On The Search for Solutions for Equilibrium Problems and Fixed Point Problems
Abstract
Fixed point theory and fixed point algorithms are significant in mathematics, and so are equilibrium theory and algorithms. Both fixed point and equilibrium theories have also important applications in economics. Equilibrium problems and the search for equilibrium points are significant as such, but interestingly an equilibrium algorithm can also be used to solving neighbouring problems like e.g., fixed point problems. Therefore, we present in this literature survey equilibrium problem types and for each type we find from the literature algorithms which solve the problems of that type and at the same time the corresponding fixed point problems. In the beginning we present some famous fixed point theorems.
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Banach, S. (1922), ‘Sur les operations dans les ensembles abstraits et leur application aux equations integrales’, Fundamenta Mathematicae 1922.
Birkhoff, G. (1913), ‘Proof of Poincare’s geometric theorem’, Transactions of the International Mathematical Society 14, pp. 14-22 (Collected Mathematical Papers of Georg David Birkhoff, Vol. 1, pp. 673-681, Dover, New York, 1968).
Bnouhachem, A. (2014), ‘A hybrid iterative method for a combination of equilibria problem, a combination of variational inequality problems and a hierarchical fixed point problem’, Fixed Point Theory and Applications 2014, 163. DOI: 10.1186/1687-1812-2014-163
Browder, F. (1965), ‘Nonexpansive nonlinear operators in a Banach space’, Proceedings of the National Academy of Sciences of the United States of America, 54(4), 1041–1044.
Brouwer, L. E. J (1912), ‘Uber Abbildungen von Mannigfaltigkeiten’, Mathematische Annalen 71, ss. 97-115.
Brown, R. (1971), ‘The Lefschetz Fixed Point Theorem’. Scott, Foresman and Co., Glenview, Ill.- London.
Buong, N. ja Duong, N. (2011), ‘A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces’. Fixed Point Theory and Applications 2011, 208434. DOI: 10.1155/2011/208434
Burton, T. (1998), ‘A Fixed-Point Theorem of Krasnoselskii’, Applied Mathematical Letters Vol. 11, No. 1, ss. 85-88.
Chen, Y. ja Zhang, Y. (2010), ‘Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problem’, Fixed Point Theory and Applications 2010, 264628. DOI: 10.1155/2010/264628
Cholamjiak, P. (2009) ‘A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Banach Spaces’, Fixed Point Theory and Applications 2009, 719360. DOI: 10.1155/2009/719360
Gebrie, A. ja Wangkeeree, R. (2018), ‘Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces’, Fixed Point Theory and Applications 2018, 5. DOI: 10.1186/s13663-018-0630-7
Göhde, D. (1965), ‘Zum Prinzip der kontraktiven Abbildung’, Mathematische Nachrichten 30(3-4), 251–258.
He, Z. ja Du, W. (2011) ‘Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications’, Fixed Point Theory and Applications 2011, 33. DOI: 10.1186/1687-1812-2011-33
Jaiboon, C. ja Kurnam, P. (2009). ‘A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings’, Fixed Point Theory and Applications Volume 2009, Article ID 374815, 32 pages. DOI: 10.1155/2009/374815
Jitpeera, T. ja Kurnam, P. (2011), ‘A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational Inclusion Problem’. Fixed Point Theory and Applications 2011, 217407. DOI: 10.1155/2011/217407
Jitpeera, T. ja Kurnam, P. (2012), ‘A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems’, Fixed Point Theory and Applications 2012, 111. DOI: 10.1186/1687-1812-2012-111
Jung, A. (2017), ‘A Fixed-Point of View on Gradient Methods for Big Data’, Frontiers in Applied Mathematics and Statistics 3(18), 11 s., DOI: 10.3389/fams.2017.00018
Kakutani, S. (1941), ‘A generalization of Brouwer’s fixed point theorem’, Duke Mathematical Journal 8(3), 457–9.
Kangtunyakarn, A. (2011), ‘Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping’, Fixed Point Theory and Applications 2011, 274820. DOI: 10.1155/2011/274820
Kim, J. (2011), ‘Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings’, Fixed Point Theory and Applications 2011, 10. DOI: 10.1186/1687-1812-2011-10
Kirk, W. (1965), ‘A fixed point theorem for mappings which do not increase distances’, The American mathematical monthly, 72(9), 1004–1006.
Krasnoselski, M., (1958) American Mathematical Society (käännös) 10 (2), ss. 345-409.
