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Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices

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Handbook of Metric Fixed Point Theory

Abstract

The aim of this chapter is to present criteria for the most important geometric properties related to the metric fixed point theory in some classes of Banach function lattices, mainly in Orlicz spaces and Cesaro sequence spaces. We also give some informations about respective results for Musielak-Orlicz spaces, Orlicz-Lorentz spaces and Calderón-Lozanovsky spaces.

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References

  1. A.G. Aksoy and M.A. Khamsi, Nonstandard methods in fixed point theory, Springer-Verlag, 1990.

    Google Scholar 

  2. G. Alherk and H. Hudzik, Uniformly non-l’ Musielak-Orlicz spaces of Bochner type, Forum Math. 1 (1989), 403–410.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.M. Ayerbe and T. Dominguez Benavides, Connections between some Banach space coefficients concerning normal structure, J. Math. Anal. Appl. 172 (1993), 53–61.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.M. Ayerbe, T. Dominguez Benavides and G. Lopez Acedo, Measures of noncompactness in metric fixed point theory, Birkhäuser Verlag, 1997.

    Google Scholar 

  5. J. Banal, On modulus of noncompact convexity and its properties, Cand. Math. Bull. 30 (2) (1987), 186–192.

    Article  MathSciNet  Google Scholar 

  6. J. Banal, Compactness conditions in the geometric theory of Banach spaces, Nonlinear Anal. 16 (7–8) (1991), 669–682.

    MathSciNet  Google Scholar 

  7. J.M. Borwein and B. Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math. 10 (1984), 339–356.

    MathSciNet  MATH  Google Scholar 

  8. Br 651 F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965), 1041–1044.

    Article  MathSciNet  MATH  Google Scholar 

  9. W.L. Bynum, Normal structure coefficient for Banach spaces, Pacific J. Math. 86 (1980), 427–436.

    MathSciNet  MATH  Google Scholar 

  10. J. Gerda, H. Hudzik, A Kaminska and M. Mastylo, Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces, Positivity, (1998), 311–337.

    Google Scholar 

  11. J. Gerda, H. Hudzik and M. Mastylo, On the geometry of some Calderón-Lozanovskii spaces, Indag. Math. (N.S.) 6 (1995), 35–49.

    Article  Google Scholar 

  12. S.T. Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.) 356, 1996.

    Google Scholar 

  13. S.T. Chen, H. Hudzik and H. Sun, Complemented copies of t l in Orlicz spaces, Math. Nachr. 158 (1992), 299–309.

    Article  MathSciNet  Google Scholar 

  14. S.T. Chen and B.-L. Lin, Banach spaces with weakly normal structure, in: Lecture Notes in Pure and Appl. Math. 213, Marcel Dekker, 2000, 115–132.

    Google Scholar 

  15. S.T. Chen and H. Sun, Reflexive Orlicz spaces have uniformly normal structure, Studia Math. 109 (1994), 197–208.

    MathSciNet  MATH  Google Scholar 

  16. Cu-II 98a1 Y.A. Cui and H. Hudzik, Maluta coefficient in Musielak-Orlicz sequence spaces equipped with the Orlicz norm,Proc. Amer. Math. Soc. 126 (1) (1998), 115–121.

    Google Scholar 

  17. Y.A. Cui and H. Hudzik, On the uniform Opial property in some modular sequence spaces, Funct. Approx. Comment. Math. 26 (1998), 93–102.

    MathSciNet  MATH  Google Scholar 

  18. Y.A. Cui and H. Hudzik, Maluta’s coefficient and Opial’s properties in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, Nonlinear Anal. 35 (1999), 475–485.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y.A. Cui and H. Hudzik, On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces, Acta Sci. Math. (Szeged) 65 (1–2) (1999), 179–187.

    MathSciNet  MATH  Google Scholar 

  20. Y.A. Cui and H. Hudzik, Some geometric properties related to fixed point theory in Cesóro spaces, Collect. Math. 50 (3) (1999), 277–288.

    MathSciNet  MATH  Google Scholar 

  21. Y.A. Cui and H. Hudzik, On the Garcia-Falset coefficient in Orlicz sequence spaces equipped with the Orlicz norm, in: “Function Spaces and Applications”, D.E. Edmunds et al, Narosa Publishing House, New Delhi, 2000, 60–68.

    Google Scholar 

  22. Y.A. Cui and H. Hudzik, Packing constant for Cesaro sequence spaces, in Nonlinear Analysis-Procced. of the Third World Congress on Nonlinear Analysis, Catania, 2000.

    Google Scholar 

  23. Y.A. Cui, H. Hudzik and Y. Li, On the Garcia-Falset coefficient in some Banach sequence spaces, in: Lecture Notes in Pure and Appl. Math. 213, Dekker, 2000, 141–148.

