Document Type: Special Issue on Fixed Point Theory

**Authors**

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

**Abstract**

In this paper, we introduce the notion of strictly (α,ψ,ξ)-G-contractive mappings in b-metric spaces endowed with a graph G. We establish a sufficient condition for existence and uniqueness of points of coincidence and common fixed points for such mappings. Our results extend and unify many existing results in the literature. Finally, we construct some examples to analyze and support our results.

**Keywords**

**Main Subjects**

[1] M. U. Ali, T. Kamran, E. Karapinar, (α, ψ, ξ)-contractive multivalued mappings, Fixed Point Theory Appl. (2014), 2014:7.

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

[3] J. H. Asl, S. Rezapour, N. Shahzad, On fixed points of α − ψ-contractive multifunctions, Fixed Point Theory and Appl. (2012), 2012:212.

[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 ope´rations dans les ensembles abstraits et leur application aux e´quations inte´grales, 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] 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.

[9] F. Bojor, Fixed point of φ-contraction in metric spaces endowed with a graph, Anal of the University of Cralova, Math. Comput. Sci. Series. 37 (2010), 85-92.

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

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

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

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

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

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

[16] K. Fallahi, G. Soleimani Rad, Fixed point results in cone metric spaces endowed with a graph, Sahand Communications in Mathematical Analysis. 6 (2017), 39-47.

[17] K. Fallahi, M. Abbas, G. Soleimani Rad, Generalized c-distance on cone b-metric spaces endowed with a graph and fixed point results, Appl. Gen. Topol. 18 (2017), 391-400.

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

[19] N. Hussain, M. A. Kutbi, P. Salimi, Fixed point theory in α-complete metric spaces with applications, Abstr. Appl. Anal. (2014), 2014:280817.

[20] N. Hussain, R. Saadati, R. P. Agrawal, On the topology and wt-distance on metric type spaces, Fixed Point Theory Appl. (2014), 2014:88.

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

[22] H. Kaneko, S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Internat. J. Math. Math. Sci. 12 (1989), 257-262.

[23] P. Kaushik, S. Kumar, Fixed point results for (α, ψ, ξ)-contractive compatible multi-valued mappings, J. Nonlinear Anal. Appl., doi:10.5899/2016/jnaa-00305.

[24] M. A. Kutbi, W. Sintunavarat, On new fixed point results for (α, ψ, ξ)-contractive multi-valued mappings on α-complete metric spaces and their consequences, Fixed Point Theory Appl. (2015), 2015:2.

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

[26] S. K. Mohanta, Coincidence points and common fixed points for expansive type mappings in b-metric spaces, Iran. J. Math Sci. Infor. 11 (2016), 101-113.

[27] S. Nadler, Multi-valued contraction mappings, Pac. J. Math. 20 (1969), 475-488.

[28] H. K. Pathak, Fixed point theorems for weak compatible multi-valued and single-valued mappings, Acta. Math. Hungar. 67 (1995), 69-78.

[29] 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.

[30] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α − ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.

[31] J. Tiammee, S. Suantal, Coincidence point theorems for graph-preserving multi-valued mappings, Fixed Point Theory Appl. (2014), 2014:70.