Ekeland's principle for vector-valued maps based on the characterization of uniform spaces via families of generalized quasi-metrics

A. Benbrik ; A. Mbarki ; S. Lahrech ; A. Ouahab

Lobachevskii journal of mathematics, Tome 21 (2006), pp. 33-44.

Résumé

Using a new characterization of uniform spaces via Families of generalized quasi-metrics, we present a variant of Ekeland's variational principle for vector-valued maps being a consequence of minimal point theorem.

Détail

Keywords: Ekeland's principle for vector-valued maps, minimal point theorem, uniform spaces, family of generalized quasi-metrics.

Bibliographie

[1] Hamel, A., Lohne, A., A Minimal Point Theorem in Uniform Spaces, manuscript, 2002

[2] Altman, M., “A Generalization of Brézis–Browder Principle on Ordered Sets”, Nonlinear Analysis, Theory, Methods and Applications, 6:2 (1982), 157–165 | DOI | MR | Zbl

[3] Brézis, H., Browder, F. E., “A General Principle on Ordered Sets in Nonlinear Functional Analysis”, Advances in Mathematics, 21 (1976), 355–364 | DOI | MR | Zbl

[4] Bishop, E., Phelps, R. R., “The support Functionals on Convex Sets”, Convexity, volume VII of Proceedings in Symposia in pure mathematics, ed. KLEE, V., American Mathematical Society, 1963, 27–35 | MR

[5] Bronstedt, A., “On a lemma of Bishop and Phelps”, Pacific Journal of Mathematics, 55:2 (1974), 335–341 | MR

[6] Caristi, J., “Fixed Point Theorems for Mappings satisfying Inwardness Conditions”, Transactions of the American Mathematical Society, 215 (1976), 241–251 | DOI | MR | Zbl

[7] Chen, G. Y., Huang, X. X., “A Unifed Approach to the Three Existing Types of Variational Principles for Vector Valued Functions”, Mathematical Methods of Operations Reasearch, 48:3 (1998), 349–358 | DOI | MR

[8] Chen, G. Y., Huang, X. X., Hou, S. H., “General Ekeland's Variationals Principle for Set-Valued Mappings”, JOTA, 106:1 (2000), 151–164 | DOI | MR | Zbl

[9] Ekeland, I., “On the Variational Principle”, Journal of Mathematics Analysis and Applications, 47 (1974), 324–353 | DOI | MR | Zbl

[10] Fang, X.-J., “The Variational Principle and Fixed Point Theorems in Certain Topological Spaces”, Journal of Mathematical Analysis and Applications, 202 (1996), 398–412 | DOI | MR | Zbl

[11] Gerth, Chr., Weidner, P., “Nonconvex Separation Theorems and Some Applications in Vector Optimization”, JOTA, 67:2 (1990), 297–320 | DOI | MR | Zbl

[12] Gerth, Chr., Iwanow, E., “Dualitat fur nichtkonvexe Vektoroptimierungsprobleme”, Wiss. Z. TH Ilmenau, 31:2 (1985), 61–81 | MR

[13] Gopfert, A., Tammer, Chr., “A New Maximal Point Theorem”, ZAA, 14:2 (1995), 379–390 | MR

[14] Gopfert, A., Tammer, Chr., Zalinescu, C., “A New Minimal Point Theorem in Product Spaces”, ZAA, 18:3 (1999), 767–770 | MR

[15] Gopfert, A., Tammer, Chr., Zalinescu, C., “On the Vectorial Ekeland's Variational Principle and Minimal Points in Product Spaces”, Nonlinear Analysis, 39 (2000), 909–922 | DOI | MR

[16] Hamel, A., Equivalents to Ekeland's Variational Principle in $F$-Type Topological Spaces, Report 09, University Halle-Wittenberg, Institut of Optimization and Stochastics, 2001

[17] Hamel, A., Phelps' Lemma, Danes Drop Theorem and Ekeland's Principle in Locally Convex Topological Vector Spaces, Report 10, University Halle-Wittenberg, Institut of Optimization and Stochastics, 2001

[18] Hamel, A., Lohne, A., Minimal Set Theorems, manuscript, 2002

[19] Hyers, D. H., “Pseudo-Normed Linear Spaces and Abelian Groups”, Duke Math. J., 5 (1939), 628–634 | DOI | MR

[20] Hyers, D. H., Isac, G., Rassias, T. M., Topics in Nonlinear Analysis and Applications, World Scientific, 1997 | MR | Zbl

[21] Isac, G., “The Ekeland's Principle and the Pareto $\varepsilon$-Efficiency”, Multi-Objective Programming and Goal Programming, Theories and Applications, Lecture Notes in Economics and Math. Systems, 432, ed. M. Tamiz, Springer, Berlin, 1996, 148–163 | Zbl

[22] Kelly, J. L., General Topology, Springer, 1995 | MR

[23] Kothe, G., Topologische lineare Raume I, Springer, 1996 | MR

[24] Li, S. J., Yang, X. Q., Chen, G., “Vector Ekeland Variational Principle”, Vector Variational inequalities and Vector Equilibria, ed. Gianessi, F., Kluwer Academic Publishers, Dordrecht, 2000 | MR

[25] Lohne, A., Minimalpunkttheoreme in uniformen Raumen und verwandte Aussagen, Diplomarbeit, MLU Halle-Wittenberg, 2001

[26] Mizoguchi, N., “A Generalization of Bronstedt's Results and its Applications”, Proc. Amer. Math. Soc., 108:3 (1990), 707–714 | DOI | MR | Zbl

[27] Nemeth, A. B., “A Nonconvex Vector Minimization Problem”, Nonlinear Analysis, 10 (1986), 669–678 | DOI | MR | Zbl

[28] Park, S., “On Generalizations of Ekeland-Type Variational Principles”, Nonlinear Analysis, 39 (2000), 881–889 | DOI | MR

[29] Phelps, R. R., “Support Cones and Their Generalization”, Convexity, volume VII of Proceedings in Symposia in pure mathematics, ed. Klee, V., American Mathematical Society, 1963, 393–401 | MR

[30] Phelps, R. R., “Support Cones in Banach Spaces and Their Applications”, Advances in Mathematics, 13 (1974), 1–19 | DOI | MR | Zbl

[31] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, 1364, Springer Verlag, 1989 | MR | Zbl

[32] Tammer, Chr., “A Generalization of Ekeland's Variational Principle”, Optimization, 25 (1992), 129–141 | DOI | MR | Zbl

[33] Tammer, Chr., “A Variational Principle and a Fixed Point Theorem”, System Modeling and Optimization, Lecture Notes in Control and Information Sciences, 197, eds. Henry, J., Yvon, J.-P., Springer, London, 1994, 248–257 | MR | Zbl

[34] Weil, A., Sur les espaces a structure uniforme et sur la topologie générale, Actualités Sci. Ind., 551, Paris, 1937 | Zbl