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  • Research Article
  • Open Access

Nonexpansive mappings defined on unbounded domains

Fixed Point Theory and Applications20062006:82080

  • Received: 18 January 2006
  • Accepted: 23 January 2006
  • Published:


We obtain fixed point theorems for nonexpansive mappings defined on unbounded sets. Our assumptions are weaker than the asymptotically contractive condition recently introduced by Jean-Paul Penot.


  • Point Theorem
  • Differential Geometry
  • Fixed Point Theorem
  • Nonexpansive Mapping
  • Contractive Condition


Authors’ Affiliations

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA


  1. Aleyner A, Reich S: A note on explicit iterative constructions of sunny nonexpansive retractions in Banach spaces. Journal of Nonlinear and Convex Analysis 2005,6(3):525–533.MathSciNetMATHGoogle Scholar
  2. Aleyner A, Reich S: An explicit construction of sunny nonexpansive retractions in Banach spaces. Fixed Point Theory and Applications 2005,2005(3):295–305. 10.1155/FPTA.2005.295MathSciNetView ArticleMATHGoogle Scholar
  3. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar
  4. Browder FE: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bulletin of the American Mathematical Society 1968, 74: 660–665. 10.1090/S0002-9904-1968-11983-4MathSciNetView ArticleMATHGoogle Scholar
  5. Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
  6. Bruck RE Jr.: A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific Journal of Mathematics 1974, 53: 59–71.MathSciNetView ArticleMATHGoogle Scholar
  7. Bula I: Some generalizations of W. A. Kirk's fixed point theorems. In Mathematics, Latv. Univ. Zināt. Raksti. Volume 595. Latv. Univ., Riga; 1994:159–166.Google Scholar
  8. Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.MathSciNetView ArticleMATHGoogle Scholar
  9. García-Falset J: Fixed points for mappings with the range type condition. Houston Journal of Mathematics 2002,28(1):143–158.MathSciNetMATHGoogle Scholar
  10. Gatica JA, Kirk WA: Fixed point theorems for contraction mappings with applications to nonexpansive and pseudo-contractive mappings. The Rocky Mountain Journal of Mathematics 1974, 4: 69–79. 10.1216/RMJ-1974-4-1-69MathSciNetView ArticleMATHGoogle Scholar
  11. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker, New York; 1984:ix+170.Google Scholar
  12. Göhde D: Zum Prinzip der kontraktiven Abbildung. Mathematische Nachrichten 1965, 30: 251–258. 10.1002/mana.19650300312MathSciNetView ArticleMATHGoogle Scholar
  13. Kato T: Nonlinear semigroups and evolution equations. Journal of the Mathematical Society of Japan 1967, 19: 508–520. 10.2969/jmsj/01940508MathSciNetView ArticleMATHGoogle Scholar
  14. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar
  15. Kirk WA: Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions. Proceedings of the American Mathematical Society 1975, 50: 143–149. 10.1090/S0002-9939-1975-0380527-7MathSciNetView ArticleMATHGoogle Scholar
  16. Kirk WA, Morales CH: Nonexpansive mappings: boundary/inwardness conditions and local theory. In Handbook of Metric Fixed Point Theory. Kluwer Academic, Dordrecht; 2001:299–321.View ArticleGoogle Scholar
  17. Kirk WA, Schöneberg R: Some results on pseudo-contractive mappings. Pacific Journal of Mathematics 1977,71(1):89–100.MathSciNetView ArticleMATHGoogle Scholar
  18. Luc DT: Recessively compact sets: properties and uses. Set-Valued Analysis 2002,10(1):15–35. 10.1023/A:1014458603461MathSciNetView ArticleMATHGoogle Scholar
  19. Luc DT, Penot J-P: Convergence of asymptotic directions. Transactions of the American Mathematical Society 2001,353(10):4095–4121. 10.1090/S0002-9947-01-02664-2MathSciNetView ArticleMATHGoogle Scholar
  20. Matsushita S-Y, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2004,2004(1):37–47. 10.1155/S1687182004310089MathSciNetView ArticleMATHGoogle Scholar
  21. Penot J-P: A fixed-point theorem for asymptotically contractive mappings. Proceedings of the American Mathematical Society 2003,131(8):2371–2377. 10.1090/S0002-9939-03-06999-5MathSciNetView ArticleMATHGoogle Scholar
  22. Penot J-P: A metric approach to asymptotic analysis. Bulletin des Sciences Mathématiques 2003,127(9):815–833. 10.1016/j.bulsci.2003.08.001MathSciNetView ArticleMATHGoogle Scholar
  23. Petryshyn WV: Structure of the fixed points sets of -set-contractions. Archive for Rational Mechanics and Analysis 1970/1971, 40: 312–328.MathSciNetMATHGoogle Scholar