Abstract and Applied Analysis

Cyclic Contractions on G -Metric Spaces

E. Karapınar, A. Yıldız-Ulus, and İ. M. Erhan

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Abstract

Conditions for existence and uniqueness of fixed points of two types of cyclic contractions defined on G -metric spaces are established and some illustrative examples are given. In addition, cyclic maps satisfying integral type contractive conditions are presented as applications.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 182947, 15 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475860

Digital Object Identifier
doi:10.1155/2012/182947

Mathematical Reviews number (MathSciNet)
MR2984023

Zentralblatt MATH identifier
1253.54043

Citation

Karapınar, E.; Yıldız-Ulus, A.; Erhan, İ. M. Cyclic Contractions on $G$ -Metric Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 182947, 15 pages. doi:10.1155/2012/182947. https://projecteuclid.org/euclid.aaa/1364475860


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References

  • Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.
  • Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete $G$-metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009.
  • Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete $G$-metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008.
  • Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete $G$-metric spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 48, no. 3, pp. 304–319, 2011.
  • Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results in $G$-metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 283028, 10 pages, 2009.
  • H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for $(\psi ,\varphi )$-weakly contractive mappings in ordered $G$-metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 298–309, 2012.
  • H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered $G$-metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2443–2450, 2011.
  • N. Van Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered G-metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 1601–1609, 2012.
  • H. Aydi, E. Karapinar, and W. Shatanawi, “Tripled fixed point results in generalized metric spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 314279, 2012.
  • H. Aydi, E. Karap\inar, and Z. Mustafa, “On common fixed points in $G$-metric spaces using (E.A) property,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1944–1956, 2012.
  • N. Tahat, H. Aydi, E. Karapinar, and W. Shatanawi, “Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in $G$-metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 48, 2012.
  • H. Aydi, E. Karap\inar, and W. Shatanawi, “Tripled coincidence point results for generalized contractions in ordered generalized metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 101, 2012.
  • H. Aydi, “Generalized cyclic contractions čommentComment on ref. [13?]: Please update the information of this reference, if possible.in G-metric spaces,” The Journal of Nonlinear Science and Application. In press.
  • W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003.
  • M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 10, pp. 3665–3671, 2009.
  • A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.
  • S. Karpagam and S. Agrawal, “Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 4, pp. 1040–1046, 2011.
  • E. Karapinar, “Best proximity points of Kannan type cylic čommentComment on ref. [18?]: Please update the information of this reference, if possible.weak phi-contractions in ordered metric spaces,” Analele stiintifice ale Universitatii Ovidius Constanta. In press.
  • E. Karapinar, “Best proximity points Of cyclic mappings,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1761–1766, 2012.
  • E. Karap\inar and \.I. M. Erhan, “Best proximity point on different type contractions,” Applied Mathematics & Information Sciences, vol. 5, no. 3, pp. 558–569, 2011.
  • M. A. Alghamdi, A. Petrusel, and N. Shahzad, “A fixed point theorem for cyclic generalized contractions in metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 122, 2012.
  • M. Păcurar, “Fixed point theory for cyclic Berinde operators,” Fixed Point Theory, vol. 12, no. 2, pp. 419–428, 2011.
  • M. Păcurar and I. A. Rus, “Fixed point theory for cyclic $\varphi $-contractions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 3-4, pp. 1181–1187, 2010.
  • G. Petruşel, “Cyclic representations and periodic points,” Universitatis Babeş-Bolyai. Studia. Mathematica, vol. 50, no. 3, pp. 107–112, 2005.
  • S. Rezapour, M. Derafshpour, and N. Shahzad, “Best proximity points of cyclic $\varphi $-contractions in ordered metric spaces,” Topological Methods in Nonlinear Analysis, vol. 37, no. 1, pp. 193–202, 2011.
  • E. Karapinar, \.I. M. Erhan, and A. Yildiz Ulus, “Fixed point theorem for cyclic maps on partial metric spaces,” Applied Mathematics & Information Sciences, vol. 6, pp. 239–244, 2012.
  • E. Karapinar and \.I. M. Erhan, “Cyclic contractions and fixed point theorems,” Filomat, vol. 26, pp. 777–782, 2012.
  • D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969.
  • Y. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in New Results in Operator Theory and Its Applications, vol. 98 of Operator Theory: Advances and Applications, pp. 7–22, Birkhauser, Basel, Switzerland, 1997.
  • B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis, vol. 47, pp. 851–861, 2001.
  • Q. Zhang and Y. Song, “Fixed point theory for generalized $\varphi $-weak contractions,” Applied Mathematics Letters, vol. 22, no. 1, pp. 75–78, 2009.
  • E. Karap\inar, “Fixed point theory for cyclic $\phi $-weak contraction,” Applied Mathematics Letters, vol. 24, no. 6, pp. 822–825, 2011.
  • J. Jachymski, “Equivalent conditions for generalized contractions on (ordered) metric spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 3, pp. 768–774, 2011.