Kurnam, P. ja Katchang, P. (2012), ‘The hybrid algorithm for the system of mixed equilibrium problems, the general system of finite variational inequalities and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings’. Fixed Point Theory and Applications 2012, 84. DOI: 10.1186/1687-1812-2012-84
Latif, A., Ceng, L. ja Ansari, Q. ‘Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems’, Fixed Point Theory and Applications 2012, 186. DOI: 10.1186/1687-1812-2012
Li, S., Li, L., Cao, L., He, X, ja Yue, X. (2013), ‘Hybrid extragradient method for generalized mixed equilibrium problems and fixed point problems in Hilbert space’, Fixed Point Theory and Applications 2013, 240. DOI: 10.1186/1687-1812-2013-240
Liu, M., Chang, S. ja Zuo, P. (2010), ‘On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi--ϕ--Asymptotically Nonexpansive Mappings in Banach Spaces’. Fixed Point Theory and Applications 2010, 157278 (2010). https://doi.org/10.1155/2010/157278
Nilsrakoo, W. (2011), ‘A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces’. Fixed Point Theory and Applications 2011, 572156. DOI: 10.1155/2011/572156
Onjai-uea, N., Jaiboon, C. ja Kurnam, P. (2011), ‘A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems’. Fixed Point Theory and Applications 2011, 32. DOI: 10.1186/1687-1812-2011-32
Osilike, M., Ofoedu, E. ja Attah, F. (2014), ‘The hybrid steepest descent method for solutions of equilibrium problems and other problems in fixed point theory’, Fixed Point Theory and Applications 2014, 156. DOI: 10.1186/ 1687-1812-2014-156
Park, S. (2004), ‘New Versions of the Fan-Browder Fixed Point Theorem and Existence of Economic Equilibria’, Fixed Point Theory and Applications 2004:2, 149–158, DOI: 10.1155/S1687182004308089
Peng, J. (2011), ‘Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings’. Fixed Point Theory and Applications 2011, 12. DOI: 10.1186/1687-1812-2011-12
Plubtieng, S. ja Sriprad, W. (2009) ‘A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces’, Fixed Point Theory and Applications Volume 2009, Article ID 567147, 20 pages, DOI: 10.1155/2009/567147
Plubtieng, S. ja Sriprad, W. (2010) ‘Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces’. Fixed Point Theory and Applications 2010, 962628. DOI: 10.1155/2010/962628
Poincare, H. (1912), ‘Sur un theoreme de geometrie’, Rendiconti del Circolo Matematico di Palermo, 33, ss. 375-407.
Qin, X., Cho, S. ja Wang, L. (2013), ‘Algorithms for treating equilibrium and fixed point problems’, Fixed Point Theory and Applications 2013, 308. DOI: 10.1186/1687-1812-2013-308
Qu, D. ja Cheng, C. (2011), ‘A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space’, Fixed Point Theory and Applications 2011, 17- DOI: 10.1186/1687-1812-2011-17
Saewan, S. ja Kurnam, P. (2011), ‘The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings’, Fixed Point Theory and Applications 2011:9. DOI 10.1186/1687-1812-2011-9
Schaefer, H. (1955), ‘Uber Die Methode Der a Priori Schranken’ , Mathematische Annalen 129, ss. 415-416.
Shang, M., Su, Y. ja Qin, X. (2007), ‘A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces’, Fixed Point Theory and Applications Volume 2007, Article ID 95412, 9 pages. DOI: 10.1155/2007/95412
Schauder, J. (1930), ’Der Fixpunktsatz in Funktionalräumen’, Studia Mathematica 2, ss. 171–180
Shukla, R. ja Panicker, R (2022), ‘Approximating Fixed Points of Enriched Nonexpansive Mappings in Geodesic Spaces’, Journal of Function Spaces vol. 2022 (Special issue “Fixed-Point Techniques and Applications to Real World Problems, ed. by S. Kumar’), Article ID 6161839, 8 s., DOI: 10.1155/2022/6161839
Suantai, S., Cholamjiak, P., Cho, Y. ja Cholamjiak, W. (2016), ‘On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces’, Fixed Point Theory and Applications 2016, 35. DOI: 10.1186/s13663-016-0509-4
Tan, J. ja Chang, S. (2011), ‘Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems’. Fixed Point Theory and Applications 2011, 915629. DOI: 10.1155/2011/915629
Ungchittrakool, K. ja Jarernsuk, A. (2012), ‘Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces’. Fixed Point Theory and Applications 2012, 147. DOI: 10.1186/1687-1812-2012-147
Voutilainen, R. (2022), ‘ Insurance Equilibria: Literature Survey’, Journal of Insurance and Financial Management 7(3), 65-85.
Wang, X., Ceng, L., Hu, H. ja Li, S. (2014), ‘General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems’, Fixed Point Theory and Applications 2014, 80. DOI: 10.1186/1687-1812-2014-80
Wang, S., Cho,Y. ja Qin, X. (2010), ‘A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings’, Fixed Point Theory and Applications Volume 2010, Article ID 165098, 18 pages. DOI: 10.1155/2010/165098
Wang, S., Gong, X., Abdou, A. ja Cho, Y. (2016), ‘Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications’, Fixed Point Theory and Applications 2016, 4. DOI: 10.1186/s13663-015-0475-2
Wang, S. and Guo, B. (2011), ‘Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces’, Fixed Point Theory and Applications 2011, 392741. DOI: 10.1155/2011/392741
Wang, S., Kang, S. ja Kwun, Y. (2011), ‘Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions’, Fixed Point Theory and Applications 2011, 859032. DOI: 10.1155/2011/859032
Wang, S. ja Zhou, C. (2011), ‘New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces’, Fixed Point Theory and Applications 2011, 372975. DOI: 10.1155/2011/372975
Witthayarat, U., Abdou, A. ja Cho, Y. (2015), ‘Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces’, Fixed Point Theory and Applications 2015, 200. DOI: 10.1186/s13663-015-0448-5
Yao, Y., Liou, Y. ja Kang, S. (2013), ‘An iterative approach to mixed equilibrium problems and fixed points problems’, Fixed Point Theory and Applications 2013, 183. DOI: 10.1186/1687-1812-2013-183
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This work is licensed under a Creative Commons Attribution 4.0 International License.