Common Fixed Point Results in C∗-Algebra Valued b-Metric Spaces Via Digraphs

  • Sushanta Kumar Mohanta Department of Mathematics, West Bengal State University,, Barasat, 24 Parganas (North), West Bengal, Kolkata 700126, India.


We discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on a C-algebra valued b-metric space endowed with a graph. Our results extend and supplement several recent results in the literature. Strength of hypotheses made in the first result have been weighted through illustrative examples.


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

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

[3] M. R. Alfuraidan, M. A. Khamsi, Caristi fixed point theorem in metric spaces with a graph, Abstract and Applied Analysis, vol. 2014, Article ID 303484.

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

[5] S. Banach, Sur les ope´rations dans les ensembles abstraits et leur application aux e´quations inte´grales, Fund. Math., 3, 1922, 133-181.

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

[7] J. A. Bondy and 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, Annala of the University of Cralova, Mathematics and Computer Science 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] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav, 1, 1993, 5-11.

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

[13] M. Cosentino, P. Salimi, P. Vetro, Fixed point results on metric-type spaces, Acta Math. Sci. Ser. B Engl. Ed., 34, 2014, 1237-1253.

[14] R. Douglas, Banach algebra techniques in operator theory, Springer, Berlin, 1998.

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

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

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

[18] N. Hussain, D. Doric´, Z. Kadelburg, S. Radenovic´, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012, 2012:126, doi:10.1186/1687-1812-2012-126.

[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] Z. Ma, L. Jiang, C∗-algebra-valued b-metric spaces and related fixed point theorems, Fixed Point Theory and Applications, 2015, 2015:222.

[22] Z. Ma, L. Jiang and H. Sun, C∗-algebra-valued metric spaces and related fixed point theorems,
Fixed Point Theory and Applications, 2014, 2014:206.

[23] G. Murphy, C∗-Algebra and operator theory, Academic Press, London, 1990.

[24] S. K. Mohanta, Some Fixed Point Theorems in Cone Modular Spaces with a Graph, Bolletino dell Unione Matematica Italiana, 2016, DOI 10.1007/s40574-016-0086-9.

[25] S. K. Mohanta, Some fixed point theorems using wt-distance in b-metric spaces, Fasciculi Mathematici, no. 54, 2015, 125-140.

[26] J. J. Nieto and R. Rodr´iguez-Lo´pez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, Englosh Ser., 2007, 2205-2212.

[27] D. Reem, S. Reich, A. J. Zaslavski, Two results in metric fixed point theory, J. Fixed Point Theory Appl., 1, 2007, 149-157.
How to Cite
KUMAR MOHANTA, Sushanta. Common Fixed Point Results in C∗-Algebra Valued b-Metric Spaces Via Digraphs. CUBO, A Mathematical Journal, [S.l.], v. 20, n. 1, p. 41–64, oct. 2018. ISSN 0719-0646. Available at: <>. Date accessed: 19 nov. 2018. doi:

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