## Densely defined selections of multivalued mappings

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- by M. M. Čoban, P. S. Kenderov and J. P. Revalski PDF
- Trans. Amer. Math. Soc.
**344**(1994), 533-552 Request permission

## Abstract:

Rather general suficient conditions are found for a given multivalued mapping $F:X \to Y$ to possess an upper semicontinuous and compact-valued selection*G*which is defined on a dense ${G_\delta }$-subset of the domain of

*F*. The case when the selection

*G*is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic non-multivaluedness of metric projections and antiprojections.

## References

- J. M. Aarts and D. J. Lutzer,
*Completeness properties designed for recognizing Baire spaces*, Dissertationes Math. (Rozprawy Mat.)**116**(1974), 48. MR**380745** - Edgar Asplund,
*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31–47. MR**231199**, DOI 10.1007/BF02391908 - Errett Bishop and R. R. Phelps,
*A proof that every Banach space is subreflexive*, Bull. Amer. Math. Soc.**67**(1961), 97–98. MR**123174**, DOI 10.1090/S0002-9904-1961-10514-4 - D. Burke, D. Lutzer, and S. Levi,
*Functional characterizations of certain $p$-spaces*, Topology Appl.**20**(1985), no. 2, 161–165. MR**800846**, DOI 10.1016/0166-8641(85)90076-8 - J. Chaber, M. M. Čoban, and K. Nagami,
*On monotonic generalizations of Moore spaces, Čech complete spaces and $p$-spaces*, Fund. Math.**84**(1974), no. 2, 107–119. MR**343244**, DOI 10.4064/fm-84-2-107-119 - Jens Peter Reus Christensen,
*Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings*, Proc. Amer. Math. Soc.**86**(1982), no. 4, 649–655. MR**674099**, DOI 10.1090/S0002-9939-1982-0674099-0 - Jens Peter Reus Christensen and Petar Stojanov Kenderov,
*Dense strong continuity of mappings and the Radon-Nikodým property*, Math. Scand.**54**(1984), no. 1, 70–78. MR**753064**, DOI 10.7146/math.scand.a-12041 - M. M. Čoban and P. S. Kenderov,
*Dense Gâteaux differentiability of the sup-norm in $C(T)$ and the topological properties of $T$*, C. R. Acad. Bulgare Sci.**38**(1985), no. 12, 1603–1604. MR**837262** - Mitrofan M. Coban and Petar S. Kenderov,
*Generic Gateaux differentiability of convex functionals in $C(T)$ and the topological properties of $T$*, Mathematics and mathematical education (Bulgarian) (Sunny Beach (SlЪnchev Bryag), 1986) Publ. House Bulgar. Acad. Sci., Sofia, 1986, pp. 141–149. MR**872913** - M. M. Čoban, P. S. Kenderov, and J. P. Revalski,
*Generic well-posedness of optimization problems in topological spaces*, C. R. Acad. Bulgare Sci.**42**(1989), no. 1, 11–14. MR**991452** - M. M. Čoban, P. S. Kenderov, and J. P. Revalski,
*Generic well-posedness of optimization problems in topological spaces*, Mathematika**36**(1989), no. 2, 301–324 (1990). MR**1045790**, DOI 10.1112/S0025579300013152 - Frank Deutsch and Petar Kenderov,
*Continuous selections and approximate selection for set-valued mappings and applications to metric projections*, SIAM J. Math. Anal.**14**(1983), no. 1, 185–194. MR**686245**, DOI 10.1137/0514015 - Ryszard Engelking,
*Topologia ogólna*, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR**0500779** - M. Fabián and N. V. Zhivkov,
*A characterization of Asplund spaces with the help of local $\epsilon$-supports of Ekeland and Lebourg*, C. R. Acad. Bulgare Sci.**38**(1985), no. 6, 671–674. MR**805439** - Zdeněk Frolík,
*Generalizations of the $G_{\delta }$-property of complete metric spaces*, Czechoslovak Math. J.**10(85)**(1960), 359–379 (English, with Russian summary). MR**116305**
M. Furi and A. Vignoli, - R. W. Hansell, J. E. Jayne, and M. Talagrand,
*First class selectors for weakly upper semicontinuous multivalued maps in Banach spaces*, J. Reine Angew. Math.**361**(1985), 201–220. MR**807260** - H. Herrlich and G. E. Strecker,
