Banach Journal of Mathematical Analysis

Quasi-contractions on symmetric and cone symmetric spaces

Z. Kadelburg and S. Radenovic

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The purpose of this paper is to introduce the concept of a cone symmetric space and to investigate relationship between (cone) metric spaces and (cone) symmetric spaces. Among other things, we shall also extend some fixed point results from metric spaces to cone metric spaces (Theorem 3.3), and to symmetric spaces (Theorems 3.2 and 3.5) under some new contraction conditions.

Article information

Banach J. Math. Anal. Volume 5, Number 1 (2011), 38-50.

First available in Project Euclid: 14 August 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Digital Object Identifier

Primary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]
Secondary: 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

normal and non-normal cone cone metric space symmetric space fixed point quasi-contraction


Radenovic, S.; Kadelburg, Z. Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5 (2011), no. 1, 38--50. doi:10.15352/bjma/1313362978.

Export citation


  • M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 416–420.
  • A. Aliouche, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J. Math. Anal. Appl. 322 (2006), no. 2, 796–802.
  • S.H. Cho, G.Y. Lee and J.S. Bae, On coincidence and fixed-point theorems in symmetric spaces, Fixed Point Theory Appl., Vol. 2008, Article ID 562130, 9 pp.
  • Lj.B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273.
  • K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
  • T.L. Hicks and B.E. Rhoades, Fixed point theory in symmetric spaces with applications to probabilistic spaces, Nonlinear Anal. 36 (1999), no. 3, 331–344.
  • L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), no. 2, 1468–1476.
  • D. Ilić and V. Rakočević, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341 (2008), no. 2, 876–882.
  • D. Ilić and V. Rakočević, Quasi-contraction on a cone metric space, Appl. Math. Lett. 22 (2009), 728–731.
  • M. Imdad, J. Ali and L. Khan, Coincidence and fixed points in symmetric spaces under strict contractions, J. Math. Anal. Appl. 320 (2006), 352–360.
  • Z. Kadelburg, S. Radenović and V. Rakočević, Remarks on “Quasi-contraction on a cone metric space”, Appl. Math. Lett. 22 (2009), 1674–1679.
  • L.V. Kantorovich, The majorant principle and Newton's method, Dokl. Akad. Nauk SSSR (N.S.), 76 (1951), 17–20.
  • W.A. Kirk and B.G. Kang, A fixed point theorem revisited, J. Korean Math. Soc. 34 (1997), 285–291.
  • M.A. Krasnosel'ski and P. P. Zabreĭ ko, Geometrical Methods in Nonlinear Analysis, Springer, 1984.
  • R.S. Palais, A simple proof of the Banach contraction principle, Fixed Point Theory Appl. 2 (2007), 221–223.
  • E. de Pascale, G. Marino and P. Pietramala, The use of E-metric spaces in the search for fixed points, Le Matematiche 48 (1993), 367–376.
  • Sh. Rezapour and R. Hamlbarani Haghi, Some notes on the paper ”Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 345 (2008), 719–724.
  • I.A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory, Cluj Univ. Press, 2008.
  • J.S. Vandergraft, Newton method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), no. 3, 406–432.
  • W.A. Wilson, On semi-metric spaces, Amer. J. Math. 53 (1931), no. 2, 361–373.
  • P.P. Zabreĭ ko, $K$-metric and $K$-normed linear spaces: survey, Fourth International Conference on Function Spaces (Zielona Góra, 1995). Collect. Math. 48 (1997), no. 4-6, 825–859.
  • J. Zhu, Y.J. Cho and S.M. Kang, Equivalent contractive conditions in symmetric spaces, Computers & Mathematics Appl. 50 (2005), 1621–1628.