Fixed Points of Almost Geraghty Contraction Type Maps/Generalized Contraction Maps With Rational Expressions in b-Metric Spaces

Document Type : Original Article


Department of Mathematics, Andhra University, Visakhapatnam-530 003, India


In this paper, we introduce almost Geraghty contraction type maps for a single self map and
prove the existence and uniqueness of fixed points. We extend it to a pair of selfmaps by defining
almost Geraghty contraction type pair of maps in which one of the maps is b-continuous in a
complete b-metric space. Further, we prove the existence of common fixed points for a pair of
selfmaps satisfying a generalized contraction condition with rational expression in which one of
the maps is b-continuous. Our results extend and generalize some of the known results that are
available in the literature. We draw some corollaries from our results and provide examples in
support of our results.


[1] M. Abbas, G. V. R. Babu, and G. N. Alemayehu, On common fixed points of weakly compatible
mappings satisfying ‘generalized condition (B)’, Filomat 25: 2(2011), 9-19.

[2] 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(4)(2014), 941-960.

[3] H. Aydi, M. F. Bota, E. Karapinar and S. Mitrovic, A fixed point theorem for set-valued quasi
contractions in b-metric spaces, Fixed Point Theory Appl., 88(2012), 8 pages.

[4] G. V. R. Babu, M. L. Sandhya and M. V. R. Kameswari, A note on a fixed point theorem of
Berinde on weak contractions, Carpathia. J. Math., 24(1)(2008), 8-12.
[5] G. V. R. Babu and P. Sudheer Kumar, Common fixed points of almost generalized (α,ψ,φ,F)-contraction
type mappings in b-metric spaces, J. Inter. Math. Virtual Inst., 9(2019), 123-137.
[6] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Func. Anal. Gos. Ped.
Inst. Unianowsk, 30(1989), 26-37.
[7] V. Berinde, Approximating fixed points weak contractions using Picard iteration, Nonlinear Anal.
Forum, 9(1)(2004), 43-53.
[8] V. Berinde, General contractive fixed point theorems for Ciric-type almost contraction in metric
spaces, Carpathia J. Math., 24(2)(2008), 10-19.
[9] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J.
Mod. Math., 4(3)(2009), 285-301.
[10] M. Boriceanu, M. Bota, and A. Petrusel, Multivalued fractals in b-metric spaces, Cent. Eur. J.
Math., 8(2)(2010), 367-377.
[11] N. Bourbaki, Topologie Generale, Herman: Paris, France, 1974.
[12] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis,
1(1993), 5-11.
[13] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti del Seminario
Matematico e Fisico (DellUniv. di Modena), 46(1998), 263-276.
[14] B. K. Dass and S. Gupta, An extension of Banach contraction principle through rational expressions,
Indian J. Pure and Appl. Math., 6(1975), 1455-1458.
[15] D. Dukic, Z. Kadelburg and S. Radenovic, Fixed points of Geraghty-type mappings in various
generalized metric spaces, Abstr. Appl. Anal.,(2011), Article ID 561245, 13 pages.
[16] H. Faraji, D. Savic, and S. Radenovic, Fixed point theorems for Geraghty contraction type mappings
in b-metric spaces and applications, Axioms, 8(34)(2019), 12 pages.
[17] H. Huang, G. Deng, and S. Radenovic, Fixed point theorems in b-metric spaces with applications
to differential equations, J. Fixed Point Theory. Appl., 2018, 24 pages.
[18] 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.
[19] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40(1973), 604-608.
[20] 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.
[21] H. Huang, L. Paunovic and S. Radenovic, On some fixed point results for rational Geraghty
contractive mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 8(2015), 800-807.
[22] 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.
[23] R. J. Shahkoohi and A. Razani, Some fixed point theorems for rational Geraghty contractive
mappings in ordered b-metric spaces, J. Inequal. Appl.,2014(1)(373), 23 pages.
[24] W. Shatanawi, Fixed and common fixed point for mappings satisfying some nonlinear contractions
in b-metric spaces, J. Math. Anal., 7(4)(2016), 1-12.
[25] F. Zabihi and A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings
in ordered b-metric spaces, J. Appl. Math., Article ID 929821, 2014, 9 pages.