    Google Scholar 

  24. Cu-H-Li] Y. Cui, H. Hudzik and Y.Li, On k-nearly uniform convexity in Orlicz spaces,submitted.

    Google Scholar 

  25. Y.A. Cui, H. Hudzik, M. Nowak and R. Pluciennik, Some geometric properties in Orlicz sequence spaces equipped with the Orlicz norm, J. Convex Anal. 6 (1) (1999), 91–113.

    MathSciNet  MATH  Google Scholar 

  26. Y.A. Cui, H. Hudzik and R. Pluciennik, Banach-Saks property in some Banach sequence spaces, Ann. Polon. Math. 65 (2) (1997), 193–202.

    MathSciNet  MATH  Google Scholar 

  27. Y.A. Cui, H. Hudzik and R. Pluciennik, On some uniform convexities and smoothness in certain sequence spaces, Z. Anal. Anwendungen 17 (4) (1998), 893–905.

    MathSciNet  MATH  Google Scholar 

  28. Y.A. Cui, H. Hudzik and R. Pluciennik, Weak orthogonality and weak property (ß) in some Banach sequence spaces, Czechoslovak Math. J. 49(124) (2) (1999), 303–316.

    Google Scholar 

  29. Cu-H-W] Y.A. Cui, H. Hudzik and T. Wang, Criteria for monotonicity properties in Musielak-Orlicz sequence spaces equipped with the Orlicz norm,to appear.

    Google Scholar 

  30. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.

    Google Scholar 

  31. D-H-L-M-S] T. Dominguez Benavides, H. Hudzik, G. López Acedo, M. Mastylo and B. Sims, Complete characterizations of Kadec-Klee properties in Orlicz spaces,to appear.

    Google Scholar 

  32. P.N. Dowling and C.J. Lennard, Every nonreflexive subspace of L1[0,1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (2) (1997), 443–446.

    Article  MathSciNet  MATH  Google Scholar 

  33. P.N. Dowling, C.J. Lennard and B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 653–662.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Foralewski and H. Hudzik, On some basic properties of generalized Calderon–Lozanovsky spaces, Collect. Math. 48 (4–5–6) (1997), 523 – 538.

    MathSciNet  MATH  Google Scholar 

  35. P. Foralewski and H. Hudzik, On some geometrical and topological properties of generalized Calderón-Lozanowskii sequence spaces, Houston J. Math. 25 (3) (1999), 523–542.

    MathSciNet  MATH  Google Scholar 

  36. C. Franchetti, Duality mapping and homomorphisms in Banach theory, Proceed. of Research Workshop on Banach Space Theory, University of Iowa, 1981.

    Google Scholar 

  37. J. Garcia-Falset, Stability and fixed points for nonexpansive mappings, Houston J. Math. 20 (1994), 495–505.

    MATH  Google Scholar 

  38. J. Garcia-Falset, The fixed point property in Banach spaces with NUS property, J. Math. Anal. Appl. 215 (1997), 532–542.

    Article  MathSciNet  MATH  Google Scholar 

  39. A.A. Gillespire and B.B. Williams, Fixed point theorem for Banach spaces with uniformly normal structure, Appl. Anal. 9 (1979), 121–124.

    Google Scholar 

  40. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.

    Google Scholar 

  41. K. Goebel and T. Sgkowski, The modulus of noncompact convexity, Ann. Univ. Mariae Curie-Sklodowska 38 (1984), 41–48.

    MATH  Google Scholar 

  42. D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258.

    Article  MathSciNet  MATH  Google Scholar 

  43. J.P. Cossez and E. Lauri Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565–573.

    Google Scholar 

  44. H. Hudzik, Uniformly non-l» Orlicz spaces with Luxemburg norm, Studia Math. 81 (1985), 271–284.

    MathSciNet  MATH  Google Scholar 

  45. H. Hudzik, Banach lattices with order isometric copies of l°°, Indag. Math. (N.S.) 9 (4) (1998), 521–527.

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Hudzik and A. Kamihska, On uniformly convexifiable and B-convex Musielak-Orlicz spaces, Comment. Math. Prace Mat. 25 (1) (1985), 59–75.

    MathSciNet  MATH  Google Scholar 

  47. H. Hudzik and A. Kamidska, Monotonicity properties of Lorentz spaces, Proc. Amer. Math. Soc. 123 (9) (1995), 2715–2721.

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Hudzik, A. Kaminska and W. Kurc, Uniformly non-l;,~ 1 Musielak-Orlicz spaces, Bull. Polish Acad. Sci. Math. 35 (1987), 411–448.

