Geodesic
On stable fixed points under several kinds of strong perturbations
Song, Qi-Qing 1 ; Luo, Ping 1
1 College of Science, Guilin University of Technology, Guilin 541004, China
Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 11, p. 699-710.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This gives new results on stable fixed points related to several kinds of strong perturbations in references. It is shown that a strong stable set of fixed points has a robust stable property. For a robust stable fixed point set of a correspondence, in its neighborhood, there is a strong stable set for any small perturbation of the correspondence. There exists a robust stable set for a correspondence, if there is at least one fixed point for the correspondence. These generalize the corresponding results in recent references and give an application in the existence of strong stable economy equilibria.
DOI : 10.22436/jnsa.012.11.01
Classification : 47H04, 91A44
Keywords: Fixed point, essential stable, robust stable, economy equilibrium

Song, Qi-Qing  1 ; Luo, Ping  1

1 College of Science, Guilin University of Technology, Guilin 541004, China 
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Song, Qi-Qing ; Luo, Ping . On stable fixed points under several kinds of strong perturbations. Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 11, p. 699-710. doi : 10.22436/jnsa.012.11.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.11.01/

[1] Agarwal, R. P.; O'Regan, D. An essential map approach for multivalued A--proper, weakly inward, DKT and weakly closed maps, Comput. Math. Appl., Volume 38 (1999), pp. 89-100 | Zbl | DOI

[2] Agarwal, R. P.; O'Regan, D. Essential and inessential maps and continuation theorems, Appl. Math. Lett., Volume 13 (2000), pp. 83-90 | Zbl

[3] Andres, J.; Górniewicz, L. On essential fixed points of compact mappings on arbitrary absolute neighborhood retracts and their application to multivalued fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 26 (2016), pp. 1-11 | Zbl | DOI

[4] Browder, F. E.; Petryshyn, W. V. Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., Volume 20 (1967), pp. 197-228

[5] Carbonell-Nicolau, O.; Wohl, N. Essential equilibrium in normal--form games with perturbed actions and payoffs, J. Math. Econom., Volume 75 (2018), pp. 108-115 | DOI | Zbl

[6] Chen, J. C.; Gong, X. H. The stability of set of solutions for symmetric vector quasi--equilibrium problems, J. Optim. Theory Appl., Volume 136 (2008), pp. 359-374 | DOI | Zbl

[7] Fort, M. K. Essential and non essential fixed points, Amer. J. Math., Volume 72 (1950), pp. 315-322 | Zbl

[8] Fudenberg, D.; Tirole, J. Game theory, MIT press, Cambridge, 1991 | Zbl

[9] Granas, A. Sur la méthode de continuité de Poincaré, C. R. Acad. Sci. Paris Sér. A-B, Volume 282 (1976), pp. 983-985 | Zbl

[10] Hillas, J. On the definition of the strategic stability of equilibria, Econometrica, Volume 58 (1990), pp. 1365-1390 | Zbl

[11] Hillas, J.; Jansen, M.; Potters, J.; Vermeulen, D. On the relation among some definitions of strategic stability, Math. Oper. Res., Volume 26 (2001), pp. 611-635 | DOI | Zbl

[12] Hung, N. V.; Kieu, P. T. On the existence and essential components of solution sets for systems of generalized quasi--variational relation problems, J. Inequal. Appl., Volume 2014 (2014), pp. 1-12 | DOI

[13] Jiang, J.-H. Essential fixed points of the multivalued mappings, Acta Math. Sinica, Volume 11 (1962), pp. 376-379 | DOI | Zbl

[14] Kinoshita, S. On essential components of the set of fixed points, Osaka Math. J., Volume 4 (1952), pp. 19-22 | Zbl

[15] Kojima, M.; Okada, A.; Shindoh, S. Strongly stable equilibrium points of $N$--person noncooperative games, Math. Oper. Res., Volume 10 (1985), pp. 650-663 | DOI | Zbl

[16] Kohlberg, E.; Mertens, J.-F. On the strategic stability of equilibria, Econometrica, Volume 54 (1986), pp. 1003-1037 | Zbl

