Remarks on the Paper ``Coupled Fixed Point Theorems for Single-Valued Operators in b-Metric Spaces''

Document Type : Original Scientific Paper

Authors

1 University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Serbia

2 University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11000 Beograd, Serbia

3 Department of Mathematics, University of Malakand, Chakdara, Dir (Lower), Khyber Pakhtunkhwa, Pakistan, 18800

Abstract

In this paper, we improve some recent coupled fixed point results for single-valued operators in the framework of ordered b-metric spaces established by Bota et al. [M-F. Bota, A. Petrusel, G. Petrusel and B. Samet, Coupled fixed point theorems for single-valued operators in b-metric spaces, Fixed Point Theory Appl. (2015) 2015:231]. Also, we prove that Perov-type fixed point theorem in ordered generalized $b$-metric spaces is equivalent with Ran-Reurings-type theorem in ordered b-metric spaces.

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1. A. Amini-Harandi, Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem, Math. Comput. Model. 57 (2013) 2343–2348.
2. I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) In: Functional Analysis, No. 30 (Russian), 26–37, Ul’yanovsk. Gos. Ped. Inst., Ul’yanovsk, 1989.
3. V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 7347– 7355.
4. T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393.
5. M.-F. Bota, A. Petrusel, G. Petrusel, B. Samet, Coupled fixed point theorems for single-valued operators in b-metric spaces, Fixed Point Theory Appl. 2015 (2015) 231, 15 pp.
6. S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993) 5–11.
7. N. V. Dung, V. T. L. Hang, On relaxations of contraction constants and Caristi’s theorem in b-metric spaces, J. Fixed Point Theory Appl. 18 (2) (2016) 267–284.
8. S. Janković, Z. Kadelburg, S. Radenović, On cone metric spaces: a survey, Nonlinear Anal. 74 (2011) 2591–2601.
9. M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed Point Theory Appl. (2010) 978121, 15 pages.
10. N. Jurja, A Perov-type fixed point theorem in generalized ordered metric spaces, Creat. Math. Inform. 17 (3) (2008) 427–430.
11. A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Približ, Metod. Rešen. Differencial’. Uravnen. Vyp. 2 (1964) 115–134.
12. S. Radenović, Remarks on some coupled coincidence point results in partially ordered metric spaces, Arab. J. Math. Sci. 20 (1) (2014) 29–39.
13. A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and  some applications to matrix eqations, Proc. Amer. Math. Soc. 132 (5) (2004) 1435–1443.
14. Gh. Soleimani Rad, S. Shukla and H. Rahimi, Some relations between n-tuple fixed point and fixed point results, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 109 (2) (2015) 471–481.
15. C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Math. Notes 14 (1) (2013) 323–333.