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Stability Results of Variational Systems Under Openness with Respect to Fixed Sets
- 作者:M. Bianchi (1)
G. Kassay (2) R. Pini (3)
1. Universit脿 Cattolica del Sacro Cuore ; Milano ; Italy 2. Babe艧-Bolyai University ; Cluj ; Romania 3. Universit脿 degli Studi di Milano-Bicocca ; Milano ; Italy
- 关键词:Linear openness ; Metric regularity ; Sum of maps ; Generalized equation ; Sensitivity analysis ; Fixed point theorem ; Ekeland鈥檚 variational principle ; 49J53 ; 49K40 ; 90C31
- 刊名:Journal of Optimization Theory and Applications
- 出版年:2015
- 出版时间:January 2015
- 年:2015
- 卷:164
- 期:1
- 页码:92-108
- 全文大小:222 KB
- 参考文献:1. Lyusternik, V.A.: On the conditional extrema of functionals. Math. Sbornik 41, 390鈥?01 (1934)
2. Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111鈥?14 (1950) 3-3" target="_blank" title="It opens in new window">CrossRef 3. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43鈥?2 (1980) 3" target="_blank" title="It opens in new window">CrossRef 4. Ioffe, A.D.: Metric regularity and subdifferential calculus (Russian). Uspekhi Mat. Nauk 55, 103鈥?62 (2000). Translation in Russian Math. Surveys 55, 501鈥?58 (2000). 3/rm292" target="_blank" title="It opens in new window">CrossRef 5. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009) 387-87821-8" target="_blank" title="It opens in new window">CrossRef 6. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics. 266. Springer, New York (2013). 7. Arutyunov, A.V.: Covering mappings in metric spaces and fixed points. Dokl. Math. 76, 665鈥?68 (2007) 34/S1064562407050079" target="_blank" title="It opens in new window">CrossRef 8. Durea, R., Strugariu, R.: Chain rules for linear openness in general Banach spaces. SIAM J. Optim. 22, 899鈥?13 (2012) 37/11082470X" target="_blank" title="It opens in new window">CrossRef 9. Durea, R., Strugariu, R.: Openness stability and implicit multifunction theorems: applications to variational systems. Nonlinear Anal. 75, 1246鈥?259 (2012) CrossRef 10. Ngai, H.V.; Tron, N.; Th茅ra, M.: Metric regularity of the sum of multifunctions and applications. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0385-6 11. Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points. Proc. Amer. Math. Soc. 139, 521鈥?34 (2011) 39-2010-10490-2" target="_blank" title="It opens in new window">CrossRef 12. Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points II. J. Convex Anal. 19, 955鈥?73 (2012) 13. Bianchi, M., Kassay, G., Pini, R.: An inverse map result and some applications to sensitivity of generalized equations. J. Math. Anal. Appl. 339, 279鈥?90 (2013) 3" target="_blank" title="It opens in new window">CrossRef 14. Nadler, S.B.: Multivalued contraction mappings. Pacific J. Math. 30, 475鈥?88 (1969) 30.475" target="_blank" title="It opens in new window">CrossRef 15. Lim, T.C.: On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J. Math. Anal. Appl. 110, 436鈥?41 (1985) 306-3" target="_blank" title="It opens in new window">CrossRef 16. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (1999) 3-662-03961-8" target="_blank" title="It opens in new window">CrossRef 17. Bianchi, M., Miglierina, E., Molho, E., Pini, R.: Some results on condition numbers in convex multiobjective optimization. Set-Valued Anal. 21, 47鈥?5 (2013) CrossRef 18. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006) 19. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998) 3-642-02431-3" target="_blank" title="It opens in new window">CrossRef
- 刊物主题:Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
- 出版者:Springer US
- ISSN:1573-2878
文摘
In this paper, we present the notions of openness and metric regularity for a set-valued map with respect to two fixed sets, proving their equivalence. By using different approaches, we show the stability, with respect to the sum of maps, of the openness property, both in the setting of Banach spaces and of metric spaces. Finally, we infer the regularity of the map solving a generalized parametric equation defined via a parametric map that is, in its turn, perturbed by the sum with another map.
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