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Approximation and fixed points for compositions of \(R_ \delta\)-maps. (English) Zbl 0793.54015

A compact-valued upper semicontinuous (u.s.c.) multimap \(\phi: X \multimap Y\) is said to be a \(J\)-map if every its value \(\phi(x)\) is such that for each neighborhood \(U(\phi(x))\) there is a neighborhood \(V(\phi(x)) \subset U\) such that the inclusion \(V \to U\) induces the trivial homomorphisms of all homotopy groups. In particular, every u.s.c. \(R_ \delta\)-valued multimap into an ANR-space is a \(J\)-map.
The main result states that every finite composition of \(J\)-maps defined on a compact subset of an AANR-space has a single-valued approximation. As a corollary it follows that every compact multimap of such class of an AAR-space into itself has a fixed point.
Reviewer’s remark. The first approximation result for multimaps of \(J\)- type was obtained by A. D. Myshkis [Mat. Sb., Nov. Ser. 34(76), 525-540 (1954; Zbl 0056.089)].

MSC:

54C65 Selections in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
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[1] Ancel, F. D., The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc., 287, 1-40 (1985) · Zbl 0507.54017
[2] Anichini, G.; Conti, G.; Zecca, P., Approximation and selection for nonconvex multifunctions in infinite dimensional spaces, Boll. Un. Mat. Ital. B, 4, 7, 411-422 (1990) · Zbl 0717.47021
[3] Armentrout, S.; Price, T. M., Decompositions into compact sets with UV properties, Trans. Amer. Math. Soc., 141, 433-442 (1969) · Zbl 0183.27902
[4] Aronszajn, N., Le correspondant topologique de l’unicité dans la théorie des équations différentielles, Ann. of Math., 43, 730-738 (1942) · Zbl 0061.17106
[5] Aubin, J.-P.; Cellina, A., Differential Inclusions (1984), Springer: Springer Berlin
[6] Borsuk, K., Fundamental retracts and extensions of fundamental sequences, Fund. Math., 64, 55-85 (1969) · Zbl 0172.48203
[7] Cannon, J. W., Taming cell-like embedding relations, (Glaser, L. C.; Rushing, T. B., Geometric Topology, 438 (1975), Springer: Springer New York), 66-118, Lecture Notes in Mathematics · Zbl 0306.57004
[8] Cellina, A., Approximation of set-valued functions and fixed point theorems, Ann. Mat. Pura Appl., 82, 17-24 (1969) · Zbl 0187.07701
[9] Clapp, M. H., On a generalization of absolute neighborhood retracts, Fund. Math., 70, 117-130 (1971) · Zbl 0231.54012
[10] De Blasi, F. S.; Myjak, J., On continuous approximations for multifunctions, Pacific J. Math., 123, 9-31 (1986) · Zbl 0595.47037
[11] Deutsch, F.; Kenderov, P., Continuous selections and approximate selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal, 14, 185-194 (1983) · Zbl 0518.41031
[12] Dugundji, J., Modified Vietoris theorems for homotopy, Fund. Math., 66, 223-235 (1970) · Zbl 0196.26801
[13] Górniewicz, L.; Granas, A.; Kryszewski, W., Sur la méthode de l’homotopie dans la théorie des points fixes pour les applications multivoques Partie 1: Transversalité topologique, C.R. Acad. Sci. Paris Sér. I Math., 307, 489-492 (1988) · Zbl 0665.54030
[14] Górniewicz, L.; Granas, A.; Kryszewski, W., Sur la méthode de l’homotopie dans la théorie des points fixes pour les applications multivoques Partie 2: L’indice dans les ANR-s compacts, C.R. Acad. Sci. Paris Sér. I Math., 308, 449-452 (1988) · Zbl 0678.54033
[15] Górniewicz, L.; Granas, A.; Kryszewski, W., On the homotopy method in the fixed point index theory of multi-valued mappings of compact Absolute Neighborhood Retracts, J. Math. Anal. Appl., 161, 457-473 (1991) · Zbl 0757.54019
[16] Granas, A., Points fixes pour les applications compactes: espaces de Lefschetz et la théorie de l’indice, (Collection SMS, 68 (1980), Presses Université de Montréal: Presses Université de Montréal Montréal, Que) · Zbl 0456.55001
[17] Hyman, D. M., On decreasing sequences of compact absolute retracts, Fund. Math., 64, 91-97 (1969) · Zbl 0174.25804
[18] Klee, V., Shrinkable neighborhoods in Hausdorff linear spaces, Math. Ann., 141, 281-285 (1960) · Zbl 0096.07902
[19] Klee, V., Leray-Schauder theory without local convexity, Math. Ann., 141, 286-296 (1960) · Zbl 0096.08001
[20] Lacher, R. C., Cell-like spaces, Proc. Amer. Math. Soc., 20, 598-602 (1969) · Zbl 0175.49902
[21] Lassonde, M., Fixed points for Kakutani factorizable multifunctions, J. Math. Anal. Appl., 152, 46-60 (1990) · Zbl 0719.47043
[22] Mas-Colell, A., A note on a theorem of F. Browder, Math. Programming, 6, 273-275 (1974)
[23] Noguchi, H., A generalization of absolute neighborhood retracts, Kodai Math. Sem. Rep., 1, 20-22 (1953) · Zbl 0052.18803
[24] von Neumann, J., Über ein Ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, Rev. Econom. Stud., 13, 1-9 (1945-46), (in English) · Zbl 0017.03901
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