# American Institute of Mathematical Sciences

March  2021, 17(2): 869-887. doi: 10.3934/jimo.2020002

## On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set

 College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing 400074, China

* Corresponding author: Zaiyun Peng

Received  June 2018 Revised  March 2019 Published  March 2021 Early access  January 2020

Fund Project: The first and the fourth authors are supported by the Graduate Innovation Foundation of Chongqing Jiaotong University (2019S0123). The second author is supported by the National Natural Science Foundation of China (11301571), the Basic and Advanced Research Project of Chongqing (cstc2018jcyjAX0337), the Program for University Innovation Team of Chongqing (CXTDX201601022), the Innovation Project for Returned Overseas Scholars in Chongqing (cx2019148), the open project funded by the Chongqing Key Lab on ORSE (CSSXKFKTZ201801) and the Education Committee Project Foundation of Bayu Scholar. The third author is supported by the National Natural Science Foundation of China (11271389)

In this paper, we mainly discuss the stability of generalized vector quasi-equilibrium problems (GVQEPs) where the ordering relations are defined by free-disposal set. Firstly, by virtue of the oriented distance function $(\triangle)$, gap functions for (GVQEPs) are given and some properties of them are studied. Then, under some types of continuity assumption, the sufficient conditions of the upper semicontinuity and the upper Painlevé-Kuratowski convergence of solutions for (GVQEPs) are talked about. Moreover, sufficient and necessary conditions of the lower semicontinuity and the lower Painlevé-Kuratowski convergence of solutions for (GVQEPs) are obtained in normed linear spaces. Some examples are given to illustrate the results, and our results are new and extend some known results in the literature.

Citation: Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial and Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
##### References:
 [1] L. Q. Anh and N. V. Hung, Gap function and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, Journal of Industrial and Management Optimization, 14 (2018), 65-79.  doi: 10.3934/jimo.2017037. [2] L. Q. Anh, T. Bantaojai and N. V. Hung, Painlevé-Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems, Computational and Applied Mathematics, 37 (2018), 3832-3845.  doi: 10.1007/s40314-017-0548-4. [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [4] C. Berge, Topological Spaces, Oliver and Boyd, London, 1963. [5] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145. [6] C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optimization Methods and Software, 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043. [7] J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric vector quasi-equilibrium problems, Journal of Optimization Theory and Applications, 136 (2008), 359-374.  doi: 10.1007/s10957-007-9309-7. [8] X. H. Gong, Lower semicontinuity of the efficient solution mapping in semi-infinite vector optimization, Journal of Systems Science and Complexity, 28 (2015), 1312-1325.  doi: 10.1007/s11424-015-3058-8. [9] J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Mémoires de la Société Mathématique de France, 60 (1979), 57-85. [10] S. H. Hou, X. H. Gong and X. M. Yang, Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions, Journal of Optimization Theory and Applications, 146 (2010), 387-398.  doi: 10.1007/s10957-010-9656-7. [11] C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, Journal of Optimization Theory and Applications, 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4. [12] S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Analysis Theory Methods Applications, 70 (2009), 1528-1535.  doi: 10.1016/j.na.2008.02.032. [13] X. J. Long, N. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Mathematical and Computer Modelling, 47 (2008), 445-451.  doi: 10.1016/j.mcm.2007.04.013. [14] J. W. Peng, S. Y. Wu and Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, Journal of Global Optimization, 52 (2012), 779-795.  doi: 10.1007/s10898-011-9711-4. [15] J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optimization Letters, 4 (2010), 501-512.  doi: 10.1007/s11590-010-0179-9. [16] Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, Journal of Optimization Theory and Applications, 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6. [17] Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6. [18] R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [19] W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031. [20] H. Yang and J. Yu, Essential components of the set of weakly Pareto-Nash equilibrium points, Applied Mathematics Letters, 15 (2002), 553-560.  doi: 10.1016/S0893-9659(02)80006-4. [21] J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E. [22] J. Yu, Z. X. Liu, D. T. Peng, D. Y. Xu and Y. H. Zhou, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.  doi: 10.1002/oca.2096. [23] A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532. [24] J. Zeng, Z. Y. Peng, X. K. Sun and X. J. Long, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems, Journal of Nonlinear Science and Applications, 9 (2016), 4104-4113. [25] Y. Zhao, Z. Y. Peng and X. M. Yang, Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints, Optimization, 65 (2016), 1397-1415.  doi: 10.1080/02331934.2016.1149711. [26] R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in reflexive banach spaces, Journal of Optimization Theory and Applications, 150 (2011), 317-326.  doi: 10.1007/s10957-011-9843-1. [27] R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Computers and Mathematics with Applications, 63 (2012), 807-815.  doi: 10.1016/j.camwa.2011.11.046. [28] Y. H. Zhong, J. Yu and S. W. Xiang, Essential stability in games with infinitely many 1pure strategies, International Journal of Game Theory, 35 (2007), 493-503.  doi: 10.1007/s00182-006-0063-0.

