Continuous selection for set-valued mappings

I. G. Tsar'kov

Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 645-669.

Résumé

We study properties of set-valued mappings $F$ admitting a continuous selection $f$ which is a continuous $\epsilon$-selection (from the set of $\epsilon$-closest points) for the images $F(x)$ $(x\in X)$. This is interpreted as an $\epsilon$-selection for continuously varying sets in a space with continuously varying norms. We deduce new fixed-point theorems from the results obtained. We also study geometric-topological properties of sets all of whose $r$-neighbourhoods possess a continuous $\epsilon$-selection for every $\epsilon>0$. We obtain a characterization of such sets.

Détail

Keywords: $\epsilon$-selection, continuous selection for set-valued mappings, $\overset{\,_\circ}{B}$-infinite connectedness, $\overset{\,_\circ}{B}$-approximative infinite connectedness, $\overset{\,_\circ}{B}$-neighbourhood infinite connectedness, fixed-point theorems.

Bibliographie

[1] I. G. Tsar'kov, “Relations between certain classes of sets in Banach spaces”, Math. Notes, 40:2 (1986), 597–610 | DOI | MR | Zbl

[2] I. G. Tsar'kov, “Continuous $\varepsilon$-selection”, Sb. Math., 207:2 (2016), 267–285 | DOI | DOI | MR | Zbl

[3] S. V. Konyagin, “O nepreryvnykh operatorakh obobschennogo ratsionalnogo priblizheniya”, Matem. zametki, 44:3 (1988), 404 | MR | Zbl

[4] I. G. Tsar'kov, “Properties of the sets that have a continuous selection from the operator $P^\delta$”, Math. Notes, 48:4 (1990), 1052–1058 | DOI | MR | Zbl

[5] I. G. Tsar'kov, “Properties of sets admitting stable $\varepsilon$-selections”, Math. Notes, 89:4 (2011), 572–576 | DOI | DOI | MR | Zbl

[6] C. S. Ryutin, “Uniform continuity of generalized rational approximations”, Math. Notes, 71:2 (2002), 236–244 | DOI | DOI | MR | Zbl

[7] E. D. Livshits, “Stability of the operator of $\varepsilon$-projection to the set of splines in $C[0,1]$”, Izv. Math., 67:1 (2003), 91–119 | DOI | DOI | MR | Zbl

[8] C. S. Ryutin, “Continuity of operators of generalized rational approximation in the space $L_1[0;1]$”, Math. Notes, 73:1 (2003), 142–147 | DOI | DOI | MR | Zbl

[9] A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci. (N.Y.), 217:6 (2016), 683–730 | DOI | MR | Zbl

[10] I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461 | DOI | DOI | MR | Zbl

[11] A. V. Fursikov, “Properties of solutions of some extremal problems connected with the Navier–Stokes system”, Math. USSR-Sb., 46:3 (1983), 323–351 | DOI | MR | Zbl

[12] M. V. Yashina, “The density of the uniqueness set of some extremal problems”, Moscow Univ. Math. Bull., 41:6 (1986), 41–44 | MR | Zbl

[13] M. V. Yashina, “K voprosu o edinstvennosti reshenii nekotorykh ekstremalnykh zadach”, Ekstremalnye zadachi, funktsionalnyi analiz i ikh prilozheniya, eds. P. L. Ulyanov, V. M. Tikhomirov, Izd-vo Mosk. un-ta, M., 1988, 10–13 | Zbl

[14] S. Eilenberg, D. Montgomery, “Fixed point theorems for multi-valued transformations”, Amer. J. Math., 68:2 (1946), 214–222 | DOI | MR | Zbl

[15] Ky Fan, “Fixed-point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad. Sci. U. S. A., 38 (1952), 121–126 | DOI | MR | Zbl

[16] L. Górniewicz, A. Granas, W. Kryszewski, “On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts”, J. Math. Anal. Appl., 161:2 (1991), 457–473 | DOI | MR | Zbl

[17] V. G. Gutev, “A fixed-point theorem for $UV^n$ usco maps”, Proc. Amer. Math. Soc., 124:3 (1996), 945–952 | DOI | MR | Zbl