Ekeland variational principle and its equivalents in \(T_1\)-quasi-uniform spaces. (English) Zbl 1520.58008
Summary: The present paper is concerned with the Ekeland Variational Principle (EkVP) and its equivalents (Caristi-Kirk fixed point theorem, Takahashi minimization principle, Oettli-Théra equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by A. Hamel and A. Löhne [in: Nonlinear analysis and applications: To V. Lakshmikantham on his 80th birthday. Vol. 1. Dordrecht: Kluwer Academic Publishers. 577–593 (2003; Zbl 1045.58013)] and A. H. Hamel [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62, No. 5, 913–924 (2005; Zbl 1093.49016)] in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of \(F\)-quasi-gauge spaces, a non-symmetric version of \(F\)-gauge spaces introduced by J.-X. Fang [J. Math. Anal. Appl. 202, No. 2, 398–412 (1996; Zbl 0859.54042)], is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by A. V. Arutyunov and B. D. Gel’man [Zh. Vychisl. Mat. Mat. Fiz. 49, No. 7, 1167–1174 (2009; Zbl 1224.58024); translation in Comput. Math., Math. Phys. 49, No. 7, 1111–1118 (2009)] and A. V. Arutyunov [Proc. Steklov Inst. Math. 291, 24–37 (2015; Zbl 1336.49005); translation from Tr. Mat. Inst. Steklova 291, 30–44 (2015)] in complete metric spaces.
MSC:
| 58E30 | Variational principles in infinite-dimensional spaces |
| 54E15 | Uniform structures and generalizations |
| 47H10 | Fixed-point theorems |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
Ekeland variational principle; Takahashi minimization principle; equilibrium problems; quasi-uniform spaces; quasi-gauge spacesReferences:
| [1] | Ekeland, I., Sur les problèmes variationnels, C R Acad Sci Paris Sér A-B, 275, 1057-1059 (1972) · Zbl 0249.49004 |
| [2] | Ekeland, I., On the variational principle, J Math Anal Appl, 47, 324-353 (1974) · Zbl 0286.49015 |
| [3] | Cobzaş, S., Fixed points and completeness in metric and generalized metric spaces, Fund Prikl Mat, 22, 1, 127-215 (2018) |
| [4] | Ekeland, I., Nonconvex minimization problems, Bull Amer Math Soc (NS.), 1, 443-474 (1979) · Zbl 0441.49011 |
| [5] | Al-Homidan, S.; Ansari, QH; Kassay, G., Takahashi’s minimization theorem and some related results in quasi-metric spaces, J Fixed Point Theory Appl, 21, 1, 38 (2019) · Zbl 1417.54011 |
| [6] | Bao, TQ; Cobzaş, S.; Soubeyran, A., Variational principles, completeness and the existence of traps in behavioral sciences, Ann Oper Res, 269, 53-79 (2018) · Zbl 1476.91113 |
| [7] | Cobzaş, S., Completeness in quasi-metric spaces and Ekeland Variational Principle, Topol Appl, 158, 1073-1084 (2011) · Zbl 1217.54026 |
| [8] | Cobzaş, SE., Takahashi and Caristi principles in quasi-pseudometric spaces, Topol Appl, 265 (2019) · Zbl 1423.58009 |
| [9] | Karapinar, E.; Romaguera, S., On the weak form of Ekeland’s variational principle in quasi-metric spaces, Topol Appl, 184, 54-60 (2015) · Zbl 1309.54012 |
| [10] | Cobzaş, S., Ekeland variational principle in asymmetric locally convex spaces, Topol Appl, 159, 2558-2569 (2012) · Zbl 1251.46039 |
| [11] | Hamel, AH. Equivalents to Ekeland’s Variational Principle in \(####\)-type topological spaces. Report no. 09. Martin-Luther-University Halle-Wittenberg, Institute of Optimization and Stochastics. 2001. |
| [12] | Hamel, AH., Equivalents to Ekeland’s variational principle in uniform spaces, Nonlinear Anal, 62, 913-924 (2005) · Zbl 1093.49016 |
| [13] | Hamel, A, Löhne, A. A minimal point theorem in uniform spaces. In: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2. Dordrecht: Kluwer Academic Publishers; 2003. p. 577-593. · Zbl 1045.58013 |
| [14] | Frigon, M., On some generalizations of Ekeland’s principle and inward contractions in gauge spaces, J Fixed Point Theory Appl, 10, 279-298 (2011) · Zbl 1252.49019 |
| [15] | Fang, J-X., The variational principle and fixed point theorems in certain topological spaces, J Math Anal Appl, 202, 398-412 (1996) · Zbl 0859.54042 |
| [16] | Fierro, R., Maximality fixed points and variational principles for mappings on quasi-uniform spaces, Filomat, 31, 5345-5355 (2017) · Zbl 1499.54169 |
| [17] | Arutyunov, AV; Gel’man, BD., The minimum of a functional in a metric space, and fixed points, Zh Vychisl Mat Mat Fiz, 49, 1167-1174 (2009) · Zbl 1224.58024 |
