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Optimization
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Volume 72, 2023 - Issue 8
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Articles

Ekeland variational principle and its equivalents in T1-quasi-uniform spaces

Pages 2123-2154 | Received 05 Jan 2021, Accepted 03 Mar 2022, Published online: 19 Mar 2022

References

  • Ekeland I. Sur les problèmes variationnels. C R Acad Sci Paris Sér A-B. 1972;275:1057–1059.
  • Ekeland I. On the variational principle. J Math Anal Appl. 1974;47:324–353.
  • Cobzaş S. Fixed points and completeness in metric and generalized metric spaces. Fund Prikl Mat. 2018;22(1):127–215 (Russian). English translation in J Math Sci (NY.) 2020;250(3):475–535 (see also arXiv:1508.05173v4 (2016), 71 p.).
  • Ekeland I. Nonconvex minimization problems. Bull Amer Math Soc (NS.). 1979;1:443–474.
  • Al-Homidan S, Ansari QH, Kassay G. Takahashi's minimization theorem and some related results in quasi-metric spaces. J Fixed Point Theory Appl. 2019;21(1):38.
  • Bao TQ, Cobzaş S, Soubeyran A. Variational principles, completeness and the existence of traps in behavioral sciences. Ann Oper Res. 2018;269:53–79.
  • Cobzaş S. Completeness in quasi-metric spaces and Ekeland Variational Principle. Topol Appl. 2011;158:1073-–1084.
  • Cobzaş SE. Takahashi and Caristi principles in quasi-pseudometric spaces. Topol Appl. 2019;265:106831.
  • Karapinar E, Romaguera S. On the weak form of Ekeland's variational principle in quasi-metric spaces. Topol Appl. 2015;184:54–60.
  • Cobzaş S. Ekeland variational principle in asymmetric locally convex spaces. Topol Appl. 2012;159:2558–2569.
  • Hamel AH. Equivalents to Ekeland's Variational Principle in F-type topological spaces. Report no. 09. Martin-Luther-University Halle-Wittenberg, Institute of Optimization and Stochastics. 2001.
  • Hamel AH. Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913–924.
  • 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.
  • Frigon M. On some generalizations of Ekeland's principle and inward contractions in gauge spaces. J Fixed Point Theory Appl. 2011;10:279–298.
  • Fang J-X. The variational principle and fixed point theorems in certain topological spaces. J Math Anal Appl. 1996;202:398–412.
  • Fierro R. Maximality fixed points and variational principles for mappings on quasi-uniform spaces. Filomat. 2017;31:5345-–5355.
  • Arutyunov AV, Gel'man BD. The minimum of a functional in a metric space, and fixed points. Zh Vychisl Mat Mat Fiz. 2009;49:1167–1174. (Translation in Comput. Math., Math. Phys. 2009;49:1111–1118).
  • 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. 2015;291:24–37. (Translation of Tr Mat Inst Steklova. 2015;291:30–44).
  • Künzi H-PA, Mršević M, Reilly IL, et al. Convergence, precompactness and symmetry in quasi-uniform spaces. Math Japon. 1993;38:239–253.
  • Fletcher P, Lindgren WF. Quasi-uniform spaces. New York (NY): M. Dekker; 1982.
  • Cobzaş S. Functional analysis in asymmetric normed spaces. Basel: Birkhäuser/Springer Basel AG; 2013. (Frontiers in mathematics).
  • 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).
  • Kelly JC. Bitopological spaces. Proc London Math Soc. 1963;13:71–89.
  • Pervin WJ. Quasi-uniformization of topological spaces. Math Ann. 1962;147:316–317.
  • Kelley JL. General topology. New York (NY): Springer-Verlag; 1975. (Graduate texts in mathematics, no. 27).
  • Reilly IL. On generating quasi uniformities. Math Ann. 1070;189:317–318.
  • Reilly IL. On quasi uniform spaces and quasi pseudo metrics. Math Chronicle. 1970;1(part 2):71–76.
  • Reilly IL. Quasi-gauge spaces. J London Math Soc. (2). 1973;6:481–487.
  • Reilly IL, Subrahmanyam PV, Vamanamurthy MK. Cauchy sequences in quasi-pseudo-metric spaces. Monatsh Math. 1982;93:127–140.
  • Brézis H, Browder FE. A general principle on ordered sets in nonlinear functional analysis. Adv Math. 1976;21:355–364.
  • Cârjă O, Ursescu C. The characteristics method for a first order partial differential equation. An Ştiinţ Univ Al I Cuza Iaşi Mat. 1993;39(4):367–396.
  • Cârjă O, Necula M. Viability, invariance and applications. Amsterdam: Elsevier Science B.V.; 2017. (North-Holland Mathematics Studies, vol. 207).
  • Turinici M. Brezis-Browder principle and dependent choice. An Ştiinţ Univ Al I Cuza Iaşi Mat (NS.). 2011;57(1):263–277.
  • 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).
  • Takahashi W. Nonlinear functional analysis. Fixed point theory and its applications. Yokohama: Yokohama Publishers; 2000.
  • Arutyunov AV, Zhukovskiy SF. Variational principles in analysis and existence of minimizers for functions on metric spaces. SIAM J Optim. 2019;29:994–1016.
  • Arutyunov AV, Zhukovskiy ES, Zhukovskiy SE. Caristi-like condition and the existence of minima of mappings in partially ordered spaces. J Optim Theory Appl. 2019;180:48–61.
  • Oettli W, Théra M. Equivalents of Ekeland's principle. Bull Austral Math Soc. 1993;48:385–392.
  • Amini-Harandi A, Ansari QH, Farajzadeh AP. Existence of equilibria in complete metric spaces. Taiwanese J Math. 2012;16:777–785.
  • Kirk WA, Saliga LM. The Brézis-Browder order principle and extensions of Caristi's theorem. Nonlinear Anal. 2001;47:2765–2778.
  • 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).
  • Lin LJ, Wang SY, Ansari QH. Critical point theorems and Ekeland type variational principle with applications. Fixed Point Theory Appl. 2011;2011:914624.

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