Equilibrium for perturbations of multifunctions by convex processes. (English) Zbl 0857.47031

This paper presents a general equilibrium theorem for the sum of an upper hemicontinuous convex-valued multifunction and a closed convex process defined on a noncompact subset of a normed space. The lack of compactness is compensated by inwardness conditions related to the existence of viable solutions of some differential inclusion.
Reviewer: J.F.Toland (Bath)


47H04 Set-valued operators
54H25 Fixed-point and coincidence theorems (topological aspects)
47J05 Equations involving nonlinear operators (general)
54C60 Set-valued maps in general topology
Full Text: EuDML EMIS


[1] B. Cornet, Paris avec handicaps et théorèmes de surjectivité de correspondances.C. R. Acad. Sc. Paris, Ser. A 281(1975), 479–482. · Zbl 0317.90087
[2] K. Fan, A minimax inequality and applications. InInequalities III (ed. byO. Shisha), 103–113;Academic Press, New York, 1972.
[3] J. P. Aubin, Analyse fonctionnelle non-linéaire et applications l’équilibre économique.Ann. Sc. Math. Québec II(1978), 5–47. · Zbl 0418.49038
[4] J. P. Aubin and H. Frankowska, Set-valued analysis.Birkhäuser, Boston, 1990. · Zbl 0713.49021
[5] G. Haddal, Monotone trajectories of differential inclusions and functional differential inclusions with memory.Israel J. Math. 39(1981), 83–100. · Zbl 0462.34048
[6] H. Ben-El-Mechaiekh, Zeros for set-valued maps with non-compact domains.C. R. Math. Rep. Acad. Sci. Canada XII(1990), 125–130. · Zbl 0734.47028
[7] J. S. Bae and S. Park, Existence of maximizable quasiconcave functions on convex spaces.J. Korean Math. Soc. 28(1991), 285–292. · Zbl 0756.47050
[8] S. Simons, An existence theorem for quasiconcave functions and applications.Nonlinear Anal. 10(1986), No. 10, 1133–1152. · Zbl 0616.47045
[9] R. T. Rockafellar, Monotone processes of convex and concave type.Mem. Amer. Math. Soc. 77(1967). · Zbl 0189.19602
[10] J. P. Aubin and R. Wets, Stable approximations of set-valued maps.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5(1988), 519–535. · Zbl 0681.54012
[11] G. N. Bergman and B. R. Halpern, A fixed point theorem for inward and outward maps.Trans. Amer. Math. Soc. 130(1968), 353–358. · Zbl 0153.45602
[12] K. Fan, Some properties of convex sets related to fixed point theorems.Math. Ann. 266(1984), 519–537. · Zbl 0515.47029
[13] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics.J. Math. Anal. Appl. 97(1983), 151–201. · Zbl 0527.47037
[14] H. Ben-El-Mechaiekh, P. Deguire and A. Granas, Une alternative non linéaire en analyse convexe et applications.C. R. Acad. Sc. Paris 295(1982), 257–259; Points fixes et coincidences pour les applications multivoques II, (applications de type {\(\pi\)} et {\(\pi\)}*), IbidemC.R. Acad. Sc. Paris 295(1982), 381–384 · Zbl 0521.47027
[15] F. Browder, The fixed point theory of multi-valued mappings in topological vector spaces.Math. Ann. 177(1968), 238–301. · Zbl 0176.45204
[16] H. Ben-El-Mechaiekh, Quelques principes topologiques en analyse convexe, parties I et II.Rapports de recherche 88-20, 21,Département de Math. et de Stat., Université de Montréal, 1988.
[17] J. C. Bellenger, Existence of maximizable quasiconcave functions on paracompact convex spaces.J. Math. Anal. Appl. 123(1987), 333–338. · Zbl 0649.46006
[18] S. Park, Generalizations of Ky Fan’s matching theorems and their applications.J. Korean Math. Soc. 28(1991), 275–283. · Zbl 0813.47063
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.