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On the Existence of Equilibria and Fixed Points of Maps under Constraints

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Handbook of Topological Fixed Point Theory

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References

  1. R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, 1992.

    Google Scholar 

  2. N. Aronszajn, Le correspondant topologique de l'unicite dans la théorie des équations differentielles, Ann. of Math. 43 (1942), 730–738.

    Article  MATH  MathSciNet  Google Scholar 

  3. J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley—Interscience, New York, 1987.

    Google Scholar 

  4. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston—Basel—Berlin, 1990.

    MATH  Google Scholar 

  5. R. Bader, The periodic problem for semilinear differential inclusions in Banach spaces, Comment. Math. Univ. Carolin. 39 (1998), 671–684.

    MATH  MathSciNet  Google Scholar 

  6. R. Bader and W. Kryszewski, On the solution set of constrained differential inclusions with applications, Set-Valued Anal. 9 (2001), 289–313.

    Article  MathSciNet  MATH  Google Scholar 

  7. _____, On the solution sets of differential inclusions and the periodic problem in Banach spaces, Nonlinear Anal. 54 (2003), 707–754.

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Barbu, Continuous perturbations of nonlinear m-accretive operators in Banach spaces, Boll. Un. Mat. Ital. (4) 6 (1972), 270–278.

    MATH  MathSciNet  Google Scholar 

  9. H. Ben-El-Mechaiekh, P. Deguire and A. Granas, Une alternative non linéaire en analyse convexe et applications, C. R. Acad. Sci. Paris Sér. I 295 (1982), 257–259.

    MathSciNet  Google Scholar 

  10. _____, Points fixex et coincidences pour les applications multivoque II, C. R. Acad. Sci. Paris Sér. I 295 (1982), 381–384.

    MathSciNet  Google Scholar 

  11. H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997), 4159–4179.

    Article  MathSciNet  MATH  Google Scholar 

  12. _____, Equilibrium for perturbations of multifunctions by convex processes, Georegian Math. J. 3 (1996), 201–215.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Bessaga and A. Pe′czyński, Selected topics in infinite-dimensional topology, Monografie Mat., vol. 58, PWN, Warszawa, 1975.

    Google Scholar 

  14. J.-M. Bonnisseau and B. Cornet, Fixed point theorems and Morse's lemma for lipschitzian functions, J. Math. Anal. Appl. 146 (1990), 318–332.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Borsuk, Theory of Retracts, Monografie Mat., vol. 44, PWN, Warszawa, 1967.

    Google Scholar 

  16. D. Bothe, Multivalued differential equations on graphs and applications, Ph. D. dissertation, Universität Paderborn (1992).

    Google Scholar 

  17. _____, Flow invariance for perturbe nonlinear evolution equations, Abstr. Appl. Anal. 1 (1996), 379–395.

    Google Scholar 

  18. H. F. Bohnenblust and S. Karlin, On a theorem of Ville, Contributions to the Game Theory, vol. 1, Princeton, 1950.

    Google Scholar 

  19. H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math. 23 (1970), 261–263.

    MATH  MathSciNet  Google Scholar 

  20. R. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Comp., Glenview Ill., London, 1971.

    MATH  Google Scholar 

  21. F. E. Browder, The fixed point theory for multivalued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301.

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Cardaliaguet, G. Gabor and M. Quincampoix, Equilibria and strict equilibria of multivalued maos on noninvariant sets, Ann. Polon. Math. 82 (2003), 19–37.

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. K. C. Chang, Variational methods for nondifferentiable functionals and its applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.

    Article  MATH  MathSciNet  Google Scholar 

  25. _____, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.

    Google Scholar 

  26. F. H. Clarke, Pointwise contractional criteria for the existence of fixed points, Bull. Canad. Math. Soc. 21 (1978), 7–11.

    MATH  Google Scholar 

  27. _____, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.

    Google Scholar 

  28. F. H. Clarke, Yu. S. Ledyaev and R. J. Stern, Fixed points and equilibria in nonconvex sets, Nonlinear Anal. 25 (1995), 145–161.

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Colombo and V. V. Goncharov, Variational inequalities and regularity properties of closed sets in Hilbert spaces, J. Convex Anal. 8 (2001), 197–221.

    MathSciNet  MATH  Google Scholar 

  30. B. Cornet, Paris avec handicaps et théoremes de surjectivité de correspondances, C. R. Acad. Sci. Paris Sér. A 281 (1975, 479–482).

