Applied Mathematics and Mechanics

, Volume 18, Issue 8, pp 771–778 | Cite as

Fixed points of a pair of asymptotically regular mappings

  • B. K. Sharma
  • B. S. Thakur


In this paper, some theorems on fixed points of pair of asymptotically regular mappings in p-uniformly convex Banach space are proved. For these mappings some fixed point theorems in a Hilbert space, in Lp spaces, in Hardy spaces Hp and in Sobolev spaces Hpk, for 1<p<+∞ and k≥0 are also established. Thus, results of Gornicki[9, 10]. Kruppel[11, 12] and others are extended.

Key words

asymptotically regular mappings p-uniformly convex Banach space asymptotic center fixed points 


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  1. [1]
    J. Barros-Neto,An Introduction to the Theory of Distribution, Dekker, New York (1973).Google Scholar
  2. [2]
    F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces,Bull. Amer. Math. Soc.,72 (1966), 671–675.Google Scholar
  3. [3]
    W. L. Bynum, Normal structure coefficients for Banach spaces,Pacific J. Math.,86 (1980), 427–436.Google Scholar
  4. [4]
    E. Casini and E. Maluta. Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure.Nonlinear Anal.,9 (1985), 103–108.Google Scholar
  5. [5]
    J. Danes. On densifying and related mappings and their applications in nonlinear functional analysis,Theory of Nonlinear Operators,Proc. Summer School, Oct. 1972, GDR, Akademie-Verlag, Berlin (1974), 15–56.Google Scholar
  6. [6]
    N. Dunford and J. Schwarz,Linear Operators, Vol.-I. Interscience, New York (1958).Google Scholar
  7. [7]
    W. L. Duren,Theory of H P Spaces, Academic Press. New York (1970).Google Scholar
  8. [8]
    K. Goebel and W. A. Kirk, Topics in metric fixed point theory,Cambridge Stud. Adv. Math.,28. Cambridge Univ. Press. London (1990).Google Scholar
  9. [9]
    J. Gornicki, Fixed point theorems for asymptotically regular mappings inL P spaces.Non. linear. Anal.,17 (1991), 153–159.Google Scholar
  10. [10]
    J. Gornicki. Fixed points of asymptotically regular mappings,Math. Slovaca,43, 3 (1993), 327–336.Google Scholar
  11. [11]
    M. Kruppel, Ein Fixpunktsatz fur asymptotisch regulare Operatoren im Hilbert-Raum. Wiss. Z. Padagog. Hochsch. “Liselotte Herrman” Gustrow,Math.-Natur. Fak.,25 (1987), 241–246.Google Scholar
  12. [12]
    M. Kruppel, Ein Fixpunktsatz fur asymptotisch regulare Operatoren im gleichmäßig konvexen Banach-Raum, Wiss. Z. Padagog. Hochsch. “Liselotte Herman” Gustrow,Math.-Natur. Fak.,27 (1989), 247–251.Google Scholar
  13. [13]
    T. C. Lim, H. K. Xu and Z. B. Xu, AnL P inequalities and its applications to fixed point theory and approximation theory,Progress in Approximation Theory, Academic Press (1991), 609–624.Google Scholar
  14. [14]
    P. K. Lin, A uniformly asymptotically regular mappings without fixed points,Canad. Math. Bull.,30 (1987), 481–483.Google Scholar
  15. [15]
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, II—Function Spaces, Springer-Verlag, New York, Berlin (1979).Google Scholar
  16. [16]
    S. A. Pichugov, , (in Russian),Mat. Zametki,43 (1988), 604–614. (Translation:Math. Notes,43 (1988), 348–354.)Google Scholar
  17. [17]
    B. Prus and R. Smarzewski, Strongly unique best approximations and centers in uniformly convex spaces,J. Math. Anal. Appl.,121 (1987), 10–21.Google Scholar
  18. [18]
    S. Prus, On Bynum's fixed point theorem,Atti Sem. Mat. Fis. Univ. Modena,38 (1990), 535–545.Google Scholar
  19. [19]
    S. Prus, Some estimates for the normal structure coefficient in Banach spaces,Rend. Circ. Mat. Palermo,40, 2 (1991), 128–135.Google Scholar
  20. [20]
    R. Smarzewski, Strongly unique best approximations in Banach spaces II,J. Approx. Theory,51 (1987), 202–217.Google Scholar
  21. [21]
    R. Smarzewski, On the inequality of Bynum and Drew,J. Math. Anal. Appl.,150 (1990), 146–150.Google Scholar
  22. [22]
    H. K. Xu, Inequalities in Banach spaces with applications,Nonlinear Anal. 16 (1991), 1127–1138.Google Scholar
  23. [23]
    C. Zalinescu, On uniformly convex function,J. Math. Anal. Appl.,95 (1983), 344–374.Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1980

Authors and Affiliations

  • B. K. Sharma
    • 1
  • B. S. Thakur
    • 1
  1. 1.School of Studies in MathematicsPt. Ravishankar Shukla UniversityRaipurIndia

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