## Abstract and Applied Analysis

### Cyclic Contractions on $G$-Metric Spaces

#### 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

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

#### 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.