Some remarks on generalized Sehgal-Guseman-like contractions and their fixed-point results

Fabiano, Nicola; Mirkov, Nikola; Radenovic, Stojan

doi:10.30495/maca.2024.2028255.1099

Some remarks on generalized Sehgal-Guseman-like contractions and their fixed-point results

10.30495/maca.2024.2028255.1099

Volume 6, Issue 2
Spring 2024

Pages 1-15

Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Authors

Nicola Fabiano ¹

Nikola Mirkov ¹

Stojan Radenovic ²

¹ Vinca Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, Mike Petrovica Alasa 12-14, 11351 Belgrade, Serbia

² Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd 35, Serbia

Abstract

The aim of this paper is to shed some light on the recently introduced generalized Sehgal-Guseman-like contractions with the help of D-functions from other aspects. For this purpose, we used the recently established connection of S-metric and b-metric spaces and thus switched to a new form of generalized Sehgal-Guseman-like contraction. Using that connection as well as the notion of a D-function, we were able to significantly improve recent results on generalized Sehgal-Guseman-like contractions and remove some of the confusion readers had when reading recent work on this topic. At the end of the paper, we listed all 120 possible generalized Sehgal-Guseman-like contractions within both S-metric and b-metric spaces. We admit that for many of them, we do not yet know whether they have a (unique) fixed point. This opens the way for further research of generalized Sehgal-Guseman-like contractions in the mentioned frameworks of certain generalized metric spaces.

Keywords

S-metric space

b-metric space

fixed point

generalized Sehgal-Guseman-like-contraction

$\mathcal{D}-$class functions

[1] J. M. Afra, M. Sabbaghan, and F.Taleghani, Some new fixed point theorems for expansive map on S-metric spaces, Theory Approx. Appl., 14(2) (2020), 57–64.

[2] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26–37.

[3] S. Chaipornjareansri, Fixed point theorems for Fw−contractions in complete S-metric spaces, Thai J. Math., 14(Special Issue) (2016), 98–109.

[4] L. Ciric, Some Recent Results in Metrical Fixed Point Theory, University of Belgrade, Beograd, Serbia, 2003.

[5] K. M. Devi, Y. Rohen, and K. A. Singh, Fixed points of modified F-contractions in S-metric spaces, J. Math. Comput. Sci., 12 (2022), 197.

[6] M. Din, U. Ishtiaq, M. Mukhtar, S.Sessa, and H. A. Ghazwani, On generalized Sehgal-Guseman-like contractions and their fixed-point results with applications to nonlinear fractional differential equations and boundary value problems for homogeneous transverse bars, Mathematics, 12 (2024), 541.

[7] T. Dosenovic, S. Radenovic, and S. Sedghi, Generalized metric spaces: Survey, TWMS. J. Pure Appl. Math., 9(1) 2018, 3–17.

[8] T. Dosenovic and S. Radenovic, A comment on “Fixed point theorems of JS-quasicontractions”, Indian J. Math. Dharma Prakash Gupta Mem., 60(1) (2018), 141–152.

[9] T. Dosenovic, M. Pavlovic, and S. Radenovic, Contractive conditions in b-metric spaces, Vojno Tehnic. Glasnik/Milit. Tech. Courier, 65(4) (2017), 851–865.

[10] N. Fabiano, Z. Kadelburg, N. Mirkov, V. Sesum Cavic, and S. Radenovic, On F-contractions: A Survey, Contemporary Mathematics, 3(3) (2022), 327.

[11] A. Gupta, Cyclic contraction on S-metric space, Int. J. Anal. Appl., 3(2) (2013), 119–130.

[12] L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc., 26 (1970), 615–618.

[13] G. E. Hardy and T.D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973), 201–206.

[14] N. T. Hieu, N. T. Ly, and N.V. Dung, A generalization of Ciric quasi-contractions for maps on S-metric spaces, Thai J. Math. 13(2) (2015), 369–380.

[15] K. Javed, F. Uddin, F. Adeel, M. Arshad, H. Alaeidizaji, and V. Parvaneh, Fixed point results for generalized contractions in S-metric spaces, Math. Anal. Contemp. Appl., 3(2) (2021), 27–39.

[16] M. Joshi, A. Tomar, and T. Abdeljawad, On fixed points, their geometry and application to satellite web coupling problem in S-metric spaces, AIMS Math., 8 (2023), 4407–4441.

[17] R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19 (2017), 2153–2163.

[18] Z. D. Mitrovic, An extension of fixed point theorem of Sehgal in b-metric spaces, Commun. Appl. Nonlinear Anal., 25(2) (2018), 54–61.

[19] Z. D. Mitrovi´c, A fixed point theorem for mappings with a contractive iterate in rectangular b-metric spaces, Mate. Vesnik, 70(3) (2018), 204–210.

[20] T. Phaneendra and K. K. Swamy, Fixed points of Chatterjea and Ciric contractions on an S-metric space, Int. J. Pure Appl. Math., 115 (2017), 361–367.

[21] T. Phaneendra, Banach and Kannan contractions on S-Metric Space, Ital. J. Pure Appl. Math., 39 (2018), 243–247.

[22] S. Radenovic, Z. Kadelburg, D. Jandrlic, and A. Jandrlic, Some results on weakly contraction maps, Bull. Iran. Math. Soc., 38(3) (2012), 625–645.

[23] Y. Rohen, T. Doˇsenovi´c, and S. Radenovi´c, A note on the paper “A fixed point theorems in Sb−metric spaces”, Filomat, 31(11) (2017), 3335–3346.

[24] G. S. Saluja, Fixed point results for generalized Fσ−contraction in complete S-metric spaces, Ann. Math. Comput. Sci., 20 (2024), 25–40.

[25] V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc., 23 (1969), 631–634.

[26] S. Sedghi, N. Shobe, and A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mate. Vesnik, 64(3) (2012), 258–266.

[27] S. Sedghi and N.V. Dung, Fixed point theorems on S-metric spaces, Mate. Vesnik, 66(1) (2014), 113–124.

[28] S. Sedghi, M.M. Rezaee, T. Doˇsenovi´c, and S. Radenovi´c, Common fixed point theorems for contractive mappings satisfying Φ-maps in S-metric spaces, Acta Univ. Sapientiae Math., 8(2) (2016), 298–311.

[29] T. Thaibema, Y. Rohen, T. Stephen, and O. Budhichandra, Fixed point of rational Fcontractions in S-metric spaces, J. Math. Comput. Sci., 12 (2022), 153.

[30] I. Vul’pe, D. Ostraih, and F. Hoiman, The topological structure of a quasimetric space, Investigations in functional analysis and differential equations, Math. Sci., Interuniv. Work Collect., Shtiintsa, 1981, pp. 13–19.

[31] M. Zhoua, X.-l. Liu, and S. Radenovi´c, S-γ-ϕ-φ-contractive type mappings in S-metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 1613–1639.

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