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April  2015, 11(2): 549-562. doi: 10.3934/jimo.2015.11.549

On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality

Zaiyun Peng 1, , Xinmin Yang 2, and  Kok Lay Teo 3,

1. 

College of Science, Chongqing JiaoTong University, Chongqing 400074, China
2. 

Department of Mathematics, Chongqing Normal University, Chongqing, 400047
3. 

Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845

Received  April 2013 Revised  April 2014 Published  September 2014

In this paper, by using a scalarization method, we establish sufficient conditions for Hölder continuity of approximate solution mapping to a class of parametric weak generalized Ky Fan Inequality with set-valued mappings. These results extend and improve some known results in the literature. Furthermore, some examples are given to illustrate the obtained results.
Keywords: approximate solution, scalarization., Hölder continuity, Ky Fan Inequality.
Mathematics Subject Classification: 90C31, 91B50, 90C4.
Citation: Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549
References:
[1]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhanser, Boston, 1990. Google Scholar

[3]

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L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim, 29 (2008), 24-42. doi: 10.1080/01630560701873068.  Google Scholar

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L. Q. Anh, P. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Analysis, 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029.  Google Scholar

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M. Bianchi and R. Pini, A note on stability for parametric equilibrium problems, Oper. Res. Lett., 31 (2003), 445-450. doi: 10.1016/S0167-6377(03)00051-8.  Google Scholar

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M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732.  Google Scholar

[8]

H. Brezis and L. Nirenberg, Stampacchia, G.: A remark on Ky Fan's minimax principle, Boll. Unione Mat. Ital., (III) VI (1972), 129-132. Google Scholar

[9]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005. Google Scholar

[10]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.  Google Scholar

[11]

K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112 (1969), 234-240. doi: 10.1007/BF01110225.  Google Scholar

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K. Fan, A minimax inequality and applications, In Shihsha, O. (ed.) Inequality III, Academic Press, New York (1972), 103-113. Google Scholar

[13]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Kluwer, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[14]

F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium problems: Nonsmooth optimization and variational inequality models, Nonconvex Optimization and Its Applications, 58 (2001), 187-205. doi: 10.1007/0-306-48026-3_12.  Google Scholar

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X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154. doi: 10.1023/A:1026418122905.  Google Scholar

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X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[17]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.  Google Scholar

[18]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[19]

K. Kimura and J. C. Yao, Semicontinuity of solutiong mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2.  Google Scholar

[20]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and the approxiamte solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[21]

S. J. Li, X. B. Li, L. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vectorequilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024.  Google Scholar

[22]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems, European J. Oper. Res., 210 (2011), 148-157. doi: 10.1016/j.ejor.2010.10.005.  Google Scholar

[23]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6.  Google Scholar

[24]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan Inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9.  Google Scholar

[25]

X. B. Li, S. J. Li and C. R. Chen, Lipschitz Continuity of an approximate solution mapping to equilibrium problems, Taiwan J. Math., 16 (2012), 1027-1040. Google Scholar

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Springer (2006). Google Scholar

[27]

Z. Y. Peng, Hölder Continuity of Solutions to Parametric Generalized Vector Quasiequilibrium Problems, Abst. Appl. Anal., 2012 (2012), Article ID 236413, 14 pages. doi: 10.1155/2012/236413.  Google Scholar

[28]

N. D. Yen, Hölder continuity of solutions to parametric variational inequalities, Appl. Math. Opim., 31 (1995), 245-255. doi: 10.1007/BF01215992.  Google Scholar

show all references

References:
[1]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. Google Scholar

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhanser, Boston, 1990. Google Scholar

[3]

M. Ait and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl, 306 (2005), 684-691. doi: 10.1016/j.jmaa.2004.10.011.  Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim, 29 (2008), 24-42. doi: 10.1080/01630560701873068.  Google Scholar

[5]

L. Q. Anh, P. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Analysis, 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029.  Google Scholar

[6]

M. Bianchi and R. Pini, A note on stability for parametric equilibrium problems, Oper. Res. Lett., 31 (2003), 445-450. doi: 10.1016/S0167-6377(03)00051-8.  Google Scholar

[7]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732.  Google Scholar

[8]

H. Brezis and L. Nirenberg, Stampacchia, G.: A remark on Ky Fan's minimax principle, Boll. Unione Mat. Ital., (III) VI (1972), 129-132. Google Scholar

[9]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005. Google Scholar

[10]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.  Google Scholar

[11]

K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112 (1969), 234-240. doi: 10.1007/BF01110225.  Google Scholar

[12]

K. Fan, A minimax inequality and applications, In Shihsha, O. (ed.) Inequality III, Academic Press, New York (1972), 103-113. Google Scholar

[13]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Kluwer, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[14]

F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium problems: Nonsmooth optimization and variational inequality models, Nonconvex Optimization and Its Applications, 58 (2001), 187-205. doi: 10.1007/0-306-48026-3_12.  Google Scholar

[15]

X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154. doi: 10.1023/A:1026418122905.  Google Scholar

[16]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[17]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.  Google Scholar

[18]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[19]

K. Kimura and J. C. Yao, Semicontinuity of solutiong mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2.  Google Scholar

[20]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and the approxiamte solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[21]

S. J. Li, X. B. Li, L. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vectorequilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024.  Google Scholar

[22]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems, European J. Oper. Res., 210 (2011), 148-157. doi: 10.1016/j.ejor.2010.10.005.  Google Scholar

[23]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6.  Google Scholar

[24]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan Inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9.  Google Scholar

[25]

X. B. Li, S. J. Li and C. R. Chen, Lipschitz Continuity of an approximate solution mapping to equilibrium problems, Taiwan J. Math., 16 (2012), 1027-1040. Google Scholar

[26]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Springer (2006). Google Scholar

[27]

Z. Y. Peng, Hölder Continuity of Solutions to Parametric Generalized Vector Quasiequilibrium Problems, Abst. Appl. Anal., 2012 (2012), Article ID 236413, 14 pages. doi: 10.1155/2012/236413.  Google Scholar

[28]

N. D. Yen, Hölder continuity of solutions to parametric variational inequalities, Appl. Math. Opim., 31 (1995), 245-255. doi: 10.1007/BF01215992.  Google Scholar

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