International Journal of Nonlinear Analysis and Applications

Coupled fixed points of generalized rational type Z-contraction maps in b-metric spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Acharya Nagarjuna University, Guntur - 522 510, India

2 Department of Mathematics, PSCMRCET, Vijayawada - 520 001, India

3 Department of Mathematics, Rayalaseema University, Kurnool - 518 007, India

4 Department of Mathematics, PBR VITS, Kavali- 524 201, India

Abstract

In this paper, we introduce generalized rational type $\mathcal{Z}$-contraction maps for a single map $f:X\times X\to X$ where $X$ is a $b$-metric space and prove the existence and uniqueness of coupled fixed points. We extend it to a pair of maps by defining generalized rational type $\mathcal{Z}$-contraction pair of maps and prove the existence of common coupled fixed points in complete $b$-metric spaces. We provide examples in support of our results.

Keywords

[1] A. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64 (2014), no. 4, 941–960.
[2] H. Aydi, M-F. Bota, E. Karapinar and S. Mitrovi¸c, A fixed point theorem for set-valued quasi contractions in b-metric spaces, Fixed Point Theory Appl. 88 (2012), 8 pages.
[3] H. Aydi, M-F. Bota, E. Karapinar and S. Moradi, A common fixed point for weak ϕ-contractions on b-metric spaces, Fixed Point Theory 13 ( 2012), no. 2, 337–346.
[4] G.V.R. Babu, T.M. Dula and P.S. Kumar, A common fixed point theorem in b-metric spaces via simulation function, J. Fixed Point Theory 12 (2018), 15 pages.
[5] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. Gos. Ped. Inst. Unianowsk 30 (1989), 26–37.
[6] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379–1393.
[7] B. Bindu and N. Malhotra, Common coupled fixed point for generalized rational type contractions in b-metric spaces, J. Nonlinear Analy. Appl. 2018 (2018), no. 2, 201–211.
[8] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math. 4 (2009), no. 3, 285–301.
[9] M. Boriceanu, M.-F. Bota and A. Petrusel, Multivalued fractals in b-metric spaces, Cent. Eur. J. Math. 8 (2010), no. 2, 367-377.
[10] N. Bourbaki, Topologie generale, Herman: Paris, France, 1974.
[11] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5–11.
[12] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 263–276.
[13] B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions, Indian J. Pure Appl. Math. 6 (1975), 1455–1458.
[14] H. Huang, L. Paunovi¸c and S. Radenovi¸c, On some fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl. 8 (2015), 800–807.
[15] N. Hussain, V. Paraneh, J.R. Roshan and Z. Kadelburg, Fixed points of cycle weakly (ψ, φ, L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl. 2013 (2013), 256, 18 pages.
[16] N. Hussain, J.R. Roshan, V. Parvaneh and M. Abbas, Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications, J. Inequal. Appl. 2013 (2013), 486, 21 pages.
[17] F. Khojasteh, S. Shukla and S. Redenovic, A new approach to the study fixed point theorems via simulation functions, Filomat 29 (2015), no. 6, 1189–1194.
[18] P. Kumam and W. Sintunavarat, The existence of fixed point theorems for partial q-set valued quasi-contractions in b-metric spaces and related results, Fixed Point Theory Appl. 2014 (2014), 226, 20 pages.
[19] V. Lakshmikantham and L. Ciric,  Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349.
[20] N. Malhotra and B. Bansal, Some common coupled fixed point theorems for generalized contraction in b-metric spaces, J. Nonlinear Sci. Appl. 8 (2015), 8–16.
[21] M. Olgun, O. Bicer and T. Alyildiz, A new aspect to Picard operators with simulation functions, Turk. J. Math. 40 (2016), 832–837.
[22] N.S. Prasad, D.R. Babu and V.A. Babu, Common coupled fixed points of generalized contraction maps in b-metric spaces, Electron. J. Math. Anal. Appl. 9 (2021), no. 1, 131–150.
[23] R.J. Shahkoohi and A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered-metric spaces, J. Inequal. Appl. 2014 (2014), no. 1, 1–23.
[24] W. Shatanawi, Fixed and common fixed point for mappings satisfying some nonlinear contractions in b-metric spaces, J. Math. Anal. 7 (2016), no. 4, 1–12.
[25] W. Shatanawi and M.B. Hani, A coupled fixed point theorem in b-metric spaces, Int. J. Pure Appl. Math. 109 (2016), no. 4, 889–897.
[26] W. Shatanawi, B. Samet and M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comp. Model. 55 (2012), 680–687.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 789-802
  • Receive Date: 05 April 2020
  • Revise Date: 08 June 2020
  • Accept Date: 30 June 2020