Homeomorphism metric space and the fixed point theorems

Volume 10, Issue 10, pp 5132--5141 Publication Date: October 12, 2017
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Authors

Yinglin Luo - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Yongfu Su - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Wenbiao Gao - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Abstract

The purpose of this paper is to introduce the concept of the homeomorphism metric space and to prove the fixed point theorems and the best proximity point theorems for generalized contractions in such spaces. The multiplicative metric space is a special form of the homeomorphism metric space. The results of this paper improve and extend the previously known ones in the literature.

Keywords

• Homeomorphism metric space
• multiplicative metric space
• metric space
• $b$-metric space
• generalized contraction
• fixed point
• best proximity point

References

• [1] R. P. Agarwal, E. Karapınar, B. Samet , An essential remark on fixed point results on multiplicative metric spaces, Fixed Point Theory Appl., 2016 (2016), 3 pages.

• [2] A. E. Bashirov, E. M. Kurplnar, A. Ozyaplcl, Multiplicative calculus and its applications , J. Math. Anal. Appl., 337 (2008), 36–48.

• [3] A. E. Bashirov, E. Misirli, Y. Tandogdu, A. Ozyaplcl , On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ., 26 (2011), 425–438.

• [4] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.

• [5] K. Fan , Extensions of two fixed point theorems of F. E. Browder, Math. Z., 112 (1969), 234–240.

• [6] L. Florack, H. Van Assen , Multiplicative calculus in biomedical image analysis, J. Math. Imaging Vision, 42 (2012), 64–75.

• [7] W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24 (2003), 851–862.

• [8] Y. Su, J.-C. Yao, Further generalized contraction mapping principle and best proximity theorem in metric spaces, Fixed Point Theory Appl., 2015 (2015), 13 pages.

• [9] J. Zhang, Y. Su, Q. Cheng, A note on ‘A best proximity point theorem for Geraghty-contractions’, Fixed Point Theory Appl., 2013 (2013), 4 pages.

• [10] J. Zhang, Y. Su, Q. Cheng , Best proximity point theorems for generalized contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2013 (2013), 7 pages.