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Ekeland-Type Variational Principles and Equilibrium Problems

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Topics in Nonconvex Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIANOIA,volume 50))

Abstract

In this survey chapter we present different forms of Ekeland’s variational principle involving τ-functions, τ-functions and fitting functions, and Q-functions, respectively. The equilibrium version of Ekeland-type variational principle is also presented. We give some equivalences of our variational principles with the Caristi—Kirk-type fixed point theorem for multivalued maps, Takahashi minimization theorem, and some other related results. As applications of our results, we derive the existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. The results of this chapter extend and generalize many results that recently appeared in the literature.

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References

  1. Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 69, 126–139 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. Amemiya, M., Takahashi, W.: Fixed point theorems for fuzzy mappings in complete metric spaces. Fuzzy Sets Syst. 125, 253–260 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  3. Ansari, Q.H., Lin, L.J., Su, L.B.: Systems of simultaneous generalized vector quasiequilibrium problems and their applications. J. Optim. Theory Appl. 127 ( 1 ), 27–44 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ansari, Q.H., Wong, N.C., Yao, J.C.: The existence of nonlinear inequalities. Appl. Math. Lett. 12 ( 5 ), 89–92 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  5. Aubin, J.-P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam, New York, Oxford (1979).

    Google Scholar 

  6. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. John Wiley, New York (1984).

    MATH  Google Scholar 

  7. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990).

    MATH  Google Scholar 

  8. Bae, J.S.: Fixed point theorems for weakly contractive multivalued maps. J. Math. Anal. Appl. 284, 690–697 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  9. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  10. Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Global Optim. 30, 121–134 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  11. Bianchi, M., Kassay, G., Pini, R.: Existence of equilibria via Ekeland’s principle. J. Math. Anal. Appl. 305, 502–512 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  12. Blum, E., Oettli,W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63 (1–4), 123–145 (1994).

    MathSciNet  MATH  Google Scholar 

  13. Bosch, C., Garcia, A., Garcia, C.L.: An extension of Ekeland’s variational principle to locally complete spaces. J. Math. Anal. Appl. 328, 106–108 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. Brézis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Un. Mat. Ital. 6, 293–300 (1972).

    MathSciNet  MATH  Google Scholar 

  15. Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215, 241–251 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  16. Caristi, J., Kirk, W.A.: Geometric ixed point theory and inwardness conditions. In: The Geonetry of Metric and Linear Spaces, Lecture Notes in Mathematics, 490, pp. 74–83. Springer-Verlag, Berlin (1975).

    Google Scholar 

  17. Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  18. Chadli, O., Wong, N.C., Yao, J.C.: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117, 245–266 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, Y., Cho, Y.J., Yang, L.: Note on the results with lower semi-continuity. Bull. Korean Math. Soc. 39 (4), 535–541 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  20. Daneš, J.: A geometric theorem useful in nonlinear analysis. Boll. Un. Mat. Ital. 6, 369–372 (1972).

    MathSciNet  MATH  Google Scholar 

  21. Daneš, J.: Equivalence of some geometric and related results of nonlinear functional analysis. Commentat. Math. Univ. Carolinae 26, 443–454 (1985).

    MATH  Google Scholar 

  22. De Figueiredo, D.G.: The Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989).

    MATH  Google Scholar 

  23. Ekeland, I.: Sur les problèms variationnels. C. R. Acad. Sci. Paris 275, 1057–1059 (1972).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Ekeland, I.: On convex minimization problems. Bull. Amer. Math. Soc. 1 ( 3 ), 445–474 (1979).

    Article  MathSciNet  Google Scholar 

  26. Facchinei, F., Pang, J.-S.: Finite Dimensional Variational Inequalities and Complementarity Problems, I. Springer-Verlag, New York (2003).

    Google Scholar 

  27. Fakhar, M., Zafarani, J.: Generalized equilibrium problems for quasimonotone and pseudomonotone bifunctions. J. Optim. Theory Appl. 123, 349–364 (2004).

    Article  MathSciNet  Google Scholar 

  28. Fakhar, M., Zafarani, J.: Equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 126, 125–136 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  29. Fan, K.: A minimax inequlity and applications, In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Feng, Y.Q., Liu, S.Y.: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings. J. Math. Anal. Appl. 317, 103–112 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  32. Flores-Bazán, F.: Existence theorems for generalized noncoercive equilibrium problems: The quasi-convex case. SIAM J. Optim. 11, 675–690 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  33. Flores-Bazán, F.: Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77, 249–297 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  34. Georgiev, P.G.: The strong Ekeland variational principle, the strong drop theorem and applications. J. Math. Anal. Appl. 131, 1–21 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  35. Göfert, A., Tammer, Chr., Riahi, H., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer-Verlag, New York (2003).

    Google Scholar 

  36. Hadjisavvas, H., Komlósi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer-Verlag, Berlin (2005).

    Book  MATH  Google Scholar 

  37. Hadjisavvas, H., Schaible, S.: From scalar to vector equilibrium problems in quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  38. Hamel, A.: Remarks to an equivalent formulation of Ekeland’s variational principle. Optimization 31 (3), 233–238 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  39. Hamel, A.: Equivalents to Ekeland’s variational principle in F-type topological spaces. Report of the Institute of Optimization and Stochastics 9, Martin Luther University, Halle-Wittenberg (2001).

