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Optimization in Science and Engineering pp 503–542Cite as

Variational Principles in Gauge Spaces

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Abstract

In Part 1, we show that the 1993 Cârjă–Ursescu Ordering Principle [An. Şt. Univ. “A. I. Cuza” Iaşi (Mat.) 39, 367–396 (1993)] is equivalent with the Dependent Choice Principle (DC); and as such, equivalent with Ekeland’s Variational Principle (EVP) [J. Math. Anal. Appl. 47, 324–353 (1974)]. This conclusion is valid for all intermediary principles; hence, in particular, for the Brezis–Browder’s (BB) [Adv. Math. 21, 355–364 (1976)]. In Part 2, it is established that the vectorial Zhu–Li Variational Principle in Fang spaces is in the logical segment between (BB) and (EVP); hence, it is equivalent with both (BB) and (EVP). In particular, the conclusion is applicable to Hamel’s Variational Principle (HVP); moreover, a proof of (HVP) ⇔ (EVP) is provided, by means of a direct approach that avoids (DC). Finally, in Part 3, we show that the gauge Brezis–Browder Principle in Turinici [Bull. Acad. Pol. Sci. (Math.) 30, 161–166 (1982)] is obtainable from (DC) and implies (EVP); hence, it is equivalent with both (DC) and (EVP). This is also true for the gauge variational principle deductible from it, including the one in Bae, Cho, and Kim [Bull. Korean Math. Soc. 48, 1023–1032 (2011)].

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References

  1. Altman, M.: A generalization of the Brezis-Browder principle on ordered sets. Nonlinear Anal. 6, 157–165 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Artzner, P., Delbean, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bae, J.-S., Cho, S.-H., Kim, J.-J.: An Ekeland type variational principle on gauge spaces with applications to fixed point theory, drop theory and coercivity. Bull. Korean Math. Soc. 48, 1023–1032 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bao, T.Q., Khanh, P.Q.: Are several recent generalizations of Ekeland’s variational principle more general than the original principle? Acta Math. Vietnam. 28, 345–350 (2003)

    MATH  MathSciNet  Google Scholar 

  5. Benbrik, A., Mbarki, A., Lahrech, S., Ouahab, A.: Ekeland’s principle for vector-valued maps based on the characterization of uniform spaces via families of generalized quasi-metrics. Lobachevskii J. Math. 21, 33–44 (2006)

    MATH  MathSciNet  Google Scholar 

  6. Bernays, P.: A system of axiomatic set theory: Part III. Infinity and enumerability analysis. J. Symb. Log. 7, 65–89 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourbaki, N.: Sur le théorème de Zorn. Archiv Math. 2, 434–437 (1949/1950)

    Google Scholar 

  8. Bourbaki, N.: General Topology (Chaps. 5–10). Springer, Berlin (1989)

    Google Scholar 

  9. Brezis, H., Browder, F.E.: A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21, 355–364 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brøndsted. A.: On a lemma of Bishop and Phelps. Pac. J. Math. 55, 335–341 (1974)

    Google Scholar 

  11. Brøndsted, A.: Fixed points and partial orders. Proc. Am. Math. Soc. 60, 365–366 (1976)

    Google Scholar 

  12. Brunner, N.: Topologische Maximalprinzipien. Zeitschr. Math. Logik Grundl. Math. 33, 135–139 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cârjă, O., Ursescu, C.: The characteristics method for a first order partial differential equation. An. Şt. Univ. “A. I. Cuza” Iaşi (Sect I-a, Mat.) 39, 367–396 (1993)

    Google Scholar 

  14. Cârjă, O., Necula, M., Vrabie, I.I.: Viability, Invariance and Applications. North Holland Mathematics Studies, vol. 207. Elsevier B.V., Amsterdam (2007)

    Google Scholar 

  15. Chang, S.S., Cho, Y.J., Lee, B.S., Jung, J.S., Kang, S.M.: Coincidence point theorems and minimization theorems in fuzzy metric spaces. Fuzzy Sets Syst. 88, 119–127 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cohen, P.J.: Set Theory and the Continuum Hypothesis. Benjamin, New York (1966)

    MATH  Google Scholar 

  17. Cristescu, R.: Topological Vector Spaces. Noordhoff Intl. Publishers, Leyden (1977)

    MATH  Google Scholar 

  18. Dodu, J., Morillon, M.: The Hahn-Banach property and the Axiom of Choice. Math. Log. Q. 45, 299–314 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  21. Fang, J.X.: The variational principle and fixed point theorems in certain topological spaces. J. Math. Anal. Appl. 202, 398–412 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Goepfert, A., Tammer, C., Zălinescu, C.: On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39, 909–922 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Goepfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Canadian Mathematical Society Books in Mathematics, vol. 17. Springer, New York (2003)

    Google Scholar 

  24. Had\(\check{\mathrm{z}}\) ić, O., Ovcin, Z.: Fixed point theorem in fuzzy metric spaces and probabilistic metric spaces. Rev. Res. Fac. Sci. Novi Sad Univ. (Math. Ser.). 24, 197–209 (1994)

    Google Scholar 

  25. Had\(\check{\mathrm{z}}\) ić, O., \(\check{\mathrm{Z}}\) ikić, T.: On Caristi’s fixed point theorem in F-type topological spaces. Novi Sad J. Math. 28, 91–98 (1998)

