Document Type: Special Issue on Fixed Point Theory


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


In this paper, we discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings satisfying some generalized contractive type conditions in $b$-metric spaces endowed with graphs and altering distance functions. Finally, some examples are provided to justify the validity of our results.


Main Subjects

[1] 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.

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

[3] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca. 64 (2014), 941-960.

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

[5] V. Berinde, Some remarks on a fixed point theorem for C´iric´-type almost contractions, Carpath. J. Math. 25 (2009), 157-162.

[6] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund.
Math. 3 (1922), 133-181.

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

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

[9] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math. 4 (2009), 285-301.

[10] I. Beg, A. R. Butt, S. Radojevic, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl. 60 (2010), 1214-1219.

[11] F. Bojor, Fixed point of φ-contraction in metric spaces endowed with a graph, An. Uni. Cralova. 37 (2010), 85-92.

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

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

[14] L. C´iric´, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput. 217 (2011), 5784-5789.

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

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

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

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

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

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

[21] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aus. Math. Soc. 30 (1984), 1-9.

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

[23] S. K. Mohanta, Common fixed points in b-metric spaces endowed with a graph, Matematic. Vesnik. 68 (2016), 140-154.

[24] J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, W. Shatanawi, Common fixed points of almost generalized (ψ, φ)s-contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. (2013), 2013:159.

[25] W. Shatanawi, A. Pitea, R. Lazovic´, Contraction conditions using comparison functions on b-metric spaces, Fixed Point Theory Appl. (2014), 2014:135.