Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 68, 2019 - Issue 8
119
Views
1
CrossRef citations to date
0
Altmetric
Articles

Maximal element with applications to Nash equilibrium problems in Hadamard manifolds

, &
Pages 1491-1520 | Received 16 Feb 2017, Accepted 16 Jul 2019, Published online: 29 Jul 2019

References

  • Yannelis NC, Prabhakar ND. Existence of maximal elements and equilibria in linear topological spaces. J Math Econom. 1983;12:233–245. doi: 10.1016/0304-4068(83)90041-1
  • Toussaint S. On the existence of equilibria in economies with infinitely many commodities and without ordered preferences. J Econ Theory. 1984;33:98–115. doi: 10.1016/0022-0531(84)90043-7
  • Tulcea CI. On the approximation of upper-semicontinuous correspondences and the equilibriums of generalized games. J Math Anal Appl. 1986;136:267–289. doi: 10.1016/0022-247X(88)90130-8
  • Kim WK. Existence of maximal element and equilibrium for a nonparacompact N-person game. Proc Amer Math Soc. 1992;116:797–807.
  • Deguire P, Tan KK, Yuan GX-Z. The study of maximal elements, fixed points for Ls-majorized mapping and their applications to minimax and variational inequalities in the product topological spaces. Nonlinear Anal. 1999;37:933–951. doi: 10.1016/S0362-546X(98)00084-4
  • Yuan GX-Z. The study of minimax inequalities and applications to economies and variational inequalities. Memoirs. 1998;132:1–132.
  • Balaj M. Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities. Nonlinear Anal. 2008;68:3962–3971. doi: 10.1016/j.na.2007.04.033
  • Balaj M, Lin LJ. Equivalent forms of a generalized KKM theorem and their applications. Nonlinear Anal. 2010;73:673–682. doi: 10.1016/j.na.2010.03.055
  • Duggan J. General conditions for the existence of maximal elements via the uncovered set. J Math Econom. 2011;47:755–759. doi: 10.1016/j.jmateco.2011.09.008
  • Kassay G, Miholca M, Vinh NT. Vector quasi-equilibrium problems for the sum of two multivalued mappings. J Optim Theory Appl. 2016;169:424–442. doi: 10.1007/s10957-016-0919-9
  • Patriche M. Applications of the KKM property to coincidence theorems, equilibrium problems, minimax inequalities and variational relation problems. Numer Func Anal Opt. doi:10.1080/01630563.2017.1287085.
  • Klingenberg W. A course in differential geometry. Berlin: Springer; 1978.
  • Udriste C. Convex functions and optimization methods on Riemannian manifolds. In: Mathematics and its applications, vol. 297. Kluwer Academic Publishers; 1994.
  • Ferreira OP, Oliveira PR. Proximal point algorithm on Riemannian manifolds. Optimization. 2002;51:257–270. doi: 10.1080/02331930290019413
  • Walter R. On the metric projections onto convex sets in Riemannian spaces. Arch Math. 1974;25:91–98. doi: 10.1007/BF01238646
  • Rapcsák T. Smooth nonlinear optimization in Rn. Dordrecht: Kluwer Academic; 1997.
  • Németh SZ. Variational inequalities on Hadamard manifolds. Nonlinear Anal. 2003;52:1491–1498. doi: 10.1016/S0362-546X(02)00266-3
  • Kristály A. Location of Nash equilibria: a Riemannian geometrical approach. Proc Amer Math Soc. 2010;138:1803–1810. doi: 10.1090/S0002-9939-09-10145-4
  • Kristály A. Nash-type equilibria on Riemannian manifolds: A variational approach. J Math Pures Appl. 2014;101:660–688. doi: 10.1016/j.matpur.2013.10.002
  • Kristály A, Li C, López-Acedo G, Nicolae A. What do ‘convexities’ imply on Hadamard manifolds? J Optim Theory Appl. 2016;170:1068–1074. doi: 10.1007/s10957-015-0780-2
  • Li SL, Li C, Liou YC, Yao JC. Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 2009;71:5695–5706. doi: 10.1016/j.na.2009.04.048
