Uniqueness of Common Fixed Points for a Family of Mappings with *Φ*-Contractive Condition in 2-Metric Spaces

Author(s)
Yong-Jie Piao

Abstract

In this paper, we will introduce a class of 5-dimensional functions Φ and prove that a family of self-mappings {T_{i,j}} _{iεN} in 2-metric space have an unique common fixed point if 1) {T_{i,j}} _{iεN} satisfies Φ_{j}-contractive condition, where Φ_{j}εΦ, for each jεN ; 2) T_{m,μ} _{n,v} for all m,n,μ,vεN with μ ≠ v . Our main result generalizes and unifies many known unique common fixed point theorems in 2-metric spaces.

In this paper, we will introduce a class of 5-dimensional functions Φ and prove that a family of self-mappings {T

Cite this paper

Y. Piao, "Uniqueness of Common Fixed Points for a Family of Mappings with*Φ*-Contractive Condition in 2-Metric Spaces," *Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 73-77. doi: 10.4236/am.2012.31012.

Y. Piao, "Uniqueness of Common Fixed Points for a Family of Mappings with

References

[1] Y. J. Piao, G. Z. Jin and B. J. Zhang, “A Family of Selfmaps Having an Unique Common Fixed Point in 2-Metric Spaces,” Yanbian University (Natural Science), Vol. 28, No. 1, 2002, pp. 1-5.

[2] H. S. Yang and D. S. Xiong, “A Common Fixed Point Theorem on p-metric Spaces,” Journal of Yunnan Normal University (Science Edition), Vol. 21, No. 1, 2001, pp. 9-12.

[3] S. L. Singh, “Some Contractive Type Principles on 2-Metric Spaces and Applications,” Mathematics Seminar Notes (Kobe University), Vol. 7, No. 1, 1979, pp. 1-11.

[4] I. S. Kim, “Common Fixed Point Theorems in 2-Metric Spaces,” Master’s Thesis, Korea Soongsil University, Seoul, 1994.

[5] Y. J. Piao and Y. F. Jin, “Unique Common Fixed Point Theorem for a Family of Contractive type Non-Commuting Selfmaps in 2-Metric Spaces,” Journal of Yanbian University (Science Edition), Vol. 32, No. 1, 2006, pp. 1-3.

[6] Y. J. Piao, “A Family of Quasi-Contractive Type Non-Commutative Self-Maps Having an Unique Common Fixed Point in 2-Metric Spaces,” Journal of Heilongjiang University (Science Edition), Vol. 23, No. 5, 2006, pp. 655-657.

[7] Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Type Contractive Condition in 2-Metric Spaces,” Analysis in Theory and Applications, Vol. 24, No. 4, 2008, pp. 316-320.
doi:10.1007/s10496-008-0316-9

[8] Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Quasi-Contractive Type Condition in 2-Metric Space,” Journal of Nanjing University (Mathematical Biquarterly), Vol. 27, No.1, 2010, pp. 82-87.