    MATH  MathSciNet  Google Scholar 

  31. _____, Euler characteristic and fixed point theorem, Positivity 6 (2002), 243–260.

    Article  MATH  MathSciNet  Google Scholar 

  32. B. Cornet and M. O. Czarnecki, Necessary and sufficient conditions for the existence of (generalized) equilibria on compact epi-Lipschitzian domain, Comm. Appl. Nonlinear Anal. 7 (2000), 21–53.

    MathSciNet  MATH  Google Scholar 

  33. M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298.

    MathSciNet  MATH  Google Scholar 

  34. A. Ćwiszewski, Zagadnienia ró.zniczkowe z ograniczeniami na stan; stopień topologiczny zaburzeń operatorw akretywnych, Ph. D. Thesis (2003), Nicolaus Copernicus University, Toruń.

    Google Scholar 

  35. A. Ćwiszewski and W. Kryszewski, Equilibria of set-valued maps: a variational approach, Nonlinear Anal. 48 (2002), 707–746.

    Article  MathSciNet  MATH  Google Scholar 

  36. _____, Approximate smoothing of locally Lipschitz functionals, Boll. Un. Mat. Ital. B 5 (2002), 289–320.

    MATH  Google Scholar 

  37. _____, Degree theory for tangent set-valued vector fields on closed sets, submitted.

    Google Scholar 

  38. K. Deimling, Fixed points of weakly inward multis, Nonlinear Anal. 10 (1986), 1261–1262.

    Article  MATH  MathSciNet  Google Scholar 

  39. _____, Multivalued differential equations, Walter de Gruyter, Berlin, New York, 1992.

    Google Scholar 

  40. K. Deimling, S. C. Hu and J. Prüss, Fixed points of weakly inward multivalued maps, Nonlinear Anal. 10 (1986), 465–469.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in L 1(μ,X), Proc. Amer. Math. Soc. 118 (1993), 447–453.

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Donchev, Properties of one-sided Lipschitz multivalued maps, Nonlinear Anal. 49 (2002), 13–20.

    Article  MATH  MathSciNet  Google Scholar 

  43. D. Downing and W. A. Kirk, Fixed point theorems for set-valued mappings in metric and Banach spaces, Math. Japonica 22 (1977), 99–112.

    MathSciNet  MATH  Google Scholar 

  44. R. Dragoni, J.W. Macki, P. Nistri and P. Zecca, Solution Sets of Differential Operators in Abstract Spaces, Addison Wesley Longman Ltd., Harlow, 1996.

    Google Scholar 

  45. J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mat., vol. 61, PWN, Warszawa, 1982.

    Google Scholar 

  46. K. Fan, Fixed points and minimax theorems in locally convex topological spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126.

    MATH  MathSciNet  Google Scholar 

  47. _____, Extensions of two fixed point theorem of F. E. Browder, Math. Z. 112 (1969), 234–240.

    Article  MATH  MathSciNet  Google Scholar 

  48. _____, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519–537.

    Article  MATH  MathSciNet  Google Scholar 

  49. G. Gabor and M. Quincampoix, On existence of equilibria of set-valued maps, Boll. Un. Mat. Ital. B (8) 6 (2003), 309–321.

    MathSciNet  MATH  Google Scholar 

  50. I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170–174.

    Article  MATH  MathSciNet  Google Scholar 

  51. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory (1990), Cambridge Univ. Press, Cambridge.

    MATH  Google Scholar 

  52. L. Górniewicz, Homological methods in fixed point theory of multi-valued maps, Dissertationes Math. 129 (1976), 1–66.

    Google Scholar 

  53. _____, Topological Fixed Point Theory, Kluwer, 1999.

    Google Scholar 

  54. _____, Topological structure of solution sets: Current results, Arch. Math. (Brno) 36 (2000), 343–382.

    MATH  Google Scholar 

  55. B. Halpern, Fixed point theorem for outward maps, Ph. D. Thesis (1965), UCLA.

    Google Scholar 

  56. _____, Fixed point theorems for set-valued maps in infinite dimensional spaces, Math. Ann. 189 (1970), 87–98.

    Google Scholar 

  57. _____, A general coincidence theory, Pacific J. Math. 77 (1978), 451–471.

    Google Scholar 

  58. B. R. Halpern and G. M. Berman, A fixed point theorem for inward and outward maps, Trans. Amer. Mat. Soc. 62 (1968), 353–358.

