Abstract

The study of common fixed point theory with different types of mappings and contractive conditions takes a significant role in present research activity. At first, the great mathematician Brouwer introduced the famous fixed-point theory in 1912. According to Brouwer, “Every continuous function from the closed and bounded subset of Rn into itself has a fixed point”. After that, the great mathematician S. Banach in 1922 invented the well-known fixed point theory, namely Banach Fixed Point Theory. “Huang & Zhang [3] have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space [3]”. The present research article is about some common fixed-point theorems for “self-mappings and commuting mappings” on Cone and Cone Rectangular Metric Space.

Keywords

Cone & Cone Rectangular Metric, Completeness, Cauchy Sequence, Fixed point.

References

[1] M. Abbas, and G. Jungck, “Common Fixed-Point Results for Non-Commuting Mappings without Continuity in Cone Metric Spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416-420, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Akbar Azam, Muhammad Arshad, and Ismat Beg, “Banach Contraction Principle on Cone Rectangular Metric Spaces,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 2, pp. 236-241, 2009.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Huang Long-Guang, and Zhang Xian, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468-1476, 2007.
 [CrossRef] [Google Scholar] [Publisher Link]

[4] Gerald Jungck, “Commuting Mappings and Fixed Points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261-263, 1976.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Gerald Jungck, “Compatible Mappings and Common Fixed Points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771-779, 1986.
[CrossRef] [Google Scholar] [Publisher Link]

[6] G. Jungck, and B.E. Rhoades, “Fixed Points for Set Valued Functions without Continuity,” Indian Journal of Pure and Applied Mathematics, vol. 29, pp. 227-238, 1998.
[Google Scholar]

[7] R.P. Pant, “Common Fixed Points of Non-Commutating Mappings,” Journal of Mathematical Analysis and Applications, vol. 188, pp. 436-440, 1994.
[CrossRef] [Google Scholar] [Publisher Link]

[8] K. Jha, “A Common Fixed Point theorem in A cone Metric Space,” Kathmandu University Journal of Science, Engineering and Technology, vol. 5, no. 1, pp. 1-5, 2009.
[Google Scholar]