show all references

##### References:
 [1] L. Q. Anh and N. V. Hung, Gap function and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, Journal of Industrial and Management Optimization, 14 (2018), 65-79.  doi: 10.3934/jimo.2017037. [2] L. Q. Anh, T. Bantaojai and N. V. Hung, Painlevé-Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems, Computational and Applied Mathematics, 37 (2018), 3832-3845.  doi: 10.1007/s40314-017-0548-4. [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [4] C. Berge, Topological Spaces, Oliver and Boyd, London, 1963. [5] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63 (1994), 123-145. [6] C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optimization Methods and Software, 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043. [7] J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric vector quasi-equilibrium problems, Journal of Optimization Theory and Applications, 136 (2008), 359-374.  doi: 10.1007/s10957-007-9309-7. [8] X. H. Gong, Lower semicontinuity of the efficient solution mapping in semi-infinite vector optimization, Journal of Systems Science and Complexity, 28 (2015), 1312-1325.  doi: 10.1007/s11424-015-3058-8. [9] J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Mémoires de la Société Mathématique de France, 60 (1979), 57-85. [10] S. H. Hou, X. H. Gong and X. M. Yang, Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions, Journal of Optimization Theory and Applications, 146 (2010), 387-398.  doi: 10.1007/s10957-010-9656-7. [11] C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, Journal of Optimization Theory and Applications, 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4. [12] S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Analysis Theory Methods Applications, 70 (2009), 1528-1535.  doi: 10.1016/j.na.2008.02.032. [13] X. J. Long, N. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Mathematical and Computer Modelling, 47 (2008), 445-451.  doi: 10.1016/j.mcm.2007.04.013. [14] J. W. Peng, S. Y. Wu and Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, Journal of Global Optimization, 52 (2012), 779-795.  doi: 10.1007/s10898-011-9711-4. [15] J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optimization Letters, 4 (2010), 501-512.  doi: 10.1007/s11590-010-0179-9. [16] Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, Journal of Optimization Theory and Applications, 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6. [17] Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6. [18] R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3. [19] W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031. [20] H. Yang and J. Yu, Essential components of the set of weakly Pareto-Nash equilibrium points, Applied Mathematics Letters, 15 (2002), 553-560.  doi: 10.1016/S0893-9659(02)80006-4. [21] J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E. [22] J. Yu, Z. X. Liu, D. T. Peng, D. Y. Xu and Y. H. Zhou, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.  doi: 10.1002/oca.2096. [23] A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532. [24] J. Zeng, Z. Y. Peng, X. K. Sun and X. J. Long, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems, Journal of Nonlinear Science and Applications, 9 (2016), 4104-4113. [25] Y. Zhao, Z. Y. Peng and X. M. Yang, Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints, Optimization, 65 (2016), 1397-1415.  doi: 10.1080/02331934.2016.1149711. [26] R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in reflexive banach spaces, Journal of Optimization Theory and Applications, 150 (2011), 317-326.  doi: 10.1007/s10957-011-9843-1. [27] R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Computers and Mathematics with Applications, 63 (2012), 807-815.  doi: 10.1016/j.camwa.2011.11.046. [28] Y. H. Zhong, J. Yu and S. W. Xiang, Essential stability in games with infinitely many 1pure strategies, International Journal of Game Theory, 35 (2007), 493-503.  doi: 10.1007/s00182-006-0063-0.
the improvement set 1
the improvement set 2
the improvement set 3
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