  • Colao V, López G, Marino G, Martín-Márquez V. Equilibrium problems in Hadamard manifolds. J Math Anal Appl. 2012;388:61–77. doi: 10.1016/j.jmaa.2011.11.001
  • Tang GJ, Zhou LW, Huang NJ. The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim Lett. 2013;7:779–790. doi: 10.1007/s11590-012-0459-7
  • Zhou LW, Huang NJ. Existence of solutions for vector optimization on Hadamard manifolds. J Optim Theory Appl. 2013;157:44–53. doi: 10.1007/s10957-012-0186-3
  • Batista EEA, Bento GC, Ferreira OP. An existence result for the generalized vector equilibrium problem on Hadamard manifolds. J Optim Theory Appl. 2015;167:550–557. doi: 10.1007/s10957-015-0761-5
  • Kim WK. A multi-valued generalization of Nemeth's fixed point theorem. Appl Math Sci. 2015;9:37–44.
  • Kim WK. Some applications of an open-valued KKM-theorem in Hadamard manifolds. Appl Math Sci. 2015;9:2627–2634.
  • Li XB, Huang NJ. Generalized vector quasi-equilibrium problems on Hadamard manifolds. Optim Lett. 2015;155:155–170. doi: 10.1007/s11590-013-0703-9
  • Noor MA, Noor KI. Some algorithms for equilibrium problems on Hadamard manifolds. J Inequal Appl. 2012;2012:230. doi:10.1186/1029-242X-2012-230.
  • Noor MA, Zainab S, Yao Y. Implicit methods for equilibrium problems on Hadamard manifolds. J Appl Math. 2012;2012. doi:10.1155/2012/437391.
  • Yang Z, Pu YJ. Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications. Nonlinear Anal. 2012;75:516–525. doi: 10.1016/j.na.2011.08.053
  • Jana S, Nahak C. Mixed equilibrium problems on Hadamard manifolds. Rend Circ Mat Palermo. 2016;65:97–109. doi: 10.1007/s12215-015-0221-y
  • Jana S, Nahak C, Ionescu C. Generalized vector equilibrium problems on Hadamard manifolds. J Nonlinear Sci Appl. 2016;9:1402–1409. doi: 10.22436/jnsa.009.03.64
  • Ding XP. Existence of solutions for quasi-equilibrium problems in noncompact topological spaces. Comput Math Appl. 2000;39:13–21. doi: 10.1016/S0898-1221(99)00329-6
  • Kelley JL. General topology. New York: Springer-Verlag; 1955.
  • Tian GQ, Zhou JX. Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization. J Math Econ. 1995;24:281–303. doi: 10.1016/0304-4068(94)00687-6
  • Farajzadeh A, Lee BS, Plubteing S. On generalized quasi-vector equilibrium problems via scalarization method. J Optim Theory Appl. 2016;168:584–599. doi: 10.1007/s10957-015-0772-2
  • Chavel I. Riemannian geometry A modern introduction. Cambridge: Cambridge University Press; 1993.
  • do Carmo MP. Riemannian geometry. Boston: Birkhäuser; 1992.
  • Sakai T. Riemannian geometry. Transl. Math. Monogr. vol. 149. Providence, RI: American Mathematical Society; 1996.
  • Dugundji J. Topology. Boston: Allyn and Bacon, Inc.; 1966.
  • O'Neill B. Semi-Riemannian geometry: with applications to relativity, pure and applied mathematics, vol. 103. New York: Academic Press; 1983.
  • Dasgupta PS, Maskin ES. Debreu's social equilibrium existence theorem. Proc Nat Acad Sci USA. 2015;112:15769–15770. doi: 10.1073/pnas.1522640113
  • Krawczyk JB. Coupled constraint Nash equilibria in environmental games. Resour Energy Econ. 2005;27:157–181. doi: 10.1016/j.reseneeco.2004.08.001
  • Ansink E, Houba H. Market power in water markets. J Environ Econ Manag. 2012;64:237–252. doi: 10.1016/j.jeem.2011.10.002
  • Smeers Y, Oggioni G, Allevi E, Schaible S. Generalized Nash equilibrium and market coupling in the European power system. EPRG Working Paper 1016, Cambridge Working Paper in Economics 1034, 2010.
  • Nikaido H, Isoda K. Note on non-cooperative convex games. Pac J Math. 1995;5:807–815. doi: 10.2140/pjm.1955.5.807

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.