Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Equilibria of set-valued maps
on nonconvex domains


Authors: H. Ben-El-Mechaiekh and W. Kryszewski
Journal: Trans. Amer. Math. Soc. 349 (1997), 4159-4179
MSC (1991): Primary 47H10, 47H04; Secondary 54C55
DOI: https://doi.org/10.1090/S0002-9947-97-01836-9
MathSciNet review: 1401763
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if $K$ is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space $E$ , and $\Phi :K\longrightarrow 2^E$ is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition

\begin{equation*}\Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*}

then there exists $x_0\in K$ such that $0\in \Phi (x_0).$ Here, $T_K^r(x)$ denotes a new concept of retraction tangent cone to $K$ at $x$ suited for compact neighborhood retracts. When $K$ is locally convex at $x,T_K^r(x)$ coincides with the usual tangent cone of convex analysis. Special attention is given to neighborhood retracts having ``lipschitzian behavior'', called $L-$retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.


References [Enhancements On Off] (What's this?)

  • [A] Arens R., Extensions of functions on fully normal spaces, Pacific J. Math. 6 (1952), 11-22. MR 14:191h
  • [AC] Aubin J. P. and A. Cellina, Differential inclusions, Springer-Verlag, Berlin, 1984. MR 85j:49010
  • [AF] Aubin J. P. and H. Frankowska, Set-valued analysis, Birkhäuser, Boston, 1990. MR 91d:49001
  • [B] Ben-El-Mechaiekh H., Continuous approximation of multifunctions, fixed points and coincidences, Proceedings of the Second International Conference on Approximation and Optimization in the Carribean, M. Florenzano et al. Eds., Approximation and Optimization Vol. 8, Peter Lang, Frankfurt, 1995, pp. 69-97. MR 96k:47091
  • [BC] Bonnisseau J-M. and B. Cornet, Fixed-point theorems and Morse's lemma for lipschitzian functions, J. Math. Anal. Appl. 146 (1990), 318-332. MR 91c:58014
  • [BK] Ben-El-Mechaiekh H. and W. Kryszewski, Equilibrium for perturbations of upper hemicontinuous set-valued maps by convex processes, Georgian Math. J. 3 (1996), 201-205. MR 97c:34025
  • [Bo] Borsuk K., Theory of retracts, Monografle Matematyczne 44, Warszawa, 1967. MR 35:7306
  • [Br] Browder F., The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. MR 37:4679
  • [C] Cornet B., Paris avec handicaps et théorèmes de surjectivité de correspondances, C. R. Acad. Sc. Paris Sér. A 281 (1975), 479-482. MR 52:7577
  • [CLS] Clarke F. H., Y. S. Ledyaev and R. J. Stern, Fixed points and equilibria in nonconvex sets, Nonlinear Anal. 25 (1995), 145-161. MR 96g:49005
  • [D] Dugundji J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. MR 13:373c
  • [F1] Fan K., Fixed point and minimax theorems in locally convex topological spaces, Proc. Nat. Acad. Sc. USA 38 (1952), 121-126. MR 13:858d
  • [F2] Fan K., Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537. MR 85i:47060
  • [G] Górniewicz L., Homological methods in fixed-point theory of multivalued maps, Diss. Math. CXXIX, Warsawa, 1976. MR 52:15438
  • [Gr] Granas A., Points fixes pour les applications compactes: espaces de Lefschetz et la théorie de l'indice, SMS, Presses de l'Université de Montréal, Montréal, 1980. MR 81i:55002
  • [H] Hu S-T., Theory of Retracts, Wayne State Univ. Press, Detroit, 1965. MR 31:6202
  • [K1] Kryszewski W., Topological and approximation methods in degree theory of set-valued maps, Diss. Math. 336 (1994). MR 95m:55005
  • [K2] Kryszewski W., Some homotopy classification and extension theorems for the class of compositions of acyclic set-valued maps, Bull. Sci. Math. 119 (1995), 21-48. MR 95m:55012
  • [L] Lassonde M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201. MR 84k:47049
  • [LR] Lasry J. M. and R. Robert, Analyse nonlinéaire multivoque, Cahiers de Math. de la Décision 7611, Université de Paris-Dauphine, Paris, 1976.
  • [M] Michael E., Local properties of topological spaces, Duke Math. J. 21 (1954), 163-171. MR 15:977c
  • [P] Plaskacz S., On the solution sets of differential inclusions, Boll. U. M. I. (7) 6-A (1992), 387-394. MR 93m:34021
  • [R] Rockafellar R. T., Clarke's tangent cones and boundaries of closed sets in ${\bf R}^n,$ Nonlinear Analysis 3 (1979), 145-154. MR 80d:49032
  • [S] Spanier E., Algebraic topology, McGraw-Hill, New York, 1966. MR 35:1007

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H10, 47H04, 54C55

Retrieve articles in all journals with MSC (1991): 47H10, 47H04, 54C55


Additional Information

H. Ben-El-Mechaiekh
Affiliation: Instytut Matematyki, Uniwersitet Mikolaja Kopernika, ul. Chopina 12/18, 87-100 Toruń, Poland
Address at time of publication: Department of Mathematics, Sultan Qaboos Universisty, P.O. Box 50, Al-Khod, Oman

W. Kryszewski
Affiliation: Instytut Matematyki, Uniwersitet Mikolaja Kopernika, ul. Chopina 12/18, 87-100 Toruń, Poland
Email: wkrysz@mat.uni.torun.pl and wkrysz@plunlo51.bitnet

DOI: https://doi.org/10.1090/S0002-9947-97-01836-9
Keywords: Equilibria, nonconvex set-valued maps, compact neighborhood retracts, normal and tangent retraction cones and inwardness, $L-$retracts
Received by editor(s): October 17, 1994
Received by editor(s) in revised form: March 18, 1996
Additional Notes: Research supported by the Natural Sciences and Engineering Research Council of Canada under grant OGP0042422
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society