Notes on Some Recent Papers Concerning $F$-Contractions in $b$-Metric Spaces

Year 2018, Volume: 1 Issue: 2, 108 - 112, 07.11.2018

Zoran Kadelburg , Stojan Radenovıć

https://doi.org/10.33205/cma.468813

Cited By: 24

Abstract

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

Keywords

$b$-Metric space, $F$-Contraction, $\alpha$-Admissible mappings

References

• [1] H. Afshari, H. Aydi and E. Karapinar, On generalized α-ψ-Geraghty contractions on b-metric spaces, Georgian Math. J., doi.org/10.1515/gmj-2017-00
• [2] M. U. Ali, T. Kamran and M. Postolache, Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem, Nonlinear Anal. Model. Control, 22(1) (2017), 17–30.
• [3] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Ulianowsk Gos. Ped. Inst., 30 (1989), 26–37.
• [4] Lj. Ciri c, V. Parvaneh and N. Hussain, Fixed point results for weakly α-admissible pairs, Filomat, 30 (14) (2016), 3697– 3713
• [5] M. Cosentino, M. Jleli, B. Samet and C. Vetro, Solvability of integrodifferential problems via fixed point theory in b-metric spaces, Fixed Point Theory Appl., 70 (2015), 1–15.
• [6] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11.
• [7] N. Hussain, M. A. Kutbi and P. Salimi, Fixed point theory in α-complete metric spaces with applications, Abstr. Appl. Anal., 2014, Article ID 280817, 1–1
• [8] A. Lukács and S. Kajántó, Fixed point theorems for various types of F-contractions in complete b-metric spaces, Fixed Point Theory, 19 (2018), 321–334.
• [9] R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19(3) (2017), 2153–2163.
• [10] H. K. Nashine, R. P. Agarwal and Z. Kadelburg, Solution to Fredholm integral inclusions via (F,δb)-contractions, Open Mathematics, 14 (2016), 1053–1064.
• [11] H. K. Nashine, R. P. Agarwal, S. Shukla and A. Gupta, Some fixed point theorems for almost (GF,δb)-contractions and applications, Fasciculi Math., 58 (2017), 123–143
• [12] M. Nazam, M. Arshad and M. Postolache, Coincidence and common fixed point theorems for four mappings satisfying (αs,F)-contraction, Nonlinear Anal. Model. Control, 23 (5) (2018), 664–69
• [13] M. Nazam, Ma Zhenhua, S. U. Khan and M. Arshad, Common fixed points of four maps satisfying F-contraction on b-metric spaces, J. Function Spaces, 2017, Art. ID 9389768, 11 pp.
• [14] S. K. Padhan, G. V. V. Jagannadha Rao, A. Al-Rawashdeh, H. K. Nashine and R. P. Agarwal, Existence of fixed points for γ-FG- conractive condition via cyclic (α,β)-admissible mappings in b-metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 5495–5508.
• [15] V. Parvaneh, N. Hussain and Z. Kadelburg, Generalized Wardowski type fixed point theorems viaα-admissible FG-contractions in b-metric spaces, Acta Math. Sci., 36 (5) (2016), 1445–1456.
• [16] H. Piri, S. Rahrovi, H. Marasi and P. Kumam, F-contraction on asymmetric metric spaces, J. Math. Comput. Sci., 17 (2017), 32–40.
• [17] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. TMA, 75(2012), 2154–216
• [18] T. Suzuki, Basic inequality on a b-metric space and its applications, J. Inequal. Appl., 256 (2017), 11pp.
• [19] D.Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 94 (2012), 6pp.