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.
Keywords: C∗ -algebra, C∗ -algebra valued b-metric, directed graph, C∗ -algebra valued G-contraction, common fixed point

Abstract

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.

References

[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.
Published
2018-10-19