Four Mappings Satisfying Ψ-Contractive Type Condition and Having Unique Common Fixed Point on 2-Metric Spaces

Hailan Jin, Yongjie Piao

Department of Mathematics, College of Science, Yanbian University, Yanji, China.

DOI: 10.4236/apm.2013.32039                

Abstract

In this paper, we introduce a class Ψ of real functions defined on the set of non-negative real numbers, and obtain a new unique common fixed point theorem for four mappings satisfying Ψ-contractive condition on a non-complete 2-metric space and give the versions of the corresponding result for two and three mappings.

Keywords

2-Metric Space; Class Ψ; Cauchy Sequence; Coincidence Point; Common Fixed Point

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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.
[2]	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.
[3]	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.
[4]	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.
[5]	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
[6]	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.
[7]	Y. J. Piao, “Uniqueness of Common Fixed Point for a Family of Mappings withφ-Contractive Condition in 2-Metric Space”, Applied Mathematics, Vol. 3, 2012, pp. 73-77. doi:10.4236/am.2012.31012
[8]	Y. J. Piao and Y. F. Jin, “New Unique Common Fixed Point Results for Four Mappings withΦ-Contractive Type in 2-Metric Spaces,” Applied Mathematics, Vol. 3, 2012, pp. 734-737. doi:10.4236/am.2012.37108
[9]	M. Abbas and G. Jungck, “Common Fixed Point Results for Noncommuting Mappings without Continuity in Cone Metric Spaces,” Journal of Mathematical Analysis and Applications, Vol. 341, No. 1, 2008, pp. 416-420. doi:10.1016/j.jmaa.2007.09.070
[10]	Y. Han and S. Y. Xu, “New Common Fixed Point Results for Four Maps on Cone Metric Spaces,” Applied Mathematics, Vol. 2, 2011, pp.1114-1118. doi:10.4236/am.2011.29153
[11]	C. D. Bari and P. Vetro, “Φ-Pairs and Common Fixed Points in Cone Metric Spaces,” Rendicontidel Circolo Matematico Palermo, Vol. 57, 2008, pp. 279-285.