    MathSciNet  Google Scholar 

  49. H. Hudzik, A. Kaminska and M. Mastylo, Geometric properties of some Calderon- Lozanovskii spaces and Orlicz-Lorentz spaces, Houston J. Math. 22 (3) (1996), 639–663.

    MathSciNet  MATH  Google Scholar 

  50. H. Hudzik, A. Kaminska and M. Mastylo, On geometric properties of Orlicz-Lorentz spaces, Canad. Math. Bull. 40 (3) (1997), 316–329.

    Article  MathSciNet  MATH  Google Scholar 

  51. H. Hudzik, A. Kaminska and M. Mastylo, Monotonicity and rotundity properties in Ba-nach lattices, Rocky Mount. J. Math. 30 (3) (2000), 1–18.

    MathSciNet  Google Scholar 

  52. H. Hudzik, A. Kaminska and J. Musielak, On the coefficient of Orlicz spaces, Math. Z. 197 (1988), 291–295.

    Article  MathSciNet  MATH  Google Scholar 

  53. H. Hudzik and W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices, J. Approx. Theory 95 (1998), 353–368.

    Article  MathSciNet  MATH  Google Scholar 

  54. H. Hudzik and T. Landes, Characteristic of convexity of Köthe function spaces, Math. Ann. 294 (1992), 117–124.

    Article  MathSciNet  MATH  Google Scholar 

  55. H. Hudzik and M. Mastylo, Almost isometric copies of l°° in some Banach spaces, Proc. Amer. Math. Soc. 119 (1) (1993), 209–215.

    Google Scholar 

  56. H. Hudzik, B.X. Wang and T.F. Wang, On the convexity characteristic of Orlicz spaces, Math. Japon. 37 (4) (1992), 691–699.

    MathSciNet  MATH  Google Scholar 

  57. H. Hudzik, C. Wu and Y. Ye, Packing constant in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, Rev. Mat. Complut. 7 (1) (1994), 13–26.

    MathSciNet  MATH  Google Scholar 

  58. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mount. J. Math. 10 (1980), 473–749.

    MathSciNet  Google Scholar 

  59. R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.

    Article  MathSciNet  MATH  Google Scholar 

  60. M.A. Japón Pineda, Some geometric coefficients in Orlicz spaces with application in fixed point theory, in: Lecture Notes in Pure and Appl. Math. 213, Dekker, 2000, 193–204.

    Google Scholar 

  61. A. Kaminska, Flat Orlicz-Musielak sequence spaces, Bull. Acad. Polon. Sér. Sci. Math. 30 (7–8) (1982), 347–352.

    MathSciNet  MATH  Google Scholar 

  62. A. Kaminska, On uniform convexity of Orlicz spaces, Nederl. Akad. Wetensch. Indag. Math. 44 (1) (1982), 27–36.

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Kamihska, On some convexity properties of Musielak-Orlicz spaces, Rend. Circ. Mat. Palermo (2), Suppl. no. 5 (1984), 63–72.

    Google Scholar 

  64. A. Kamihska, P.-K. Lin and H. Sun, Uniformly normal structure of Orlicz-Lorentz spaces, in: Lecture Notes in Pure and Appl. Math. 175, Dekker, 1996, 229–238.

    Google Scholar 

  65. L. Kantorovich and G. Akilov, Functional Analysis, 2nd ed., in Russian, Nauka, 1977.

    Google Scholar 

  66. E. Katirtzoglou, Normal structure of Musielak–Orlicz spaces, Collect. Math. 48 (4–5–6) (1997), 571 – 585.

    MathSciNet  MATH  Google Scholar 

  67. W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.

    Article  MathSciNet  MATH  Google Scholar 

  68. Ko-a] P. Kolwicz, On the property (ß) in Banach lattices, Calderón-Lozanovskii spaces and OrliczLorentz spaces,to appear.

    Google Scholar 

  69. Ko-b] P. Kolwicz, The property (ß) in Orlicz-Bochner sequence spaces,Comment. Math. Univ. Carolin., to appear.

    Google Scholar 

  70. Ko-c] P. Kolwicz, The property (ß) in Musielak-Orlicz sequence spaces of Bochner type,to appear.

    Google Scholar 

  71. Ko-d] P. Kolwicz, The uniformly Kadec-Klee and the nearly uniformly convex property in KötheBochner sequence spaces,submitted.

    Google Scholar 

  72. M.A. Krasnoselskii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Nordhoff, 1961.

    Google Scholar 

  73. K. Kuratowski, Sur les espaces completes, Fund. Math. 15 (1930), 301–309.

    MATH  Google Scholar 

  74. W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory 69 (2) (1992), 173–187.

    Article  MathSciNet  Google Scholar 

  75. D.N. Kutzarova, k — ß and k-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162 (1991), 322–338.

    Article  MathSciNet  MATH  Google Scholar 

  76. D.N. Kutzarova, E. Maluta and S. Prus, Property (ß) implies normal structure of dual space, Rend. Circ. Mat. Palermo (2) 41 (3) (1992), 353–368.

