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Fragmentability of sequences of set-valued mappings with applications to variational principles
Author(s):
Marc
Lassonde;
Julian
P.
Revalski
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2637-2646.
MSC (2000):
Primary 49J53;
Secondary 46B20, 46B22, 54C60
Posted:
March 15, 2005
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Abstract:
We propose to study fragmentability of set-valued mappings not only for a given single mapping, but also for a sequence of mappings associated with the initial one. It turns out that this property underlies several variational principles, such as for example the Deville-Godefroy-Zizler variational principle and the Stegall variational principle, by providing a common path for proof.
References:
-
- [BPr]
- J.M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and differentiability of convex functions, Trans. Amer. Math. Soc. 303(1987), 517-527. MR 0902782 (88k:49013)
- [BCFR]
- J.M. Borwein, L. Cheng, M. Fabian and J.P. Revalski, A one perturbation variational principle and applications, Set-Valued Anal., 12(2004), 49-60. MR 2069351
- [ChrK]
- J.P.R. Christensen and P.S. Kenderov, Dense strong continuity of mappings and the Radon-Nikodým property, Math. Scand., 54(1984), 70-78. MR 0753064 (85k:46015)
- [CK]
- M.M. Coban and P.S. Kenderov, Dense Gâteaux differentiability of the
 -norm in  and the topological properties of  , Compt. Rend. Acad. Bulg. Sci., 38(1985), 1603-1604. MR 0837262 (87h:46070) - [CKR1]
- M.M. Coban, P.S. Kenderov and J.P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika, 36(1989), 301-324. MR 1045790 (91c:90119)
- [CKR2]
- M.M. Coban, P.S. Kenderov and J.P. Revalski, Densely defined selections of multivalued mappings, Trans. Amer. Math. Soc., 344(1994), 533-552. MR 1154539 (94k:54039)
- [CKR3]
- M.M. Coban, P.S. Kenderov and J.P. Revalski, Topological spaces related to the Banach-Mazur game and to the generic well-posedness of optimization problem, Set-Valued Anal., 3(1995), 263-279. MR 1353413 (96h:90172)
- [DGZ1]
- R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal., 111 (1993), 197-212.MR 1200641 (94b:49010)
- [DGZ2]
- R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman monographs and Surveys in Pure and Appl. Math., Longman Scientific & Technical, 1993.MR 1211634 (94d:46012)
- [DR]
- R. Deville and J.P. Revalski, Porosity of ill-posed problems. Proc. Amer. Math. Soc., 128 (2000), 1117-1124.MR 1636942 (2000i:49033)
- [Ek1]
- I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47(1974), 324-353.MR 0346619 (49:11344)
- [Ek2]
- I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1(1979), 443-474. MR 0526967 (80h:49007)
- [HJT]
- R.W. Hansell, J.E. Jayne and M. Talagrand, First class selectors for weakly upper semi-continuous multivalued maps in Banach spaces, J. Reine Angew. Math., 361(1985), 201-220. MR 0807260 (87m:54059a)
- [ILR]
- A.D. Ioffe, R. Lucchetti and J.P. Revalski, A variational principle for problems with functional constraints, SIAM J. Optim., 12, No.2(2001), 461-478. MR 1885571 (2002m:49042)
- [IZa]
- A.D. Ioffe and A.J. Zaslavski, Variational principles and well-posedness in optimization and calculus of variations. SIAM J. Control Optim., 38 (2000), 566-581. MR 1741153 (2000m:49031)
- [JNRo]
- J. E. Jayne, I. Namioka and C. A. Rogers, Topological properties of Banach spaces, Proc. London Math. Soc., (3) 66(1993), 651-672.MR 1207552 (94c:46041)
- [JRo]
- J.E Jayne and C.A. Rogers, Borel selectors for upper semi-continuous set-valued maps, Acta Math., 155(1985), 41-79.MR 0793237 (87a:28011)
- [KMo]
- P. S. Kenderov and W. B. Moors, Fragmentability and sigma-fragmentability of Banach spaces, J. London Math. Soc., 60(1999), 203-223.MR 1721825 (2001f:46025)
- [KR]
- P.S. Kenderov and J.P. Revalski, The Banach-Mazur game and generic existence of solutions to optimization problems, Proc. Amer. Math. Soc., 118(1993), 911-917.MR 1137224 (93i:49012)
- [LW]
- P.D. Loewen, X.F. Wang, A generalized variational principle, Can. J. Math., 53(6)(2001), 1174-1193.MR 1863847 (2002g:49029)
- [Ph]
- R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math. # 1364, Springer Verlag, Berlin, 1993.MR 1238715 (94f:46055)
- [Ri1]
- N.K. Ribarska, Internal characterization of fragmentable spaces, Mathematika, 34(1987), 243-257. MR 0933503 (89e:54063)
- [Ri2]
- N.K. Ribarska, The dual of a Gâteaux smooth Banach space is weak star fragmentable, Proc. Amer. Math. Soc., 114(1992), 1003-1008. MR 1101992 (92g:46020)
- [St]
- C. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Ann., 236(1978), 171-176.MR 0503448 (80a:46022)
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Additional Information:
Marc
Lassonde
Affiliation:
Laboratoire AOC, Département de Mathématiques, Université des Antilles et de la Guyane, 97159 Pointe-à-Pitre, France
Email:
marc.lassonde@univ-ag.fr
Julian
P.
Revalski
Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria
Email:
revalski@math.bas.bg
DOI:
10.1090/S0002-9939-05-07865-2
PII:
S 0002-9939(05)07865-2
Received by editor(s):
April 20, 2004
Posted:
March 15, 2005
Additional Notes:
The second author's research was supported by a Marie Curie Fellowship of the European Community program IHP under contract HPMF-CT-2002-01874
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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