A survey of $\mathcal{C}-$class and pair upper-class functions in fixed point theory

Document Type : Review articles

Authors

1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Government Degree College Thatyur (Tehri Garhwal) Uttarakhand, India

3 S. G. R. R. (P. G.) College Dehradun, India

Abstract

We demonstrate  that the $\mathcal{C}-$class functions, pair $(h,f)$ upper-class functions, cone $\mathcal{C}-$class functions, $1-1-$ up-class functions, multiplicative $\mathcal{C}-$class functions, inverse$-\mathcal{C}-$class functions, $\mathcal{C}_{F}-$simulation functions, $\mathcal{C}^{*}-$class functions are powerful and fascinating weapons for the generalization, improvement, and extension of considerable conclusions obtained in the fixed point theory. Towards the end, we point out some open problems whose answers could be interesting.

Keywords

[1] C.T. Aage and J.N. Salunke, Fixed points of (ψ −ϕ)−weak contractions in cone metric spaces, Ann. Funct. Anal. 2(1) (2003) 59—71.
[2] R.P. Agarwal, E. Karapınar and B. Samet, An essential remark on fixed point results on multiplicative metric spaces, Fixed Point Theory Appl. 21 (2016).
[3] Ya.I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, New Results in Operator Theory and its applications, Springer (1997) 7–22.[4] A. Amini-Harandi and M. Fakhar, Fixed point theory in cone metric spaces obtained via the scalarization method, Comput. Math. Appl. 59(11) (2010) 3529–3534.
[5] C. Ampadu and A.H. Ansari, Fixed point heorems in complete multiplicative metric spaces with application to multiplicative analogue of C-class functions, J. Fixed Point Theory Appl. 11(2) (2016) 113–124.
[6] A.H. Ansari, Note on φ − ψ-contractive type mappings and related fixed point, The 2nd Regional Conf. Math. Appl. Payame Noor University (2014) 377–380.
[7] A.H. Ansari, Note on α−admissible mappings and related fixed point theorems, The 2nd Regional Conf. Math. Appl. Payame Noor University (2014) 373–376.
[8] A.H. Ansari, S. Chandok and C. Ionescu, Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, J. Inequal. Appl. 429 (2014) 1–17.
[9] A.H. Ansari, M. Berzig and S. Chandok, Some Fixed Point Theorems for (CAB)-contractive Mappings and Related Results, Math. Moravica 19(2) (2015) 97–112.
[10] A.H. Ansari and S. Shukla, Some fixed point theorems for ordered F-(F, h)-contraction and subcontractions in 0-f-orbitally complete partial metric spaces, J. Adv. Math. Stud. 9(1) (2016) 37–53.
[11] A.H. Ansari, S. Chandok, N. Hussain and L. Paunovi´c, Fixed points of (ψ, φ)−weak contractions in regular cone metric spaces via new function, J. Adv. Math. Stud. 9(1) (2016) 72–82.
[12] A,H. Ansari, G.K. Jacob, M. Marudai and P. Kumam, On the C-class functions of fixed point and best proximity point results for generalised cyclic-coupled mappings, Cogent Math. Stat. 3(1) (2016) 1235354.
[13] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equation int´egrales, Fund. Math. 3 (1992) 133–181.
[14] V. Berinde, Contract¸ii generalizate ¸si aplicat¸ii, vol. 2, Editura Cub Press, Baia Mare, Romania, 1997
[15] M. Berzig, E. Karapınar and A. Rold´an., Discussion on generalized-(αψ, βφ)-contractive mappings via generalized altering distance function and related fixed point theorems, Abstr. Appl. Anal. 2014 (2014) Article ID 634371.
[16] M. Berzig and E. Karapınar, Fixed point results for (αψ, βϕ)-contractive mappings for a generalized altering distance, Fixed Point Theory Appl. 205(1) (2013) 1–18.
[17] S. Chandok, D. Kumar and C. Park, C∗−algebra valued partial metric space and fixed point theorems, Proc. Math. Sci. 129(3) (2019) 1–9.
[18] S.K. Chatterjea, Fixed point theorem, C. R. Acad. Bulgare Sci. 25 (1972) 727–730.
[19] P.N. Dutta and B.S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. 2008 (2008) 205.
[20] J. Dixmier, C ∗−Algebras, North-Holland Publ. Co., Amsterdam, New York, Oxford, 1977.
[21] M. Geraghty, On contractive mappings, Proc. Am. Math. Soc. 40 (1973) 604–608.
[22] L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332(2) (2007) 1468–1476.
[23] N. Hussain, E. Karapinar, P. Salimi and F. Akbar, α−admissible mappings and related fixed point theorems, J. Inequal. Appl. 1 (2013) 1–11.
[24] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal. 74 (2011) 768–774.