*$H$-closed spaces and reflective subcategories*, Math. Ann.**177**(1968), 302–309. MR**234427**, DOI 10.1007/BF01350722 - J. E. Jayne and C. A. Rogers,
*Borel selectors for upper semicontinuous set-valued maps*, Acta Math.**155**(1985), no. 1-2, 41–79. MR**793237**, DOI 10.1007/BF02392537 - Petar Kenderov,
*Dense strong continuity of pointwise continuous mappings*, Pacific J. Math.**89**(1980), no. 1, 111–130. MR**596921** - Petar S. Kenderov,
*Continuity-like properties of set-valued mappings*, Serdica**9**(1983), no. 2, 149–160. MR**731839** - P. S. Kenderov and J. R. Giles,
*On the structure of Banach spaces with Mazur’s intersection property*, Math. Ann.**291**(1991), no. 3, 463–473. MR**1133343**, DOI 10.1007/BF01445220
S. V. Konyagin, - S. V. Konjagin,
*Approximation properties of closed sets in Banach spaces and the characterization of strongly convex spaces*, Dokl. Akad. Nauk SSSR**251**(1980), no. 2, 276–280 (Russian). MR**565493** - Ka Sing Lau,
*Almost Chebyshev subsets in reflexive Banach spaces*, Indiana Univ. Math. J.**27**(1978), no. 5, 791–795. MR**510772**, DOI 10.1512/iumj.1978.27.27051
S. Mazur, - Ernest Michael,
*Continuous selections. I*, Ann. of Math. (2)**63**(1956), 361–382. MR**77107**, DOI 10.2307/1969615 - E. Michael,
*A theorem on semi-continuous set-valued functions*, Duke Math. J.**26**(1959), 647–651. MR**109343** - E. Michael,
*Complete spaces and tri-quotient maps*, Illinois J. Math.**21**(1977), no. 3, 716–733. MR**467688** - E. Michael,
*Almost complete spaces, hypercomplete spaces and related mapping theorems*, Topology Appl.**41**(1991), no. 1-2, 113–130. MR**1129701**, DOI 10.1016/0166-8641(91)90103-S - I. Namioka,
*Separate continuity and joint continuity*, Pacific J. Math.**51**(1974), 515–531. MR**370466** - B. B. Panda and O. P. Kapoor,
*On farthest points of sets*, J. Math. Anal. Appl.**62**(1978), no. 2, 345–353. MR**473794**, DOI 10.1016/0022-247X(78)90131-2 - B. A. Pasynkov,
*Zero-dimensional, open, dimension-raising mappings*, Uspehi Mat. Nauk**18**(1963), no. 5 (113), 183–190 (Russian). MR**0156320** - Robert R. Phelps,
*Convex functions, monotone operators and differentiability*, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR**984602**, DOI 10.1007/BFb0089089 - N. K. Ribarska,
*The dual of a Gâteaux smooth Banach space is weak star fragmentable*, Proc. Amer. Math. Soc.**114**(1992), no. 4, 1003–1008. MR**1101992**, DOI 10.1090/S0002-9939-1992-1101992-3 - Haskell P. Rosenthal,
*The heredity problem for weakly compactly generated Banach spaces*, Compositio Math.**28**(1974), 83–111. MR**417762**
S. B. Stečkin, - Charles Stegall,
*Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of $C(K)$ Banach spaces*, Mathematika**34**(1987), no. 1, 101–107. MR**908845**, DOI 10.1112/S0025579300013334 - A. N. Tihonov,
*Stability of a problem of optimization of functionals*, Ž. Vyčisl. Mat i Mat. Fiz.**6**(1966), 631–634 (Russian). MR**198308** - Luděk Zajíček,
*On the points of multivaluedness of metric projections in separable Banach spaces*, Comment. Math. Univ. Carolin.**19**(1978), no. 3, 513–523. MR**508958** - Nikolai V. Zhivkov,
*Continuity and nonmultivaluedness properties of metric projections and antiprojections*, Serdica**8**(1982), no. 4, 378–385 (1983). MR**694960**

*About well-posed minimization problems for functionals in metric spaces*, J. Optim. Theory Appl.

**5**(1970), 225-290.

*Approximative properties of subsets in Banach spaces*, Dokl. Akad. Nauk SSSR

**239**(1978), 261-264.

*Über konvexe Mengen in linearen normierten Räumen*, Studia Math.

**4**(1933), 70-84.

*Approximative properties of Banach spaces subsets*, Rev. Roumaine Math. Pures Appl.

**8**(1963), 5-8.

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**344**(1994), 533-552 - MSC: Primary 54C65; Secondary 46G99, 47H04, 49J40, 54C60
- DOI: https://doi.org/10.1090/S0002-9947-1994-1154539-6
- MathSciNet review: 1154539