Common Fixed Point Theorems for Compatible and Weakly Compatible Maps in G-Metric Spaces

Asha  Rani; Sanjay  Kumar; Naresh  Kumar; S. K.  Garg

doi:10.4236/am.2012.310166

Asha Rani^*, Sanjay Kumar, Naresh Kumar, S. K. Garg

Abstract: In this paper first we prove common fixed point theorems for compatible and weakly compatible maps. Secondly, we prove common fixed point theorems for weakly compatible maps along with property (E.A.) and (CLRg) property respectively.

Keywords: G-metric Spaces; Compatible Maps; Weakly Compatible Maps; Property (E.A.); (CLRg) Property

Cite this paper: A. Rani, S. Kumar, N. Kumar and S. Garg, "Common Fixed Point Theorems for Compatible and Weakly Compatible Maps in G-Metric Spaces," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1128-1134. doi: 10.4236/am.2012.310166.

References

[1] S. Gahler, “2-Metrices Raume und Ihre Topologische Struktur,” Mathematische Nachrichten, Vol. 26, 1963, pp. 115-148. doi:10.1002/mana.19630260109

[2] C. R. Hsiao, “A Property of Contraction Type Mappings in 2-Metric Spaces,” Janabha, Vol. 16, 1986, pp. 223-229.

[3] B. C. Dhage, “Generalized Metric Spaces and Mappings with Fixed Point,” Bulletin of Calcutta Mathematical Society, Vol. 84, 1992, pp. 329-336.

[4] Z. Mustafa and B. Sims, “Some Remarks Concerning D-Metric Spaces,” Proceedings of International Conference on Fixed Point Theory and Applications, Yokohama Publishers, Valencia, Vol. 13, No. 19, 2004, pp. 189-198.

[5] Z. Mustafa and B. Sims, “A New Approach to a Generalized Metric Spaces,” Journal of Nonlinear and Convex Analysis, Vol. 7, No. 2, 2006, pp. 289-297.

[6] Z. Mustafa, H. Obiedat and F. Awawdeh, “Some Fixed Point Theorems for Mappings on Complete G-Metric Spaces,” Fixed Point Theory and Applications, 2008, Article ID: 18970, 12 p. doi:10.1155/2008/189870

[7] Z. Mustafa, W. Shatanawi and M. Bataineh, “Existence of Fixed Points Results in G-Metric Spaces,” International Journal of Mathematics and Mathematical Sciences, 2009, Article ID: 283028, 10 p. doi:10.1155/2009/283028

[8] G., Jungck, “Commuting Mappings and Fixed Point,” American Mathematical Monthly, Vol. 83, 1976, pp. 261263. doi:10.2307/2318216

[9] G. Jungck, “Compatible Mappings and Common Fixed Points,” International Journal of Mathematics and Mathematical Sciences, Vol. 9, No. 4, 1986, pp. 771-779. doi:10.1155/S0161171286000935

[10] B. S. Choudhury, S. Kumar, Asha and K. Das, “Some Fixed Point Theorems in G-Metric Spaces,” Nonlinear Science Letters, Vol. 1, No. 1, 2012, pp. 25-31.

[11] G., Jungck, “Common fxed Points for Non-Continuous Non-Self Mappings on Non-Metric Spaces,” Far East Journal of Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199-215.

[12] M. Aamri and D. E. I. Moutawakil, “Some New Common Fixed Point Theorems under Strict Contractive Conditions,” Journal of Mathematical Analysis and Applications, Vol. 270, 2002, pp. 181-188. doi:10.1016/S0022-247X(02)00059-8

[13] H. K. Pathak, R. Rodriguez-Lopez and R. K. Verma, “A Common Fixed Point Theorem Using Implicit Relation and E.A. Property in Metric Spaces,” Filomat, Vol. 21, No. 2, 2007, pp. 211-234. doi:10.2298/FIL0702211P

[14] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces,” Journal of Applied Mathematics, 2011, Article ID: 637958, 14 p. doi:10.1155/2011/637958