[25] M. Joshi, A. Tomar, H.A. Nabwey and R. George, On unique and nonunique fixed points and fixed circles in Mb
v−metric space and application to cantilever beam problem, J. Funct. Spaces 2021 (2021) Article ID 6681044.
[26] M. Joshi and A. Tomar, On unique and non-unique fixed points in metric spaces and application to chemical sciences, J. Funct. Spaces 2021 (2021) Article ID 5525472.
[27] M. Joshi, A. Tomar and S.K. Padaliya, On geometric properties of non-unique fixed points in b−metric spaces, Chapter in a book “Fixed Point Theory and its Applications to Real World Problem” Nova Science Publishers, New York, USA. (2021) 33–50.
[28] M. Joshi, A. Tomar and S.K. Padaliya, Fixed point to fixed disc and application in partial metric spaces, Chapter in a book “Fixed point theory and its applications to real world problem” Nova Science Publishers, New York, USA. (2021) 391–406.
[29] M. Joshi, A. Tomar and S.K. Padaliya, Fixed point to fixed ellipse in metric spaces and discontinuous activation function, Appl. Math. E-Notes 21 (2021) 225–237.
[30] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71–76.
[31] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distancces between the points, Bull. Aust. Math. Soc. 30 (1984) 1–9.
[32] F. Khojasteh, S. Shukla, S. Radenovi´c, A new approach to the study of fixed point theory for simulation functions, Filomat 29(6) (2015) 1189–1194.
[33] X.L. Liu, A. H. Ansari, S. Chandok and S. Radenovi´c, On some results in metric spaces using auxiliary simulationfunctions, J. Comput. Anal. Appl. 24(6) (2018) 1103–1114.
[34] Z. Ma, L. Jiang and H. Sun, C ∗−Algebra valued metric spaces and related fixed point theorems, Fixed Point Theory
Appl. 206 (2014).
[35] S. Moradi and E. Analoei, Common fixed point of generalized (ψ, ϕ)−weak contraction mappings, Int. J. Nonlinear Anal. Appl. 3(1) (2012) 24–30.
[36] N.Y. Ozg¨ur, N. Ta¸s and U. C¸ elik, ¨ New fixed-circle results on S−metric spaces, Bull. Math. Anal. 1134 Appl. 9(2) (2017) 10–23.
[37] N.Y. Ozg¨ur and N. Ta¸s, ¨ Some fixed-circle theorems and discontinuity at fixed circle, AIP Conf. Proc. 1926 (2018) 020048.
[38] N.Y. Ozg¨ur and N. Ta¸s, ¨ Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 4 (42) (2019) 1433–1449.
[39] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47(4) (2001) 2683–2693.
[40] A.F. Rold´an-L´opez-de Hierro, E. Karapınar, C. Rold´an-L´opez-de Hierro and J. Mart´ınez-Morenoa, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math. 275 (2015) 345–355.
[41] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.
[42] N. Saleem, A.H. Ansari and M.K. Jain, Some fixed point theorems of inverse C- class function under weak semi compatibility, J. Fixed Point Theory 9 (2018).
[43] P. Salimi, C. Vetro, P. Vetro, Fixed point theorems for twisted (α, β) ψ−contractive type mappings and applications, Filomat 27 (2013) 605–615.
[44] B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α − ψ contractive type mappings, Nonlinear Anal. 75 (2012) 2154–2165.
[45] S. Shukla and S. Radenovi´c, Some common fixed point theorems for F−contraction type mappings in 0−complete partial metric spaces, J. Math. 2013 (2013) Article ID 878730.
[46] S. Shukla, S. Radenovi´c and Z. Kadelburg, Some fixed point theorems for ordered F−generalized contractions in 0 − f−orbitally complete partial metric spaces, Theory Appl. Math. Comput. Sci. 4(1) (2014) 87–98.
[47] A. Tomar and M. Joshi, Note on results in C ∗−algebra valued metric spaces, Electron. J. Math. Anal. Appl. 9(2)
(2021) 262–264.
[48] A. Tomar, M. Joshi, A. Deep, Fixed points and its applications in C ∗−algebra valued partial metric space, TWMS
J. App. And Eng. Math. 11(2) (2021) 329–340.
[49] A. Tomar, M. Joshi, and S. K. Padaliya, Fixed Point to Fixed Circle and Activation Function in Partial Metric Space, J. Appl. Anal. 28(1) (2021).
[50] A. Tomar and M. Joshi, Near fixed point, near fixed interval circle and near fixed interval disc in metric interval space, Chapter in a book “Fixed Point Theory and its Applications to Real World Problem” Nova Science Publishers, New York, USA. (2021) 131-150.
[51] Q. Zhang and Y. Song, Fixed point theory for generalized φ−weak contractions, Appl. Math. Lett. 22 (2009) 75–78.
Volume 13, Issue 1
March 2022
Pages 1879-1896
  • Receive Date: 20 August 2020
  • Revise Date: 09 July 2021
  • Accept Date: 23 August 2021