International Journal of Nonlinear Analysis and Applications

Conditional reciprocal continuity and a common fixed point in a $b$-metric space

Document Type : Research Paper

Authors

1 Department of Mathematics, Vellore Institute of Technology, Vellore, Tamilnadu, India

2 Department of Mathematics, MVSR Engineering College, Rangareddy, Hyderabad-501510, Telangana State, India

Abstract

A unique common fixed point is obtained for compatible and non-compatible self-maps on a $b$-metric space, through the notion of conditional reciprocal continuity, due to Pant and Bist.

Keywords

[1] M. A. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002), 181–188.
[2] K. Afassinou, A. A. Mebawondu, H. A. Abbas, and O. K. Narain, Existence of solution of differential and Riemann-Liouville equation via fixed point approach in complex valued b-metric spaces, Austr. J. Math. Anal. and Appl. 18 (2011), no. 1, 15.
[3] A. Azam, B. Fisher, and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim. 32 (2011), 243–253.
[4] I. A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst. 30 (1989), 26–37.
[5] B. C. Dhage, Generalized metric spaces and mappings with fixed point, Bull. Calcutta Math. Soc. 84 (1992), no. 4, 329–336.
[6] Jungck Gerald, Compatible self-maps and common fixed point, Int. J. Math. & Math. Sci. 9 (1986), 771–779.
[7] Jungck Gerald, Common fixed points for non-continuous and non-self-mappings on a metric space, Far East J. Math. Sci. 4 (1996), no. 2, 199–212.
[8] A. A. Mebawondu, H. A. Abass, M. O. Aibinu, and O. K. Narain, Existence of solution of differential equation via fixed point in complex valued b-metric spaces, Nonlin. Fun. Anal. & Appl. 26 (2021), no. 1, 1–30.
[9] A. A. Mukheimer, Some common fixed point theorems in complex valued b-metric spaces, Sci. World J. (2014), 1–6.
[10] R. P. Pant, Common fixed points of non-commuting mappings, J. Math. Anal. Appl. 188 (1994), 436–440.
[11] , A common fixed point theorem under a new condition, Indian J. Pure Appl. Math. 30 (1999), no. 2, 147–152.
[12] R. P. Pant and R. K. Bist, Common fixed point theorems under a new continuity condition, Ann Univ. Ferrara.
[13] R. P. Pant, R. K. Bist, and D. Arora, Weak reciprocal continuity and fixed point theorems, Ann Univ. Ferrara 57 (2011), 181–190.
[14] H. K. Pathak, Y. J. Cho, and S. M. Kang, Remarks on r-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc. 17 (1997), 247–257.
[15] H. K. Pathak and M. S. Khan, Two generalized common fixed point theorems involving compatibility and property E.A., Demon. Math. 28 (2014), no. 2-3, 449–458.
[16] T. Phaneendra and V. Sivarama Prasad, A comparison of various types of compatible maps and common fixed points, Indian. J. pure appl. Math. 28 (1997), no. 4, 477–485.
[17] J. R. Roshan, N. Shobkolaei, S. Sedghi, and M. Abbas, Common fixed point of four maps in b-metric spaces, Hacettepe Journal of Mathematics and Statistics 43 (2014), no. 3, 613–624.
[18] S. Sessa, On weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. Debre. 32 (1982), 149–153.
[19] S. L. Singh and Anita Tomar, Weaker forms of commuting maps and existence of fixed points, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 10 (2003), no. 3, 145–161.
[20] B. Wu, F. He, and T. Xu, Common fixed point theorems for Ciric type mappings in b-metric spaces without any completeness assumption, J. Nonlinear Sci. Appl. 10 (2017), 3180–3190.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 1219-1227
  • Receive Date: 04 July 2021
  • Revise Date: 18 August 2021
  • Accept Date: 21 August 2021