Equilibria of set-valued maps on nonconvex domains

Ben-El-Mechaiekh, H.; Kryszewski, W.

doi:10.1090/S0002-9947-97-01836-9

Equilibria of set-valued maps on nonconvex domains
HTML articles powered by AMS MathViewer

by H. Ben-El-Mechaiekh and W. Kryszewski PDF

Trans. Amer. Math. Soc. 349 (1997), 4159-4179 Request permission

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

P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
H. Ben-El-Mechaiekh, Continuous approximations of multifunctions, fixed points and coincidences, Approximation and optimization in the Caribbean, II (Havana, 1993) Approx. Optim., vol. 8, Peter Lang, Frankfurt am Main, 1995, pp. 69–97. MR 1368166
Jean-Marc Bonnisseau and Bernard Cornet, Fixed-point theorems and Morse’s lemma for Lipschitzian functions, J. Math. Anal. Appl. 146 (1990), no. 2, 318–332. MR 1043103, DOI 10.1016/0022-247X(90)90305-Y
H. Ben-El-Mechaiekh and W. Kryszewski, Equilibrium for perturbations of multifunctions by convex processes, Georgian Math. J. 3 (1996), no. 3, 201–215. MR 1388669, DOI 10.1007/BF02280004
Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
Felix E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301. MR 229101, DOI 10.1007/BF01350721
Bernard Cornet, Paris avec handicaps et théorèmes de surjectivité de correspondances, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 12, Aiii, A479–A482 (French, with English summary). MR 386726
F. H. Clarke, Yu. S. Ledyaev, and R. J. Stern, Fixed points and equilibria in nonconvex sets, Nonlinear Anal. 25 (1995), no. 2, 145–161. MR 1333819, DOI 10.1016/0362-546X(94)00215-4
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519–537. MR 735533, DOI 10.1007/BF01458545
Lech Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 71. MR 394637
Andrzej Granas, Points fixes pour les applications compactes: espaces de Lefschetz et la théorie de l’indice, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 68, Presses de l’Université de Montréal, Montreal, Que., 1980 (French). With an appendix, “Infinite-dimensional cohomology and bifurcation theory”, by Kazimierz Gęba. MR 569745
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
Wojciech Kryszewski, Topological and approximation methods of degree theory of set-valued maps, Dissertationes Math. (Rozprawy Mat.) 336 (1994), 101. MR 1307460
Wojciech Kryszewski, Some homotopy classification and extension theorems for the class of compositions of acyclic set-valued maps, Bull. Sci. Math. 119 (1995), no. 1, 21–48. MR 1313856
Marc Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), no. 1, 151–201. MR 721236, DOI 10.1016/0022-247X(83)90244-5
Lasry J. M. and R. Robert, Analyse nonlinéaire multivoque, Cahiers de Math. de la Décision 7611, Université de Paris-Dauphine, Paris, 1976.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Sławomir Plaskacz, On the solution sets for differential inclusions, Boll. Un. Mat. Ital. A (7) 6 (1992), no. 3, 387–394 (English, with Italian summary). MR 1196133
R. T. Rockafellar, Clarke’s tangent cones and the boundaries of closed sets in $\textbf {R}^{n}$, Nonlinear Anal. 3 (1979), no. 1, 145–154 (1978). MR 520481, DOI 10.1016/0362-546X(79)90044-0
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112