    Google Scholar 

  40. Hamel, A.: Phelps’ lemma, Daneš’ drop theorem and Ekeland principle in locally convex spaces. Proc. Amer. Math. Soc. 131, 3025–3038 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  41. Hamel, A.: Equivalents to Ekeland’s variational principle in uniform spaces. Nonlinear Anal. 62(5), 913–924 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  42. Isac. G.: Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces. Optimization 53, 253–268 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  43. Isac, G.: Ekeland principle and nuclear cones: A geometrical aspect. Math. Computat.Modell. 26 (11), 111–116 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  44. Jing-Hui, Q.: Ekeland’s variational principle in locally complete spaces. Math. Nachr. 257, 55–58 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  45. Kada, O., Suzuki, T., Takahashi, W.: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Japonica 44 ( 2 ), 381–391 (1996).

    MathSciNet  MATH  Google Scholar 

  46. Kas, P., Kassay, G., Sensoy, Z.B.: On generalized equilibrium points. J. Math. Anal. Appl 296, 619–633 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  47. Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. John Willy & Sons, New York (2001).

    Google Scholar 

  48. Lin, L.J.: System of coincidence theorems with applications. J. Math. Anal. Appl. 285, 408–418 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  49. Lin, L.J., Du, W.S.: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 323, 360–370 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  50. Lin, L.J., Du, W.S.: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal. 68, 1246–1262 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  51. McLinden, L.: An application of Ekeland’s theorem to minimax problems. Nonlinear Anal. Theory, Meth. Appl. 6, 189–196 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  52. Nadler, S.B.Jr.: Multivalued contraction mappings. Pacific J. Math. 30, 475–488 (1969).

    MathSciNet  MATH  Google Scholar 

  53. Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pacific J. Math. 5, 807–815 (1955).

    MathSciNet  Google Scholar 

  54. Oettli, W., Théra, M.: Equivalents of Ekeland’s principle. Bull. Austral. Math. Soc. 48, 385–392 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  55. Park, S.: Equivalent formulations of Ekeland’s variational principle for approximate solutions of minimization problems and their applications. In: S.P. Singh, V.M. Sehgal, and J.H.W. Burry (Eds.) Operator Equations and Fixed Point Theorems, pp. 55–68. MSRI-Korea Publishers 1, Soul (1986).

    Google Scholar 

  56. Park, S.: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. 39, 881–889 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  57. Penot, J.-P.: The drop theorem, the petal theorem and Ekeland’s variational principle. Nonlinear Anal. Theory, Meth. Appl. 10, 813–822 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  58. Rockafellar, R.T.: Directionally Lipschitz functions and subdifferential calculus. Proc. London Math. Soc. 39, 331–355 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  59. Sullivan, F.: A characterization of complete metric spaces. Proc. Amer. Math. Soc. 83, 345–346 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  60. Shioji, N., Suzuki, T., Takahashi, W.: Contractive mappings, Kannan mappings and metric completeness. Proc. Amer. Math. Soc. 126, 3117–3124 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  61. Suzuki, T.: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253, 440–458 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  62. Suzuki, T.: On Downing-Kirk’s theorem. J. Math. Anal. Appl. 286, 453–458 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  63. Suzuki, T.: Generalized Caristi’s fixed point theorems by Bae and others. J. Math. Anal. Appl. 302, 502–508 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  64. Suzuki, T.: The strong Ekeland variational principle. J. Math. Anal. Appl. 320, 787–794 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  65. Suzuki, T., Takahashi, W.: Fixed point theorems and characterizations ofmetric completeness. Top. Meth. Nonlinear Anal. 8, 371–382 (1996).

    MathSciNet  MATH  Google Scholar 

  66. Takahashi, W.: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Baillon, J.-B., Théra, M. (Eds.) Fixed Point Theory and Applications, pp. 397–406. Pitman Research Notes in Mathematics, 252, Longman, Harlow (1990).

    Google Scholar 

  67. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000).

    MATH  Google Scholar 

  68. Tataru, D.: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163, 345–392 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  69. Yuan, G.X.-Z.: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker, New York (1999).

    MATH  Google Scholar 

  70. Zhong, C.-K.: On Ekeland’s variational principle and a minimax theorem. J. Math. Anal. Appl. 205, 239–250 (1997).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, 202 002, Aligarh, India

    Qamrul Hasan Ansari

  2. Department of Mathematics, National Changhua University for Education, 50058, Changhua, Taiwan, ROC

    Lai-Jiu Lin

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  1. Qamrul Hasan Ansari
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  2. Lai-Jiu Lin
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Correspondence to Qamrul Hasan Ansari .

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Editors and Affiliations

  1. Faculty of Science, Dept. of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    Shashi Kant Mishra

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Ansari, Q.H., Lin, LJ. (2011). Ekeland-Type Variational Principles and Equilibrium Problems. In: Mishra, S. (eds) Topics in Nonconvex Optimization. Springer Optimization and Its Applications(), vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9640-4_10

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