[17] Lin, Z.; Yang, H.; Yu, J. On existence and essential components of the solution set for the system of vector quasi--equilibrium problems, Nonlinear Anal., Volume 63 (2005), pp. 2445-2452 | DOI | Zbl

[18] McLennan, A. Fixed points of parameterized perturbations, J. Math. Econom., Volume 55 (2014), pp. 186-189 | DOI | Zbl

[19] O'Neill, B. Essential sets and fixed points, Amer. J. Math., Volume 75 (1953), pp. 497-509 | Zbl

[20] O'Neill, B. Semi-Riemannian geomerty with applications to relativity, Academic Press, London, 1983

[21] Scalzo, V. On the existence of essential and trembling--hand perfect equilibria in discontinuous games, Econ. Theory Bull. 2, Volume 2 (2014), pp. 1-12 | DOI

[22] Song, Q.-Q.; Guo, M.; Chen, H.-Z. Essential sets of fixed points for correspondences with application to Nash equilibria, Fixed Point Theory, Volume 17 (2016), pp. 141-150 | Zbl

[23] Song, Q.-Q.; Tang, G. Q.; Wang, L. S. On Essential Stable Sets of Solutions in Set Optimization Problems, J. Optim. Theory Appl., Volume 156 (2013), pp. 591-599 | DOI | Zbl

[24] Tarafdar, E. U.; Chowdhury, M. S. R. Topological Methods for Set--Valued Nonlinear Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, 2008 | Zbl | DOI

[25] Wills, M. D. Hausdorff distance and convex sets, J. Convex Anal., Volume 14 (2007), pp. 109-117 | Zbl

[26] Wu, W.-T.; Jiang, J.-H. Essential equilibrium points of $n$-person non-cooperative games, Sci. Sinica, Volume 11 (1962), pp. 1307-1322 | DOI | Zbl

[27] Xiang, S.-W.; Liu, G. D.; Zhou, Y. H. On the strongly essential components of Nash equilibria of infinite $n$-person games with quasiconcave payoffs, Nonlinear Anal., Volume 63 (2005), pp. 2639-2647 | Zbl | DOI

[28] Xiang, S.-W.; Xia, S. Y.; He, J. H.; Yang, Y. L.; Liu, C. W. Stability of fixed points of set--valued mappings and strategic stability of Nash equilibria, J. Nonlinear Sci. Appl., Volume 10 (2017), pp. 3599-3611 | Zbl

[29] Xiang, S.-W.; Zhang, D.-J.; Li, R.; Yang, Y.-L. Strongly essential set of Ky Fan's points and the stability of Nash equilibrium, J. Math. Anal. Appl., Volume 459 (2018), pp. 839-851 | DOI | Zbl

[30] Yang, Z. Essential stability of $\alpha$-core, Internat. J. Game Theory, Volume 46 (2017), pp. 13-28 | Zbl | DOI

[31] Yang, H.; Xiao, X. C. Essential components of Nash equilibria for games parametrized by payoffs and strategies, Nonlinear Anal., Volume 71 (2009), pp. 2322-2326 | DOI | Zbl

[32] Yang, Z.; Zhang, H. Essential stability of cooperative equilibria for population games, Optim. Lett., Volume 2018 (2018), pp. 1-10 | DOI

[33] Yao, Y. H.; Cho, Y. J.; Liou, Y.-C.; Agarwal, R. P. Constructed nets with perturbations for equilibrium and fixed point problems, J. Inequal. Appl., Volume 2014 (2014), pp. 1-14 | Zbl | DOI

[34] Yu, J. Essential equilibria of $n$-person noncooperative games, J. Math. Econom., Volume 31 (1999), pp. 361-372 | DOI | Zbl

[35] Yu, J.; Wang, N.-F.; Yang, Z. Equivalence results between Nash equilibrium theorem and some fixed point theorems, Fixed Point Theory Appl., Volume 2016 (2016), pp. 1-10 | DOI | Zbl

[36] Yu, J.; Yang, Z.; Wang, N.-F. Further results on structural stability and robustness to bounded rationality, J. Math. Econom., Volume 67 (2016), pp. 49-53 | DOI | Zbl

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