    Article  MathSciNet  MATH  Google Scholar 

  77. T. Landes, Normal structure and weakly normal structure of Orlicz sequence spaces, Trans. Amer. Math. Soc. 285 (1984), 523–534.

    Article  MathSciNet  MATH  Google Scholar 

  78. T. Landes, Normal structure and the sum-property, Pacific J. Math. 123 (1986), 127–147.

    MathSciNet  MATH  Google Scholar 

  79. P.Y. Lee, Cesdro sequence spaces, Math. Chronicle 13 (1984), 29–45.

    MathSciNet  MATH  Google Scholar 

  80. W.A.J. Luxemburg, Banach function spaces, Thesis, Technische Hogeschool te Delft, 1955.

    MATH  Google Scholar 

  81. L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics, Campines SP (Brasil ), 1989.

    MATH  Google Scholar 

  82. E. Maluta, Uniformly normal structre and related coefficients, Pacific J. Math. 111 (2) (1984), 357–369.

    MathSciNet  MATH  Google Scholar 

  83. J. Musielak, Criiez spaces and modular spaces, Lecture Notes in Math. 1034, Springer-Verlag, 1983.

    Google Scholar 

  84. H. Nakano, Modulared semi-ordered linear spaces, Maruzen, Tokyo, 1950.

    MATH  Google Scholar 

  85. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.

    Article  MathSciNet  MATH  Google Scholar 

  86. [Pr 89] S. Prus, Nearly uniformly smooth Banach spaces,Boll. Unione Mat. Ital. (7) 3B (1989), 507–521.

    Google Scholar 

  87. S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697–704.

    Article  MathSciNet  MATH  Google Scholar 

  88. S. Prus, On infinite dimensional uniform smoothness of Banach spaces, Comment. Math. Univ. Carolin. 40 (1) (1999), 97–105.

    MathSciNet  MATH  Google Scholar 

  89. M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, 1991.

    Google Scholar 

  90. S. Rolewicz, On drop property, Studia Math. 87 (1987), 181–191.

    MathSciNet  MATH  Google Scholar 

  91. T. Sgkowski and A. Stachura, Noncompact smoothness and noncompact convexity, Atti Sem. Mat. Fis. Univ. Modena 36 (1982), 239–338.

    Google Scholar 

  92. J.S. Shue, On the Cesdro sequence spaces, Tamkang J. Math. 1 (1970), 143–150.

    Google Scholar 

  93. B. Sims, Orthogonality and fixed points of nonexpansive mappings, Proc. Centre for Math. Anal., Austral. Nat. Uni. 20 (1988), 178–186.

    MathSciNet  Google Scholar 

  94. B. Sims, Geometric conditions sufficient for the weak and weak* -fixed point theory, in: Proc. Second International Conference on Fixed Point Theory and Appl., Ed. K.K. Tan, World Scientific Publisher, River Edge NJ 1992, 278–290.

    Google Scholar 

  95. A.E. Taylor and B.C. Lay, Introduction to Functional Analysis (second edition), John Wiley and Sons, 1980.

    Google Scholar 

  96. B. Turett, Rotundity of Orlicz spaces, Nederl. Akad. Wetensch. Proc. Ser. A 79 (5) (1976), 462–468.

    MathSciNet  MATH  Google Scholar 

  97. C. Wu and H. Sun, Calculation of Orlicz norm and complex convexity in Musielak-Orlicz sequence spaces, Chinese Ann. Math. Ser. A 12 (1991), 98–102.

    MathSciNet  MATH  Google Scholar 

  98. G. Zhang, Weakly convergent sequence coefficient of product spaces, Proc. Amer. Math. Soc. 117 (3) (1992), 637–643.

    Article  Google Scholar 

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Authors and Affiliations

  1. Harbin University of Science and Technology, Harbin, 150080, Ṕ.R. China

    S. Chen & Y. Cui

  2. Institute of Mathematics, Poznań University of Technology, Piotrowo 3a, 60-965, Poznań, Poland

    H. Hudzik

  3. Mathematics, School of Mathematical and Physical Sciences, The University of Newcastle, NSW, 2308, Australia

    B. Sims

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  1. S. Chen
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  2. Y. Cui
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Editors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA, USA

    William A. Kirk

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia

    Brailey Sims

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Chen, S., Cui, Y., Hudzik, H., Sims, B. (2001). Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices. In: Kirk, W.A., Sims, B. (eds) Handbook of Metric Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1748-9_12

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