Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An approach to fixed-point theorems on uniform spaces


Author: E. Tarafdar
Journal: Trans. Amer. Math. Soc. 191 (1974), 209-225
MSC: Primary 54H25; Secondary 47H10
DOI: https://doi.org/10.1090/S0002-9947-1974-0362283-5
MathSciNet review: 0362283
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Diaz and Metcalf [2] have some interesting results on the set of successive approximations of a self mapping which is either a nonexpansion or a contraction on a metric space with respect to the set of fixed points of the mapping. We have extended most of these results to a Hausdorff uniform space. We have also proved a Banach's contraction mapping principle on a complete Hausdorff uniform space and indicated some applications in locally convex linear topological spaces.


References [Enhancements On Off] (What's this?)

  • [1] W. W. Taylor, Fixed-point theorems for nonexpansive mappings in linear topological spaces, J. Math. Anal. Appl. 40 (1972), 164-173. MR 0322612 (48:974)
  • [2] J. B. Diaz and F. T. Metcalf, On the set of subsequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969), 459-485. MR 0234327 (38:2644)
  • [3] F. Tricomi, Un teorema sulla convergenza delle successoni formate delle successive iterate di una funzione di una variabile reale, Giorn. Mat. Battaglini 54 (1916), 1-9.
  • [4] D. G. Defigueiredo, Topics in nonlinear functional analysis, Lecture series No. 48, University of Maryland Press, College Park, Md. 1967.
  • [5] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [6] W. J. Thron, Topological structures, Holt, Rinehart and Winston, New York, 1966. MR 34 #778. MR 0200892 (34:778)
  • [7] H. Schubert, Topologie. Eine Einführung, Teubner, Stuttgart, 1964; English transl., Allyn and Bacon, Boston, Mass., 1968. MR 30 #551; 37 #2160. MR 0170313 (30:551)
  • [8] M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79. MR 24 #A2936. MR 0133102 (24:A2936)
  • [9] H . H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [10] G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1960; English transl., Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR 24 #A411; 40 #1750.
  • [11] F. F. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571-575. MR 32 #8155b. MR 0190745 (32:8155b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H25, 47H10

Retrieve articles in all journals with MSC: 54H25, 47H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0362283-5
Keywords: Uniform space, locally convex linear topological space, family of pseudometrics, family of seminorms, nonexpansion, contraction, asymptotically regular maps, contraction mapping principle, starshaped subset
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society