (1966).

[15] X. Long, N. Huang and K. Teo, "Levitin-Polyak well-posedness for equilibrium problems with functional constraints", J. Inequal, Appl, 2008 Article ID 657329 , pp. 1–14. (2008).

[16] J. Peng, Y. Wang and S. Wu, "Levitin-Polyak well-posedness of generalized vector equilibrium problems", J. Math. Taiwan, Vol. 15, pp. 2311–2330. (2011).

[17] J. Salamon, "Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems", J. Glob, Optim, Vol. 47, pp. 173–183. (2010).

[18] L. SJ L. MH,"Levitin-Polyak well-posedness of vector equilibrium problems", Math, Methods Oper. Res. Vol. 69, pp. 125-140. (2009).

[19] A. N. Tykhonov, "On the stability of the functional optimization problem". USSR J. Comput. Math, Phys, Vol. 6, pp. 631–634. (1966).

[20] A. Zaslavski, "Generic well-posedness for a class of equilibrium problems", J. Inequal, Appl. 2008, Article ID 581917, pp. 1–9. (2008).

[21] Y. Zhanga and T. Chen, "A note on well-posedness of Nash-type games problems with set payoff", J. Nonlinear Sci, Appl, Vol. 9, pp. 486–492. (2016).

[22] K. Zhang, Z,Quan HE and D. Peng GAO, "Extended well- posedness for quasivariational inequality", Vol.10, pp. 1-10. (2009).

Department of mathematics, University of Thi- Qar, Iraq.

Department of Mathematics, University of Mazandaran, Babolsar, Iran.

DOI: https://doi.org/10.17656/jzs.10598

**Abstract**The main goal of this paper is to establish some well-posedness results for a nonstandard equilibrium problem (for short ) and for optimization problem involving α-monotone bifunction, whose solution is sought in a subset K of a real reflexive Banach space X. Moreover, we establish some metric characterizations of well-posedness for a nonstandard equilibrium problems and for an optimization problem.

**Key Words:**Equilibrium problem; Well-posed optimizat -ion problems; bifunction monotone; Metric char- acterizeations.

**References**

[1] L. Q. Anh, P. Q. Khanh, and D. T. M. Van, "Well-posedness under relaxed semicontinuouty for bilevel equilibrium and optimization problems with equilibrium constraint", Journal of Optimization Theory and Application, Vol. 153, pp. 42-59. (2012).

[2] E. Bednarczuk and J. P. Penot, "Metrically well-set minimization problems", Appl. Math. Optim. Vol. 26, pp. 273-285. (1992).

[3] M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems, Math. Methods Oper. Res. Vol. 70, pp. 171-182. (2009).

[4] M. Bianchi and R. Pini, "Sensitivity for parametric vector equilbria", Optimization Vol. 55, pp. 221-130. (2006).

[5] J.W. Chen, Y. J. Cho and X.Q. Qu, "Levitin-polyak well-posedness for set-valued optimization problems with constraints", Filomat, Vol. 28, pp. 1345-1352. (2014).

[6] Y. P. Fang, N. J. Huang and J.C. Yao, "Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems", Journal of Global Optimization, Vol. 41, pp. 117-133. (2008).

[7] Y. P. Fang, R. Hu and N. J. Huang, "Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints", Comput. Math. Appl. Vol. 55, pp. 89-100. (2008).

[8] A.E. Hashoosh, and M. Alimohammady, "On well-posedness of generalized equilibrium problems involving α- monotone bifunction", Journal of Hyperstructures (2016) accepted.

[9] A. E. Hashoosh, M. Alimohammady and H. M. Buite, "Existence results for nonstandard equilibrium problems with generalized monotone bifunctions", Journal of Applied Mathematics, (2016) accepted.

[10] A. E. Hashoosh, M. Alimohammady and M. K. Kalleji, "Existence Results for Some Equilibrium Problems involving α-Monotone Bifunction", International Journal of Mathematics and Mathematical Sciences, 2016, pp. 1–5. (2016).

[11] X. X. Huang, "Extended and strongly extended well-posedness of set-valued optimization problems", Math. Methods. Oper. Res. Vol. 53, pp. 101–116. (2001).

[12] K. Kimura, Y.C. Liou, S. Y. Wu and J. C. Yao, "Well-posedness for parametric vector equilibrium with applications, J. Ind, Manag. Optim. Vol. 4, pp. 313–327. (2008).

[13] K. Kuratowski, "Topology", vols, Academic Press, New York, NY, 1–2, (1989).

[14] E. S. Levitin and B. T. Polyak, "Convergence of minimizing sequences in conditional extremum problems Soviet" Math. Dokl. Vol. 7, pp. 764–767. (1966).

[15] X. Long, N. Huang and K. Teo, "Levitin-Polyak well-posedness for equilibrium problems with functional constraints", J. Inequal, Appl, 2008 Article ID 657329 , pp. 1–14. (2008).

[16] J. Peng, Y. Wang and S. Wu, "Levitin-Polyak well-posedness of generalized vector equilibrium problems", J. Math. Taiwan, Vol. 15, pp. 2311–2330. (2011).

[17] J. Salamon, "Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems", J. Glob, Optim, Vol. 47, pp. 173–183. (2010).

[18] L. SJ L. MH,"Levitin-Polyak well-posedness of vector equilibrium problems", Math, Methods Oper. Res. Vol. 69, pp. 125-140. (2009).

[19] A. N. Tykhonov, "On the stability of the functional optimization problem". USSR J. Comput. Math, Phys, Vol. 6, pp. 631–634. (1966).

[20] A. Zaslavski, "Generic well-posedness for a class of equilibrium problems", J. Inequal, Appl. 2008, Article ID 581917, pp. 1–9. (2008).

[21] Y. Zhanga and T. Chen, "A note on well-posedness of Nash-type games problems with set payoff", J. Nonlinear Sci, Appl, Vol. 9, pp. 486–492. (2016).

[22] K. Zhang, Z,Quan HE and D. Peng GAO, "Extended well- posedness for quasivariational inequality", Vol.10, pp. 1-10. (2009).