×

zbMATH — the first resource for mathematics

Hukuhara’s topological degree for non compact valued multifunctions. (English) Zbl 1048.47045
The aim of this paper is to construct a variant of the Leray-Schauder topological degree for maps of the form \(I- F\), where \(F: \overline D\to 2^E\) is a set-valued map defined on a bounded open subset of a Banach space \(E\), taking nonempty closed bounded (and possibly) nonconvex values in the hyperspace \(B(E)\) of all such subsets of \(E\) equipped with the Hausdorff metric and being \(h\)-compact in the sense that the set \(\{F(x)\mid x\in\overline D\}\) is precompact in \(B(E)\) (contrary to the usual constructions of Cellina, Hukuhara, Górniewicz and others, the map \(F\) is not compact in the usual sense: the set \(F(D)\) is not assumed to be precompact in \(E\)). The presented construction relies on some approximation techniques. After some preliminaries, the authors consider set-valued maps \(F\) having so-called regular representation (roughly speaking, \(F(x)= \overline{r\circ \phi(x)}\), where \(r: E\to E\) is Lipschitz continuous and \(\phi:\overline D\to 2^E\) is Hausdorff upper semicontinuous with closed bounded and convex values and admits arbitrarily close graph approximations) and provide a construction of the degree for \(I- F\). This degree satisfies the usual properties: homotopy invariance, existence and normalization. The standard use of topological degree methods allows to obtain some appropriate variants of the well-known results of fixed-point theory, such as versions of the nonlinear and Leray-Schauder alternative and the Borsuk antipodal theorem.

MSC:
47H11 Degree theory for nonlinear operators
55M25 Degree, winding number
54C60 Set-valued maps in general topology
47H04 Set-valued operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Altman, M., A fixed point theorem in Banach spaces, Bull. Acad. Polon. Sci. Cl III, 5 (1957), 89-92. · Zbl 0078.11703
[2] Borisovitch, Y. G., Gelman, B. D., Myshkis, A. D. and Obukhovskii, V. V., Topological methods in the fixed point theory of multivalued mappings, Russian Math. Surveys, 35 (1980), 65-143. · Zbl 0464.55003
[3] Cellina, A. and Lasota, A., A new approach to the definition of topological degree of multivalued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 47 (1969), 434-440. · Zbl 0194.44801
[4] Dawidowicz, A., Méthodes homologiques dans la théorie des applications et des champs de vecteurs sphériques dans les espaces de Banach, Dissertationes Math. (Rozprawy Mat.), 326 (1993), 1-50. · Zbl 0795.55001
[5] De Blasi, F. S., Characterizations of certain classes of semicontinuous multifunctions by continuous approximations, J. Math. Anal. Appl., 106 (1985), 1-18. · Zbl 0574.54012
[6] De Blasi, F. S. and Myjak, J., A remark on the definition of topological degree for set-valued mappings, J. Math. Anal. Appl., 92 (1983), 445-551. · Zbl 0524.47038
[7] Fan, Ky Fixed points and minimax theorems in locally convex spaces, Proc. Nat. Acad. Sc. U.S., 38 (1952), 121-126. · Zbl 0047.35103
[8] Fitzpatrick, P. M. and Petryshyn, W. V., Fixed point theorems and fixed point index for multivalued mappings in cones, J. London Math. Soc. (2), 12 (1975), 75-85. · Zbl 0329.47022
[9] G \? eba, K. and Granas, A., Infinite dimensional cohomology theory, J. Math. Pures Appl., 5 (1973), 147-270. · Zbl 0275.55009
[10] Górniewicz, L., Homological methods in fixed point theory of multi-valued maps, Dis- sertationes Math. (Rozprawy Mat.), 129 (1976), 1-71. · Zbl 0324.55002
[11] Górniewicz, L., Granas, A. and 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 (1991), 457-473. · Zbl 0757.54019
[12] Górniewicz, L. and Lassonde, M., Approximation and fixed points for compositions of R\delta -maps, Topology Appl., 55 (1994), 239-250. · Zbl 0793.54015
[13] Granas, A., Sur la notion de degré topologique pour une certaine classe de transforma- tions multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci., Série Sc. Math. Astronom. Phys., 7 (1959), 191-194. · Zbl 0087.32303
[14] —, Theorem on antipodes and theorems on fixed points for a certain class of multi- valued mappings in Banach spaces, Bull. Acad. Polon. Sci., Série Sc. Math. Astronom. Phys., 7 (1959), 271-275. · Zbl 0089.11202
[15] Hu, S. and Papageorgiou, N. S., Handbook of multivalued analysis, Vol. I, Kluwer, Dor- drecht, 1997. · Zbl 0887.47001
[16] Hukuhara, M., Sur l’application semi-continue dont la valeur est un compact convexe, Funkcial. Ekvac., 10 (1967), 43-66. · Zbl 0155.19402
[17] Istr\? at\?escu, V. I., Fixed point theory, Reidel, Dordrecht, 1981.
[18] Kakutani, S., A generalization of Brouwer’s fixed point theorem, Duke Math. J., 8 (1941), 457-459. · Zbl 0061.40304
[19] Lasry, J. M. and Robert, R., Analyse nonlineaire multivoque, Cahier de Mathématiques de la décision, No 7611, CEREMADE, Université de Paris IX, Dauphine, 1976.
[20] Leray, J. and Schauder, J., Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. Sér. 3, 51 (1934), 45-78. i i i i i Hukuhara’s Topological Degree 203 · Zbl 0009.07301
[21] Lloyd, N. G., Degree theory, Cambridge University Press, Cambridge, 1978. · Zbl 0367.47001
[22] Ma, T. W., Topological degree for set-valued compact vector fields in locally convex spaces, Dissertationes Math. (Rozprawy Mat.), 92 (1972), 1-43. · Zbl 0211.25903
[23] Michael, E., Continuous selections I, Ann. of Math., 63 (1956), 361-382. · Zbl 0071.15902
[24] Petryshyn, W. V. and Fitzpatrick, P. M., A degree theory, fixed point theorems and mappings theorems for multivalued noncompact mappings, Trans. Amer. Math. Soc., 194 (1974), 1-25. · Zbl 0297.47049
[25] Rothe, E., Zur Theorie der topologischen Ordnung und der Vectorfelder in Banachshen Räumen, Compositio Math., 5 (1937), 177-197. · Zbl 0018.13304
[26] Schwartz, J. T., Nonlinear functional analysis, Gordon and Breach, New York, 1969. i i i i · Zbl 0203.14501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.