Abstract and Applied Analysis

Volume 2012 (2012), Article ID 736214, 19 pages

http://dx.doi.org/10.1155/2012/736214

Research Article

## Common Fixed Point Theorems for a Class of Twice Power Type Contraction Maps in *G*-Metric Spaces

Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036, China

Received 5 February 2012; Accepted 27 July 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Hongqing Ye and Feng Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked 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. View at Google Scholar · View at Zentralblatt MATH - M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 215, no. 1, pp. 262–269, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar - Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete $G$-metric spaces,”
*Fixed Point Theory and Applications*, Article ID 917175, 10 pages, 2009. View at Google Scholar - Z. Mustafa, F. Awawdeh, and W. Shatanawi, “Fixed point theorem for expansive mappings in $G$-metric spaces,”
*International Journal of Contemporary Mathematical Sciences*, vol. 5, no. 49–52, pp. 2463–2472, 2010. View at Google Scholar - Z. Mustafa and H. Obiedat, “A fixed point theorem of Reich in $G$-metric spaces,”
*Cubo*, vol. 12, no. 1, pp. 83–93, 2010. View at Google Scholar - 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. View at Publisher · View at Google Scholar - S. Manro, S. S. Bhatia, and S. Kumar, “Expansion mappings theorems in $G$-metric spaces,”
*International Journal of Contemporary Mathematical Sciences*, vol. 5, no. 49–52, pp. 2529–2535, 2010. View at Google Scholar - W. Shatanawi, “Fixed point theory for contractive mappings satisfying $\mathrm{\Phi}$-maps in $G$-metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 181650, 9 pages, 2010. View at Google Scholar - R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, “Property $P$ in $G$-metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 401684, 12 pages, 2010. View at Google Scholar - M. Abbas, T. Nazir, and S. Radenović, “Some periodic point results in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 8, pp. 4094–4099, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Obiedat and Z. Mustafa, “Fixed point results on a nonsymmetric $G$-metric spaces,”
*Jordan Journal of Mathematics and Statistics*, vol. 3, no. 2, pp. 65–79, 2010. View at Google Scholar - M. Abbas, T. Nazir, and P. Vetro, “Common fixed point results for three maps in
*G*-metric spaces,”*Filomat*, vol. 25, no. 4, pp. 1–17, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. P. R. Rao, A. Sombabu, and J. Rajendra Prasad, “A common fixed point theorem for six expansive mappings in $G$-metric spaces,”
*Kathmandu University Journal of Science, Engineering and Technology*, vol. 7, no. 1, pp. 113–120, 2011. View at Google Scholar - M. Abbas, T. Nazir, and R. Saadati, “Common fixed point results for three maps in generalized metric space,”
*Advances in Difference Equations*, vol. 49, pp. 1–20, 2011. View at Google Scholar - S. Manro, S. Kumar, and S. S. Bhatia, “$R$-weakly commuting maps in $G$-metric spaces,”
*Polytechnica Posnaniensis*, no. 47, pp. 11–18, 2011. View at Google Scholar - L. Gajić and Z. Lozanov-Crvenković, “A fixed point result for mappings with contractive iterate at a point in $G$-metric spaces,”
*Filomat*, vol. 25, no. 2, pp. 53–58, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. K. Vats, S. Kumar, and V. Sihag, “Some common fixed point theorem for compatible mappings of type (A) in complete $G$-metric space,”
*Advances in Fuzzy Mathematics*, vol. 6, no. 1, pp. 27–38, 2011. View at Google Scholar - H. Aydi, “A fixed point result involving a generalized weakly contractive condition in
*G*-metric spaces,”*Bulletin Mathematical Analysis and Applications*, vol. 3, no. 4, pp. 180–188, 2011. View at Google Scholar - M. Abbas, S. H. Khan, and T. Nazir, “Common fixed points of $R$-weakly commuting maps in generalized metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 41, 2011. View at Publisher · View at Google Scholar - H. Aydi, W. Shatanawi, and C. Vetro, “On generalized weak $G$-contraction mapping in $G$-metric spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 11, pp. 4222–4229, 2011. View at Publisher · View at Google Scholar - W. Shatanawi, “Coupled fixed point theorems in generalized metric spaces,”
*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 3, pp. 441–447, 2011. View at Google Scholar · View at Zentralblatt MATH - R. K. Vats, S. Kumar, and V. Sihag, “Fixed point theorems in complete $G$-metric space,”
*Fasciculi Mathematici*, no. 47, pp. 127–139, 2011. View at Google Scholar - M. Abbas, A. R. Khan, and T. Nazir, “Coupled common fixed point results in two generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 13, pp. 6328–6336, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Kaewcharoen, “Common fixed point theorems for contractive mappings satisfying $\mathrm{\Phi}$-maps in $G$-metric spaces,”
*Banach Journal of Mathematical Analysis*, vol. 6, no. 1, pp. 101–111, 2012. View at Google Scholar - M. Abbas, T. Nazir, and D. Dorić, “Common fixed point of mappings satisfying (E.A) property in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7665–7670, 2012. View at Publisher · View at Google Scholar - R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, “Fixed point theorems in generalized partially ordered $G$-metric spaces,”
*Mathematical and Computer Modelling*, vol. 52, no. 5-6, pp. 797–801, 2010. View at Publisher · View at Google Scholar - B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 1-2, pp. 73–79, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered $G$-metric spaces,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1601–1609, 2012. View at Publisher · View at Google Scholar - M. Abbas, M. Ali Khan, and S. Radenović, “Common coupled fixed point theorems in cone metric spaces for $w$-compatible mappings,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 195–202, 2010. View at Publisher · View at Google Scholar - I. Beg, M. Abbas, and T. Nazir, “Generalized cone metric spaces,”
*Journal of Nonlinear Science and Applications*, vol. 3, no. 1, pp. 21–31, 2010. View at Google Scholar · View at Zentralblatt MATH - F. Gu and Z. He, “The common fixed point theorems for a class of twice power type $\mathrm{\Phi}$-contraction mapping,”
*Journal of Shangqiu Teachers College*, vol. 22, no. 5, pp. 27–323, 2006. View at Google Scholar