    Article  Google Scholar 

  59. _____, A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353–358.

    Article  MathSciNet  MATH  Google Scholar 

  60. P. Hartman, Ordinary differential eqiuations (1982), Birkhaüser, Boston.

    Google Scholar 

  61. C. Himmelberg and F. Van Vleck, On the topological triviality of solution sets, Rocky Mountain J. Math. 10 (1980), 247–252.

    MathSciNet  MATH  Google Scholar 

  62. _____, A note on the solution sets of differential inclusions, Rocky Mountain J. Math. 12 (1982), 621–625.

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Hirsch, Springer-Verlag, New York, Heidelberg, Berlin (1976).

    Google Scholar 

  64. S. Hu and N. S. Papageorgiou, Handbook of Set-Valued Analysis, vol. I, Kluwer Academic Publishers, 1999; vol. II, 2001.

    Google Scholar 

  65. S. Hu and Y. Sun, Fixed point index for weakly inward maps, J. Math. Anal. Appl. 172 (1993), 266–273.

    Article  MathSciNet  MATH  Google Scholar 

  66. D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91–97.

    MATH  MathSciNet  Google Scholar 

  67. M. KamenskiÄ­, V. ObukhovskiÄ­ and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, Walter de Gruyter (to appear).

    Google Scholar 

  68. S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke J. Math. 8 (1941), 457–459.

    Article  MATH  MathSciNet  Google Scholar 

  69. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508–520.

    Article  MATH  MathSciNet  Google Scholar 

  70. W. A. Kirk, History and methods of metric fixed point theory, Antipodal Points and Fixed Points, Lecture Notes, vol. 28, Seul National University, Research Institute of Mathematics, 1995.

    Google Scholar 

  71. _____, Transfinite methods in metric fixed-point theory, Abstr. Appl. Anal. 5 (2003), 311–324.

    Article  MATH  MathSciNet  Google Scholar 

  72. W. A. Kirk and S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990), 357–364.

    MathSciNet  MATH  Google Scholar 

  73. H. N. Ko, Fixed points theorems for point-to-set mappinga and the set of fixed points, Pacific J. Math. 42 (1972), 369–379.

    MATH  MathSciNet  Google Scholar 

  74. M. A. Krasnosel'skiÄ­ and P. P. ZabreÄ­ko, Geometrical Methods in Nonlinear Analysis, Nauka, Moscow, 1975. (Russian)

    Google Scholar 

  75. W. Kryszewski, Topological and approximation methods in the degree theory of set-valued maps, Diss. Math. 336 (1994), 1–102.

    MathSciNet  Google Scholar 

  76. _____, Homotopy properties of set-valued mappings, Habilitation Thesis (1997), Nicolaus Copernicus University, Torunń.

    Google Scholar 

  77. E. Lami Dozo, Multivalued nonexpansive mappings and Opial's condition, Proc. Amer. Math. Soc. 38 (1973), 286–292.

    Article  MATH  MathSciNet  Google Scholar 

  78. M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151–201.

    Article  MATH  MathSciNet  Google Scholar 

  79. T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 124 (1996), 3345–3349.

    MATH  Google Scholar 

  80. _____, A fixed point theorem for weakly inward multivalued contractions, J. Math. Anal. Appl. 247 (2000), 323–327.

    Article  MATH  MathSciNet  Google Scholar 

  81. R. H. Martin, Differential equations on closed sets in Banach spaces, Trans. Amer. Math. Soc. 179 (1973), 399–414.

    Article  MATH  MathSciNet  Google Scholar 

  82. R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976..

    MATH  Google Scholar 

  83. C. Martinez-Yanez, A remark on weakly inward multivalued mappings, Nonlinear Anal. 16 (1991), 847–848.

    Article  MATH  MathSciNet  Google Scholar 

  84. G. Minty, A theorem on monotone sets in Hibert spaces, J. Math. Anal. Appl. 11 (1967), 434–439.

    Article  MathSciNet  Google Scholar 

  85. N. Mizoguchi and W. Takahashi, Fixed point for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989), 177–188.

    Article  MathSciNet  MATH  Google Scholar 

  86. D. Motreanu and N. H. Pavel, Tangency, flow invariance for differential equations and optimization problems (1999), Marcel Dekker Inc., New York.

