Document Type: Original Article

**Authors**

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

**Abstract**

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.

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.

**Keywords**

[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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

(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.

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.

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.

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.

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.

in ordered b-metric spaces, J. Appl. Math., Article ID 929821, 2014, 9 pages.

Volume 6, Issue 1

Summer and Autumn 2019

Pages 40-59