## Taiwanese Journal of Mathematics

### COMMON FIXED POINTS OF NONCOMMUTING DISCONTINUOUS WEAKLY CONTRACTIVE MAPPINGS IN CONE METRIC SPACES

Vasile Berinde

#### Abstract

In this paper we prove the existence of coincidence points and common fixed points for a large class of noncommuting discontinuous contractive type mappings in cone metric spaces. These results generalize, extend and unify several well-known recent related results in literature.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 5 (2010), 1763-1776.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406015

Digital Object Identifier
doi:10.11650/twjm/1500406015

Mathematical Reviews number (MathSciNet)
MR2724132

Zentralblatt MATH identifier
1235.54023

#### Citation

Berinde, Vasile. COMMON FIXED POINTS OF NONCOMMUTING DISCONTINUOUS WEAKLY CONTRACTIVE MAPPINGS IN CONE METRIC SPACES. Taiwanese J. Math. 14 (2010), no. 5, 1763--1776. doi:10.11650/twjm/1500406015. https://projecteuclid.org/euclid.twjm/1500406015

#### 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.
• V. Berinde, Abstract $\phi$-contractions which are Picard mappings, Mathematica $($Cluj$)$, 34(57) (1992), no. 2, 107-111.
• V. Berinde, A fixed point theorem of Maia type in $K$-metric spaces, Seminar on Fixed Point Theory, 7-14, Preprint, 91-3, "Babeş-Bolyai" Univ., Cluj-Napoca, 1992.
• V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19(1) (2003), 7-22.
• V. Berinde, Picard iteration converges faster than the Mann iteration in the class of quasi-contractive operators, Fixed Point Theory Appl., 2004(2) (2004), 97-105.
• V. Berinde, On the convergence of Ishikawa iteration for a class of quasi contractive operators, Acta Math. Univ. Comen., 73(1) (2004), 119-126.
• V. Berinde, A convergence theorem for some mean value fixed point iterations in the class of quasi contractive operators, Demonstratio Math., 38(1) (2005), 177-184.
• V. Berinde, Error estimates for approximating fixed points of discontinuous quasi-contractions, General Mathematics, 13(2) (2005), 23-34.
• V. Berinde and M. Berinde, On Zamfirescu's fixed point theorem, Rev. Roumaine Math. Pures Appl., 50(5-6) (2005), 443-453.
• V. Berinde, Iterative Approximation of Fixed Points, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, 2007.
• V. Berinde, Approximating common fixed points of noncommuting dicontinuous contractive type mappings in metric spaces, (submitted).
• S. K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci., 25 (1972) 727-730.
• E. De Pascale, G. Marino and P. Pietromala, The use of $E$-metric spaces in the search for fixed points, Le Mathematiche, 48 (1993), 367-376.
• L.-G. Huang and Z. Xian, Cone metric spaces and fixed point theorems of contractive mapings, J. Math. Anal. Appl., 332 (2007), 1468-1476.
• G. Jungck, Commuting maps and fixed points, Amer. Math Monthly, 83 (1976), 261-263.
• G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci., 4 (1996), 199-215.
• R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 10 (1968), 71-76.
• J. Meszaros, A comparison of various definitions of contractive type mappings, Bull. Calcutta Math. Soc., 84(2) (1992), 167-194.
• Sh. Rezaapour and R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mapings, J. Math. Anal. Appl., (2008), doi:10.1016/jmaa2008.04.049.
• B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257-290.
• B. E. Rhoades, Contractive definitions revisited, Contemporary Mathematics, 21 (1983), 189-205.
• B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics, 72 (1988), 233-245.
• I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.
• I. A. Rus, Picard operators and applications, Scientiae Math. Japon., 58(1) (2003), 191-219.
• I. A. Rus, A. Petruşel and M. A. \c Serban, Weakly Picard operators: equivalent definitions, applications and open problems, Fixed Point Theory, 7(1) (2006), 3-22.
• P. P. Zabreiko, $K$-metric and $K$-normed linear spaces: survey, Collect. Math., 48 (1997), 825-859.
• P. P. Zabreiko, The fixed point theory and the Cauchy problem for partial differential equations, Gajshun, I. V. (ed.) et al., Nonlinear analysis and applications. Minsk: Natsional'na Akademiya Nauk Belarusi. Tr. Inst. Mat., Minsk. 1, 1998, pp. 93-106.
• T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. $($Basel$)$, 23 (1972), 292-298.