| [18] | Arutyunov, AV., Caristi’s condition and existence of a minimum of a lower bounded function in a metric space, applications to the theory of coincidence points, Proc Steklov Inst Math, 291, 24-37 (2015) · Zbl 1336.49005 |
| [19] | Künzi, H-PA; Mršević, M.; Reilly, IL, Convergence, precompactness and symmetry in quasi-uniform spaces, Math Japon, 38, 239-253 (1993) · Zbl 0783.54022 |
| [20] | Fletcher, P.; Lindgren, WF., Quasi-uniform spaces (1982), New York (NY): M. Dekker, New York (NY) · Zbl 0501.54018 |
| [21] | Cobzaş, S. Functional analysis in asymmetric normed spaces. Basel: Birkhäuser/Springer Basel AG; 2013. (Frontiers in mathematics). · Zbl 1266.46001 |
| [22] | Künzi, H-PA. An introduction to quasi-uniform spaces. In: Mynard F, Pearl E, editors. Beyond topology. Providence (RI): American Mathematical Society; 2009. p. 239-304. (Contemp. Math.; vol. 486). · Zbl 1193.54014 |
| [23] | Kelly, JC., Bitopological spaces, Proc London Math Soc, 13, 71-89 (1963) · Zbl 0107.16401 |
| [24] | Pervin, WJ., Quasi-uniformization of topological spaces, Math Ann, 147, 316-317 (1962) · Zbl 0101.40501 |
| [25] | Kelley, JL. General topology. New York (NY): Springer-Verlag; 1975. (Graduate texts in mathematics, no. 27). · Zbl 0306.54002 |
| [26] | Reilly, IL., On generating quasi uniformities, Math Ann, 189, 317-318 (1070) · Zbl 0194.23704 |
| [27] | Reilly, IL., On quasi uniform spaces and quasi pseudo metrics, Math Chronicle, 1, part 2, 71-76 (1970) · Zbl 0215.51803 |
| [28] | Reilly, IL., Quasi-gauge spaces, J London Math Soc. (2), 6, 481-487 (1973) · Zbl 0257.54034 |
| [29] | Reilly, IL; Subrahmanyam, PV; Vamanamurthy, MK., Cauchy sequences in quasi-pseudo-metric spaces, Monatsh Math, 93, 127-140 (1982) · Zbl 0472.54018 |
| [30] | Brézis, H.; Browder, FE., A general principle on ordered sets in nonlinear functional analysis, Adv Math, 21, 355-364 (1976) · Zbl 0339.47030 |
| [31] | Cârjă, O.; Ursescu, C., The characteristics method for a first order partial differential equation, An Ştiinţ Univ Al I Cuza Iaşi Mat, 39, 4, 367-396 (1993) · Zbl 0842.34021 |
| [32] | Cârjă, O, Necula, M. Viability, invariance and applications. Amsterdam: Elsevier Science B.V.; 2017. (North-Holland Mathematics Studies, vol. 207). |
| [33] | Turinici, M., Brezis-Browder principle and dependent choice, An Ştiinţ Univ Al I Cuza Iaşi Mat (NS.), 57, 1, 263-277 (2011) · Zbl 1265.49011 |
| [34] | Takahashi, W. Existence theorems generalizing fixed point theorems for multivalued mappings. In: Thećra MA, Baillon J-B, editors. Fixed point theory and applications. (Marseille, 1989). Harlow: Longman Scientific & Technical Publisher; 1991. p. 397-406. (Pitman Res. Notes Math. Ser., vol. 252). · Zbl 0760.47029 |
| [35] | Takahashi, W., Nonlinear functional analysis. Fixed point theory and its applications (2000), Yokohama: Yokohama Publishers, Yokohama · Zbl 0997.47002 |
| [36] | Arutyunov, AV; Zhukovskiy, SF., Variational principles in analysis and existence of minimizers for functions on metric spaces, SIAM J Optim, 29, 994-1016 (2019) · Zbl 1451.58007 |
| [37] | Arutyunov, AV; Zhukovskiy, ES; Zhukovskiy, SE., Caristi-like condition and the existence of minima of mappings in partially ordered spaces, J Optim Theory Appl, 180, 48-61 (2019) · Zbl 1466.49024 |
| [38] | Oettli, W.; Théra, M., Equivalents of Ekeland’s principle, Bull Austral Math Soc, 48, 385-392 (1993) · Zbl 0793.54025 |
| [39] | Amini-Harandi, A.; Ansari, QH; Farajzadeh, AP., Existence of equilibria in complete metric spaces, Taiwanese J Math, 16, 777-785 (2012) · Zbl 1241.49012 |
| [40] | Kirk, WA; Saliga, LM., The Brézis-Browder order principle and extensions of Caristi’s theorem, Nonlinear Anal, 47, 2765-2778 (2001) · Zbl 1042.54506 |
| [41] | Goubault-Larrecq, J. Non-Hausdorff topology and domain theory: Selected topics in point-set topology. Cambridge: Cambridge University Press; 2013. (New mathematical monographs, vol. 22). · Zbl 1280.54002 |
| [42] | Lin, LJ; Wang, SY; Ansari, QH., Critical point theorems and Ekeland type variational principle with applications, Fixed Point Theory Appl, 2011 (2011) · Zbl 1213.49015 |
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