Article ID: MTJPAM-D-20-00008

Title: On Sequences of Certain Contractive Mappings and Their Fixed Points

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00008; Volume 3 / Issue 1 / Year 2021, Pages 70-77

Document Type: Research Paper

Author(s): Mohamed Akkouchi a

aFaculty of Sciences-Semlalia, Cadi Ayyad University, Marrakech, Morocco

Received: 18 April 2020, Accepted: 8 December 2020, Available online: 7 January 2021.

Corresponding Author: Mohamed Akkouchi (Email address:

Full Text: PDF


In 1988, M. Imdad, M.S. Khan and S. Sessa have introduced a general contractive condition of Reich-Hardy-Rogers type in a (complete) metric space (X, d). Let (Tn)n be a sequence of selfmaps of X satisfying this general condition. They proved that each selfmap Tn has a unique fixed point (say) zn in X. Suppose that the sequence (Tn)n converges pointwise on X to a selfmap T. M. Imdad, M.S. Khan and S. Sessa sudied the convergence problem for the sequence (zn)n in X. They established the convergence of this sequence under a regularity assumption, and they raised the question whether this regularity assumption is necessary or not ?
  The aim of this paper is to answer positively to that question. Precisely, we prove that the main results of M. Imdad, M.S. Khan and S. Sessa are still valid without requiring the regularity assumption upon the sequence of fixed points (zn)n of the sequence (Tn)n.

Keywords: Complete metric space, fixed point, sequences of mappings, selfmaps of Reich-Hardy-Rogers type

  1. J. Achari, Fixed point theorems in metric spaces, Tamkang J. of Math. 7, 71–74, 1976.
  2. Y.I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, New Results in Operator Theory and Its Applications, Springer, 7-22, 1997.
  3. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3, 133-181, 1922.
  4. V. Berinde, Iterative approximation of fixed points, Lecture Notes in Mathematics, Springer, 2006.
  5. F.F. Bonsall, Lectures on Some Fixed Point Theorems of Functional Analysis, Tata Institute of Fundamental Research, Bombay, 1962.
  6. D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20, 458-464, 1969.
  7. A. Branciari, A fixed point theorem for mapping satisfying a general contractive condition of integral type, Int. J. M. and M. Sciences 29 (9), 531-536, 2002.
  8. R. Caccioppoli, Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale, Rend. Accad. Naz. Lincei. 11, 794-799, 1930.
  9. S. K. Chatterjea, Fixed Point Theorems, Comptes Rend. Acad. Bulgare Sci. 25, 727-730, 1972.
  10. Lj.B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) (NS) 12, 19-26, 1971.
  11. Lj.B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45, 267-273, 1974.
  12. R.B. Fraser and S.B. Nadler, Sequences of Contractive Maps and Fixed Points, Pacific J. Math. 51, 659-667, 1969.
  13. L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Annal. Appl. 332, 1468-1476, 2007.
  14. G.E. Hardy and T.D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16, 201-206, 1973.
  15. M. Imdad, M.S. Khan and S. Sessa, On sequences of contractive mappings and their fixed points, Internat. J. Math & Math. Sci. 11 (3), 527-534, 1988.
  16. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60, 71-76, 1968.
  17. S.G. Matthews, Partial metric topology. In, General Topology and its Applications, Proc. 8th Summer Conf., Queen’s College (1992). Annals of the New York Academy of Sciences 728, 183-197, 1994.
  18. A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl.28, 326-329, 1969.
  19. R. N. Mukherjee, On some fixed point theorems, Kyungpook Math. J. 14, 37-44, 1974.
  20. N. Muresan, Families of mappings and fixed points, Studia Univ. Babes-Bolyai Ser. Math.-Mech. 19 (1), 13-15, 1974.
  21. S. B. Nadler, Sequences of contractions and fixed points, Pacific J, Math. 27, 579-585, 1968.
  22. Jr., S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30, 475-488, 1969.
  23. R.S. Palais, A simple proof of the Banach contraction principle, Journal of Fixed Point Theory and Applications 2, 221-223, 2007.
  24. S. Park, Sequences of Quasi-contractions and Fixed points, J. Korean Math. Soc. 14 (1), 99-103, 1977.
  25. E. Rakotch, A Note on Contractive Mappings, Proc. Amer. Math. Soc. 13, 459-465, 1962.
  26. S. Reich, Some Remarks Concerning Contraction Mappings, Canad. Math. Bull. 14, 121-124, 1971.
  27. S. Reich, Kannan’s Fixed Point Theorem, Boll. Un. Mat. Ital. 4 (4), 1-11, 1971.
  28. S. Reich, Fixed Points of Contractive Functions, Boll. Un. Mat. Ital. 5 (4), 26-42, 1972.
  29. B.E. Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc. 226, 257-290, 1977.
  30. B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47, 2683-2693, 2001.
  31. S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd)(N.S.) 32 (46), 14-153, 1982.
  32. S.P. Singh, On Sequence of Contraction Mappings, Riv. Mat. Univ. Parma 11 (2), 227-231, 1970.
  33. T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23, 292–298, 1972.