    Google Scholar 

  26. Hamel, A.: Variational Principles on Metric and Uniform Spaces. Habilitation Thesis, Martin-Luther University, Halle-Wittenberg, Germany (2005)

    Google Scholar 

  27. Hamel, A.: Equivalents to Ekeland’s variational principle in uniform spaces. Nonlinear Anal. 62, 913–924 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hamel, A., Loehne, A.: A minimal point theorem in uniform spaces. In: Agarwal, R.P., O’Regan, D. (eds.) Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th birthday, vol. 1, pp. 577–593. Kluwer, Dordrecht (2003)

    Google Scholar 

  29. Hicks, T.L.: Some fixed point theorems. Radovi Math. 5, 115–119 (1989)

    MATH  MathSciNet  Google Scholar 

  30. Hyers, D.H., Isac, G., Rassias, T.M.: Topics in Nonlinear Analysis and Applications. World Scientific Publishing, Singapore (1997)

    Book  MATH  Google Scholar 

  31. Isac, G.: Sur l’existence de l’optimum de Pareto. Rivista Mat. Univ. Parma (Serie IV) 9, 303–325 (1983)

    Google Scholar 

  32. Isac, G.: The Ekeland’s principle and Pareto \(\varepsilon\)-efficiency. In: Tamiz, M. (ed.) Multi-Objective Programming and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol. 432, pp. 148–163. Springer, Berlin (1996)

    Chapter  Google Scholar 

  33. Jung, C.F.K.: On generalized complete metric spaces. Bull. Am. Math. Soc. 75, 113–116 (1969)

    Article  MATH  Google Scholar 

  34. Kang, B.G., Park, S.: On generalized ordering principles in nonlinear analysis. Nonlinear Anal. 14, 159–165 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kasahara, S.: On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  36. Luxemburg, W.A.J.: On the convergence of successive approximations in the theory of ordinary differential equations (II). Indagationes Math. 20, 540–546 (1958)

    MathSciNet  Google Scholar 

  37. Mizoguchi, N.: A generalization of Brøndsted’s result and its applications. Proc. Am. Math. Soc. 108, 707–714 (1990)

    MATH  MathSciNet  Google Scholar 

  38. Moore, G.H.: Zermelo’s Axiom of Choice: Its Origin, Development and Influence. Springer, New York (1982)

    Book  Google Scholar 

  39. Moskhovakis, Y.: Notes on Set Theory. Springer, New York (2006)

    Google Scholar 

  40. Nachbin, L.: Topology and Order. D. van Nostrand Comp. Inc., Princeton (1965)

    MATH  Google Scholar 

  41. Schechter, E.: Handbook of Analysis and Its Foundation. Academic Press, New York (1997)

    Google Scholar 

  42. Tarski, A.: Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fundam. Math. 35, 79–104 (1948)

    MATH  MathSciNet  Google Scholar 

  43. Turinici, M.: A generalization of Brezis-Browder’s ordering principle. An. Şt. Univ. “A. I. Cuza” Iaşi (S I-a: Mat) 28, 11–16 (1982)

    Google Scholar 

  44. Turinici, M.: Mapping theorems via contractor directions in metrizable locally convex spaces. Bull. Acad. Pol. Sci. (Ser. Sci. Math.) 30, 161–166 (1982)

    Google Scholar 

  45. Turinici, M.: Metric variants of the Brezis-Browder ordering principle. Demonstr. Math. 22, 213–228 (1989)

    MATH  MathSciNet  Google Scholar 

  46. Turinici, M.: A monotone version of the variational Ekeland’s principle. An. Şt. Univ. “A. I. Cuza” Iaşi (S. I-a: Mat.) 36, 329–352 (1990)

    Google Scholar 

  47. Turinici, M.: Vector extensions of the variational Ekeland’s result. An. Şt. Univ. “A. I. Cuza” Iaşi (S I-a: Mat) 40, 225–266 (1994)

    Google Scholar 

  48. Turinici, M.: Minimal points in product spaces. An. Şt. Univ. “Ovidius” Constanţa (Ser. Math.) 10, 109–122 (2002)

    Google Scholar 

  49. Turinici, M.: Variational statements on KST-metric structures. An. Şt. Univ. “Ovidius” Constanţa (Mat.) 17, 231–246 (2009)

    Google Scholar 

  50. Turinici, M.: Functional variational principles and coercivity over normed spaces. Optimization 59, 199–222 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  51. Turinici, M.: Brezis-Browder principle and dependent choice. An. Şt. Univ. “Al. I. Cuza” Iaşi (S. N.) Mat. 57, 263–277 (2011)

    Google Scholar 

  52. Wolk, E.S.: On the principle of dependent choices and some forms of Zorn’s lemma. Canad. Math. Bull. 26, 365–367 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  53. Zhu, J., Li, S.J.: Generalization of ordering principles and applications. J. Optim. Theory Appl. 132, 493–507 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. “A. Myller” Mathematical Seminar, “A. I. Cuza” University, 700506, Iaşi, Romania

    Mihai Turinici

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  1. Mihai Turinici
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Correspondence to Mihai Turinici .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Princeton University Dept. Chemical Engineering, Princeton, New Jersey, USA

    Christodoulos A. Floudas

  3. Dept. Industrial & Systems Engineering, Texas A&M University, College Station, Texas, USA

    Sergiy Butenko

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Turinici, M. (2014). Variational Principles in Gauge Spaces. In: Rassias, T., Floudas, C., Butenko, S. (eds) Optimization in Science and Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0808-0_24

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