Abstract
Using the concept of 0-epi mapping defined by Furi, Martelli, and Vignoli, we define the concept of 0-epi family of quasiconvex mappings. We prove that the theory of 0-epi families is a good substitute of the degree theory in the localization of Nash equilibrium points, because it is more refined and very simple.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Van, L. C.,Topological Degree of a Family of Convex Functions and Localization of Nash Equilibrium Points, Journal of Optimization Theory and Applications, Vol. 30, pp. 33–43, 1980.
Van, L. C.,Calcul sur Ordinateur du Degré Topologique sur R 2 et R 3, Université Paris IX-Dauphin, Thèse du 3ème Cycle, 1978.
Furi, M., Martelli, M., andVignoli, A.,On the Solvability of Nonlinear Operator Equations in Normed Spaces, Annali di Matematica Pura ed Applicata, Vol. 124, pp. 321–343, 1980.
Furi, M., andPera, M. P.,An Elementary Approach to Boundary-Value Problems at Resonance, Seminari dell' Instituto di Matematica Applicata Giovanni Sansone, Universitá di Firenze, Report, 1979.
Furi, M., andPera, M. P.,On the Existence of an Unbounded Connected Set of Solutions for Nonlinear Equations in Banach Spaces, Seminari dell Instituto di Matematica Applicata Giovanni Sansone, Universita di Firenze, Report, 1979.
Furi, M., andPera, M. P.,On Unbounded Branches of Solutions for Nonlinear Operator Equations in the Nonbifurcating Case, Seminari dell Instituto di Matematica Applicata Giovanni Sansone, Universitá di Firenze, Report, 1981.
Pera, M. P.,Sulla Risolubilita' di Equazioni Nonlineari in Spazi di Banach Ordinati, Seminari dell Instituto di Matematica Applicata Giovanni Sansone, Universitá di Firenze, Report, 1981.
Greenberg, H. J., andPierskalla, W. P.,Quasiconjugate Functions and Surrogate Duality, Cahiers du Centre d'Étude de Recherche Opérationnelle, Vol. 15, pp. 437–448, 1973.
Greenberg, H. J., andPierskalla, W. P.,A Review of Quasiconvex Functions, Operations Research, Vol. 19, pp. 1553–1570, 1971.
Crouzeix, J. P.,Contributions à l'Étude des Fonctions Quasiconvexes, Université Clermont-Ferrand, Thèse, 1977.
Auslender, A.,Problèmes de Minimax via l'Analyse Convexe et les Inégalités Variationnelles: Théorie et Algorithmes, Springer-Verlag, New York, New York, 1972.
Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
Amann, H.,Saddle Points and Multiple Solutions of Differential Equations, Mathematische Zeitschrift (to appear).
Corley, H. W.,Duality Theory for Maximizations with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 84, pp. 560–568, 1981.
Schwartz, J. T.,Nonlinear Functional Analysis, Gordon and Breach, New York, New York, 1969.
Allgower, E., andGeorg, K.,Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations, SIAM Review, Vol. 22, pp. 28–85, 1980.
Cronin, J.,Fixed Points and Topological Degree in Nonlinear Analysis, American Mathematical Society, Providence, Rhode Island, 1964.
Erdelsky, P. J.,Computing the Brouwer Degree in R, Mathematics of Computation, Vol. 27, pp. 133–137, 1973.
Kearfott, B.,An Efficient Degree-Computation Method for a Generalized Method of Bisection, Numerische Mathematik (to appear).
O'Neill, T., andThomas, J. W.,The Calculation of the Topological Degree by Quadrature, SIAM Journal on Numerical Analysis, Vol. 12, pp. 673–680, 1975.
Peitgen, H. O., andPrüfer, M.,The Leray-Schauder Continuation Method as a Constructive Element in the Numerical Study of Nonlinear Eigenvalue and Bifurcation Problems, Functional Differential Equations and Approximation of Fixed Points, Edited by H. O. Peitgen and H. O. Walther, Springer-Verlag, New York, New York, 1979.
Prüfer, M., andSiegberg, H. W.,On Computational Aspects of Degree in R n,Functional Differential Equations and Approximation of Fixed Points, Edited by H. O. Peitgen and H. O. Walther, Springer-Verlag, New York, New York, 1979.
Prüfer, M., andSiegberg, H. W.,Complementary Pivoting and the Hopf Degree Theorem, Journal of Mathematical Analysis and Applications, Vol. 84, pp. 133–149, 1981.
Siegberg, H. W.,Brouwer Degree: History and Numerical Computation (to appear).
Siegberg, H. W.,Abbildungsgrade in Analysis and Topology, University of Bonn, Diplomarbeit, 1977.
Stenger, F.,Computing the Topological Degree of a Mapping in R nNumerische Mathematik, Vol. 25, pp. 23–28, 1975.
Stynes, M. J.,An Algorithm for the Numerical Calculation of the Degree of a Mapping, Oregon State University, PhD Thesis, 1977.
Bourbaki, N.,Topologie Générale, Chapter 9, Hermann, Paris, France, 1971.
Dugundji, J.,An Extension of Tietze's Theorem, Pacific Journal of Mathematics, Vol. 1, pp. 353–367, 1959.
Vainberg, M. M.,Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco, California, 1964.
White, R. E.,Value Set at x ε Ω for an Arbitrary Distribution with Applications to Local Extrema of f ε C (Ω) and a Maximum Principle for Ordinary Differential Equations, SIAM Journal on Mathematical Analysis, Vol. 11, pp. 61–72, 1980.
Mirica, S.,A Note on the Generalized Differentiability of Mappings, Nonlinear Analysis, Theory, Methods, and Applications, Vol. 4, pp. 567–575, 1980.
Lasry, J. M., andRobert, R.,Degré et Théorème du Point Fixe pour les Applications Multivoques et Applications, Université Paris IX-Dauphin, France, Cahier de Mathématiques de Décision, 1975.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
Rights and permissions
About this article
Cite this article
Isac, G. 0-Epi families of mappings, topological degree, and optimization. J Optim Theory Appl 42, 51–75 (1984). https://doi.org/10.1007/BF00934133
Issue Date:
DOI: https://doi.org/10.1007/BF00934133