    MATH  Google Scholar 

  87. S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.

    MATH  MathSciNet  Google Scholar 

  88. D. O'Regan, A continuation theory for weakly inward maps, Glasgow Math. J. 40 (1998), 311–321.

    Article  MATH  MathSciNet  Google Scholar 

  89. _____, Fixed points and random fixed points for weakly inward approximable maps, Proc. Amer. Math. Soc. 38 (1999), 89–100.

    MATH  Google Scholar 

  90. S. Park, Eighty years of the Brouwer fixed point theorem, Lecture Notes, vol. 28, Seul National University, Research Institute of Mathematics, 1995.

    Google Scholar 

  91. A. Pazy, Springer-Verlag 1983.

    Google Scholar 

  92. J.-P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mém. 60 (1979), 129–152.

    MATH  MathSciNet  Google Scholar 

  93. S. Plaskacz, On the solution sets of differential inclusions, Boll. Un. Mat. Ital. A (7) 6 (1992), 387–394.

    MATH  MathSciNet  Google Scholar 

  94. R. Precup, On some fixed point theorems of Deimling, Nonlinear Anal. 23 (1994), 1315–1320.

    Article  MATH  MathSciNet  Google Scholar 

  95. S. Reich, Fixed points in locally convex spaces, Math. Z. 125 (1972), 17–31.

    Article  MATH  MathSciNet  Google Scholar 

  96. _____, The range and sums of accretive and monotone operators, J. Math. Anal. Appl. 68 (1979), 310–317.

    Article  MATH  MathSciNet  Google Scholar 

  97. _____, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973), 460–467.

    Article  MATH  MathSciNet  Google Scholar 

  98. _____, Approximate selections, best approximations, fixed point and invariant sets, J. Math. Anal. Appl. 62 (1978), 104–113.

    Article  MATH  MathSciNet  Google Scholar 

  99. R. T. Rockafellar, Clarke's tangent cones and boundaries of closed sets in \(\mathbb{R}^n \) , Nonlinear Analysis 3 (1070), 145–154.

    Article  MathSciNet  Google Scholar 

  100. P. Saveliev, A Lefschetz type coincidence theorem.

    Google Scholar 

  101. M. Sion, On general minimax theorem, Pacific J. Math. (1958), 171–176.

    Google Scholar 

  102. G. V. Smirnov, Introduction to the theory of differential inclusions, Grad. Texts in Maths. 41 (2002), Amer. Math. Soc. Providence.

    Google Scholar 

  103. W. Song, A generalization of Clarke's fixed point theorem, Appl. Math. J. Chinese Univ. Ser. 10 B (1995), 13–23.

    Google Scholar 

  104. E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966; Glenview, Illinois, 1972.

    MATH  Google Scholar 

  105. R. J. Stern, On zeros of multifunctions in non-lipschitzian sets, manuscript (1996).

    Google Scholar 

  106. D. W. Walkup and R. Wets, Continuity of some convex-cone-valued mapping, Proc. Amer. Math. Soc. 18, 229–235.

    Google Scholar 

  107. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

    MATH  Google Scholar 

  108. H.-K. Xu, Metric fixed point theory for multivalued mappings, Diss. Math. 389 (2000), 1–39.

    Article  Google Scholar 

  109. _____, Metric fixed point theory for multivalued mappings, Diss. Math. 389 (2000), 1–39.

    Google Scholar 

  110. _____, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal. 43 (2001), 693–706.

    Article  MATH  MathSciNet  Google Scholar 

  111. K. Yanagi, On some fixed point theorems for multivalued mappings, Pacific. J. Math. 87 (1980), 233–240.

    MATH  MathSciNet  Google Scholar 

  112. K. Yosida, Functinal Analysis, Springer-Verlag, Berlin, 1966.

    Google Scholar 

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Authors
  1. Wojciech Kryszewski
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Editors and Affiliations

  1. University of California, Los Angeles, USA

    R. F. Brown

  2. Department of Applied Mathematics ‘G. Sansone', Florence, Italy

    M. Furi

  3. Juliusz Schauder Center of the Nicolaus Copernicus University, Poland

    L. Górniewicz

  4. Department of Mathematics, Peking University, Beijing, China

    B. Jiang

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Kryszewski, W. (2005). On the Existence of Equilibria and Fixed Points of Maps under Constraints. In: Brown, R.F., Furi, M., Górniewicz, L., Jiang, B. (eds) Handbook of Topological Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3222-6_21

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