Common fixed point results for graph preserving mappings in parametric $N_b$-metric spaces

Subject Areas : Fixed point theory

S. Kumar Mohanta ¹ , R. Kar ²

1 - Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), Kolkata-700126, West Bengal, India
2 - Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), Kolkata-700126, West Bengal, India

Received: 2019-07-27 Accepted : 2020-06-21 Published : 2020-06-01

Keywords: fixed point, digraph, Parametric $N_b$-metric, coincidence point,

Abstract :

In this paper, we discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of graph preserving mappings in parametric $N_b$-metric spaces. As some consequences of this study, we obtain several important results in parametric $b$-metric spaces, parametric $S$-metric spaces and parametric $A$-metric spaces. Finally, we provide some illustrative examples to justify the validity of our main result.

References:

[1] M. Abbas, B. Ali, Y. I. Suleiman, Generalized coupled common fixed point results in partially ordered A-
metric spaces, Fixed Point Theory and Appl. (2015), 2015:64.

[2] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416-420.

[3] S. Aleomraninejad, S. Rezapour, N. Shahzad, Fixed point results on subgraphs of directed graphs, Math. Sci. (2013), 7:41.

[4] M. R. Alfuraidan, M. A. Khamsi, Caristi fixed point theorem in metric spaces with a graph, Abstr. Appl. Anal. (2014), 2014:303484.

[5] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Gos. Ped. Inst.
Unianowsk. 30 (1989), 26-37.

[6] S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fund. Math. 3 (1922), 133-181.

[7] J. A. Bondy, U. S. R. Murty, Graph theory with applications, American Elsevier Publishing Co., Inc., New York, 1976.

[8] F. Bojor, Fixed point of φ-contraction in metric spaces endowed with a graph, Annals of the University of Craiova, Math. Comp. Sci. Series. 37 (2010), 85-92.

[9] F. Bojor, Fixed points of Kannan mappings in metric spaces endowed with a graph, An. St. Univ. Ovidius Constanta. 20 (2012), 31-40.

[10] V. Berinde, Approximating common fixed points of noncommuting discontinuous weakly contractive mappings in metric spaces, Carpathian J. Math. 25 (2009), 13-22.

[11] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5-11.

[12] G. Chartrand, L. Lesniak, P. Zhang, Graph and digraph, CRC Press, New York, USA, 2011.

[13] F. Echenique, A short and constructive proof of Tarski’s fixed point theorem, Internat. J. Game Theory. 33 (2) (2005), 215-218.

[14] R. Espinola, W. A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topology Appl. 153 (2006), 1046-1055.

[15] J. I. Gross, J. Yellen, Graph theory and its applications, CRC Press, New York, USA, 1999.

[16] N. Hussain, S. Khaleghizadeh, P. Salimi, A. A. N. Abdou, A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces, Abstr. Appl. Anal. (2014), 2014:690139.

[17] N. Hussain, P. Salimi, V. Parvaneh, Fixed point results for various contractions in parametric and fuzzy b-metric spaces, J. Nonlinear Sci. and its Appl. 8 (2015), 719-739.

[18] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (4) (2008), 1359-1373.

[19] G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci. 4 (1996), 199-215.

[20] R. Krishnakumar, N. P. Sanatammappa, Fixed point theorems in parametric metric space, Int. J. Math. Research. 8 (3) (2016), 213-220.

[21] W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory. 4 (1) (2003), 79-89.

[22] X. Liu, M. Zhou, L. N. Mishra, V. N. Mishra, B. Damjanovi´ c, Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property, Open Mathematics. 16 (2018), 1423-1434.

[23] S. K. Mohanta, A fixed point theorem via generalized w-distance, Bull. Math. Anal. Appl. 3 (2) (2011), 134-139.

[24] S. K. Mohanta, Common fixed points for mappings in G-cone metric spaces, J. Nonlinear Anal. Appl. 2012 (2012), 1-13.

[25] S. K. Mohanta, Generalized w-distance and a fixed point theorem, Int. J. Contemp. Math. Sci. 6 (18) (2011), 853-860.

[26] S. K. Mohanta, S. Patra, Coincidence points and common fixed points for hybrid pair of mappings in b-metric spaces endowed with a graph, J. Linear. Topological. Algebra. 06 (4) (2017), 301-321.

[27] S. K. Mohanta, D. Biswas, Common fixed points for a pair of mappings in b-metric spaces via digraphs and altering distance functions, J. Linear. Topological. Algebra. 07 (3) (2018), 201-218.

[28] L. N. Mishra, V. N. Mishra, P. Gautam, K. Negi, Fixed point theorems for cyclic-Ciric-Reich-Rus contraction mapping in quasi-partial b-metric spaces, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech. 12 (1) (2020), 47-56.

[29] L. N. Mishra, S. K. Tiwari, V. N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, J. Appl. Anal. Comput. 5 (4) (2015), 600-612.

[30] L. N. Mishra, S. K. Tiwari, V. N. Mishra, I. A. Khan, Unique fixed point theorems for generalized contractive mappings in partial metric spaces, J. Function Spaces. (2015), 2015:960827.

[31] N. Priyobarta, Y. Rohen, S. Radenovic, Fixed point theorems on parametric A-metric space, Amer. J. Appl. Math. Stat. 6 (1) (2018), 1-5.

[32] A. G. Sanatee, M. Iranmanesh, L. N. Mishra, V. N. Mishra, Generalized 2-proximal C-contraction mappings in complete ordered 2-metric space and their best proximity points, Scientific Publications of the State University of Novi Pazar Ser. A: Appl. Math. Inform. and Mech. 12 (1) (2020), 1-11.

[33] B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal. 13 (2) (2012), 82-97.

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

[35] N.Tas, N. Y. Ozgur, On parametric S-metric spaces and fixed-point type theorems for expansive mappings, J. Math. (2016), 2016:4746732.

[36] N.Tas, N. Y. Ozgur, Some fixed point results on parametric Nb-metric spaces, Commun. Korean Math. Soc. 33 (3) (2018), 943-960.