Banach Journal of Mathematical Analysis

A Fixed point theorem on cone metric spaces with new type contractivity

Mujahid Abbas, Ishak Altun, and Hakan Simsek

Full-text: Open access

Abstract

In the present work, a common fixed point theorem for self maps on cone metric spaces is proved. Also two examples, which shows that our main theorem is generalized version of main theorems of [A. Branciari, Int. J. Math. Math. Sci., 29 (2002), no. 9, 531-536] and [L.G. Huang and X. Zhang, J. Math. Anal. Appl. 332 (2007), no. 2, 1468-1476] are given.

Article information

Source
Banach J. Math. Anal., Volume 5, Number 2 (2011), 15-24.

Dates
First available in Project Euclid: 14 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362998

Digital Object Identifier
doi:10.15352/bjma/1313362998

Mathematical Reviews number (MathSciNet)
MR2780865

Zentralblatt MATH identifier
1223.54053

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

Keywords
fixed point cone metric space integral type contraction

Citation

Altun, Ishak; Abbas, Mujahid; Simsek, Hakan. A Fixed point theorem on cone metric spaces with new type contractivity. Banach J. Math. Anal. 5 (2011), no. 2, 15--24. doi:10.15352/bjma/1313362998. https://projecteuclid.org/euclid.bjma/1313362998


Export citation

References

  • M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl.341 (2008), 416–420.
  • M. Abbas and B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2008), 511–515.
  • M. Abbas and B.E. Rhoades, Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, Math. Commun. 13 (2) (2008), 295–301.
  • 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 (2) (2006), 796–802.
  • I. Altun, D. Turkoglu and B.E. Rhoades, Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed Point Theory and Appl. 2007 (2007), Article ID 17301.
  • I. Altun, B. Damjanović and D. Djorić, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. 23 (2010), 310–316.
  • I. Altun and G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo 58 (2009), 319–325.
  • M. Arshad, A. Azam and P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 493965.
  • A. Azam, M. Arshad and I. Beg, Common fixed points of two maps in cone metric spaces, Rend. Circ. Mat. Palermo 57 (2008), 433–441.
  • A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (9) (2002), 531–536.
  • C. Di Bari and P. Vetro, $\varphi $-pairs and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 57 (2008), 279–285.
  • C. Di Bari and P. Vetro, Weakly $\varphi $-pairs and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 58 (2009), 125–132.
  • R.H. Haghi and Sh. Rezapour, Fixed points of multifunctions on regular cone metric spaces, Expo. Math. 28 (2010), 71–77.
  • 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) 876–882.
  • G. Jungck, S. Radenović, S. Radojević and V. Rako čević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 643840.
  • D. Klim and D. Wardowski, Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Alal. 71 (2009), 5170–5175.
  • J. Matkowski, Remarks on Lipschitzian mappings and some fixed point theorems, Banach J. Math. Anal. 1 (2) (2007), 237–244.
  • S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl. 58 (6) (2009), 1273–1278.
  • P. Raja and S.M. Vaezpour, Some extensions of Banach's contraction principle in complete cone metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 768294.
  • Sh. Rezapour and R. H. Hagli, Fixed point of multifunctions on cone metric spaces, Numer. Funct. Anal. Opt. 30 (7-8) (2009), 1–8.
  • Sh. Rezapour and R. Hamlbarani, Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719–724.
  • B.E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 2003 (63) (2003), 4007–4013.
  • D. Turkoglu and I. Altun, A common fixed point theorem for weakly compatible mappings in symetric spaces satisfying an implicit relation, Bol. Soc. Mat. Mexicana 13 (1) (2007), 195–205.
  • P. Vetro, Common fixed points in cone metric spaces, Rend. Circ. Math. Palermo 56 (2007), 464–468.
  • P. Vetro, A. Azam and M. Arshad, Fixed point results in cone metric spaces, Int.J. Modern Math. 5 (1) (2010), 101–108.
  • X. Zhang, Common fixed point theorems for new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2007), 780–786.