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Images of connected sets by semicontinuous multifunctions. (English) Zbl 0578.54013

The aim of this paper is to prove the following two theorems (T: \(X\to Y\) is a multifunction): (1) The image of a compact set \(K\subset X\) by an upper semicontinuous (u.s.c.) compact-valued multifunction T is compact and the graph \(T_ K\) is also compact. (2) The image of a connected set \(K\subset X\) by an u.s.c. (or l.s.c.) multifunction T whose values are non-empty connected sets is connected. If T is either l.s.c. or u.s.c. and compact-valued, then the graph \(T_ K\) is also connected.
As some applications, the mean-value theorem for locally Lipschitzian functions, an existence theorem for the Lagrange problem in the calculus of variations, the compactness and connectedness of the reachable set for a general nonlinear differential inclusion and for a differential equation without uniqueness and connectedness of the set of non-dominated outcomes in multicriteria optimization are presented.
Reviewer: Z.Wyderka

MSC:

54C60 Set-valued maps in general topology
93B03 Attainable sets, reachability
49J05 Existence theories for free problems in one independent variable
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[1] Aubert, G.; Tahraoui, R., Théorèmes d’existence pour des problèmes du calcul des variations du type Inf ∝\(_0^Lf(x, u\)′\((x)) dx\) et Inf ∝\(_0^Lf(x, u(x), u\)′\((x)) dx \), J. Differential Equations, 33, 1-15 (1979) · Zbl 0404.49001
[2] Aubin, J.-P, Set-valued maps, (mimeographed lectures notes (1980), University of Paris-Dauphine)
[4] Berge, C., Espaces topologiques—Fonctions multivoques (1966), Dunod: Dunod Paris · Zbl 0164.52902
[5] Bitran, G. R.; Magnanti, T. L., The structure of admissible points with respect to cone dominance, J. Optim. Theory Appl., 29, 573-614 (1979) · Zbl 0389.52021
[6] Cellina, A., The role of approximation in the theory of multivalued mappings, (Kuhn, H. W.; Szegö, G. P., Differential Games and Related Topics (1971), North-Holland: North-Holland Amsterdam) · Zbl 0233.34071
[7] Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc., 205, 247-262 (1975) · Zbl 0307.26012
[8] Clarke, F. H., On the inverse function theorem, Pacific J. Math., 64, 97-102 (1976) · Zbl 0331.26013
[10] Davy, J. L., Properties of the solution set of a generalized differential equation, Bull. Austral. Math. Soc., 6, 379-398 (1972) · Zbl 0239.49022
[11] Eggleston, H. G., Convexity (1958), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0086.15302
[12] Flett, T. M., Differential Analysis (1980), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0442.34002
[13] Haddad, G., Topological properties of the sets of solutions for functional differential inclusions, Nonlinear Anal. TMA, 5, 1349-1366 (1981) · Zbl 0496.34041
[14] Hahn, H., Relle Funktionen (1948), Chelsea: Chelsea New York
[15] Hanner, O.; Radström, H., A generalization of a theorem of Fenchel, (Proc. Amer. Math. Soc., 2 (1951)), 589-593 · Zbl 0043.16203
[16] Hiriart-Urruty, J.-B, Mean value theorems in nonsmooth analysis, Numer. Funct. Anal. Optim., 2, 1-30 (1980) · Zbl 0462.49032
[17] Hiriart-Urruty, J.-B, On the Extension of Two Theorems in General Topology. Applications, (Technical Note (1978), University of Kentucky: University of Kentucky Lexington) · Zbl 0526.90068
[18] Karlin, S., Mathematical Methods and Theory in Games, Programming and Economics (1960), McGraw-Hill: McGraw-Hill New York · Zbl 0139.12704
[19] Kikuchi, N., On some fundamental theorems of contingent equations in connection with the control problems, Publ. Res. Inst. Math. Ser. A, 3, 177-201 (1967) · Zbl 0189.47301
[20] Kisielewicz, M., Compactness and upper semicontinuity of solution set of generalized differential equation in separable Banach space, Demonstratio Math., 15, 753-761 (1982) · Zbl 0524.34063
[21] Lebourg, G., Valeur moyenne pour gradient généralisé, C. R. Acad. Sci. Paris, 281, 795-797 (1975) · Zbl 0317.46034
[22] Naccache, P. H., Connectedness of the set of nondominated outcomes in multicriteria optimization, J. Optim. Theory Appl., 25, 459-467 (1978) · Zbl 0363.90108
[23] Nieuwenhuis, J. W., Some results about nondominated solutions, J. Optim. Theory Appl., 36, 289-301 (1982) · Zbl 0452.90074
[24] (Huard, P., Point-to-Set Maps and Mathematical Programming. Point-to-Set Maps and Mathematical Programming, Math. Programming Study 10 (1979))
[25] Penot, J.-P, (Mimeographed lecture notes (1978), University of Pau)
[26] Penot, J.-P, Continuity properties of performance functions, (Hiriart-Urruty, J.-B; Oettli, W.; Stoer, J., Optimization: Theory and Algorithms. Optimization: Theory and Algorithms, Lecture Notes in Pure and Applied Mathematics, Vol. 86 (1983), Dekker: Dekker New York) · Zbl 0886.90180
[27] Robert, R., Contributions à l’Analyse Non Linéaire, (Thèse de Doctorat ès Sciences Mathématiques (1976), Université de Grenoble I)
[28] Smithson, R. E., Multifunctions, Nieuw Arch. Wisk., 20, 31-53 (1972) · Zbl 0236.54013
[29] Szufla, S., On the structure of solution sets of differential and integral equations in Banach spaces, Ann. Polon. Math., 34, 165-177 (1977) · Zbl 0384.34038
[30] Valadier, M., Deux propriétés des trajectoires d’une différentielle multivoque, (Séminaire d’Analyse Convexe de Montpellier (1975)), Exposé No. 12 · Zbl 0356.49024
[31] Vidossich, G., On Peano phenomenon, Bull. Un. Mat. Ital., 3, 33-42 (1970) · Zbl 0179.47101
[32] Vidossich, G., On the structure of the set of solutions of nonlinear equations, J. Math. Anal. Appl., 34, 602-617 (1971)
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