Some new coincidence point results for single-valued and multi-valued mappings in $b$-metric spaces via digraphs

S. K. Mohanta; R. Kar

doi:10.30970/ms.53.1.69-84

S. K. Mohanta Department of Mathematics, West Bengal State University, Barasat, Kolkata-700126, India
R. Kar Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), Kolkata-700126, West Bengal, India

DOI: https://doi.org/10.30970/ms.53.1.69-84

Keywords: b-metric, digraph, generalized F − G-contractions, coincidence point

Abstract

We introduce the concept of generalized $F$-$G$-contraction and prove some new coincidence point results for single-valued and multi-valued mappings in $b$-metric spaces endowed with a digraph $G$. Our results generalize and extend several well-known comparable results including Nadler's fixed point theorem for multi-valued mappings. Moreover, we give some examples to justify the validity of our main result.

References

J. Ahmad, A. Al-Rawashdeh, A. Azam, New fixed point theorems for generalized F -contractions in complete metric spaces, Fixed Point Theory and Applications, 80 (2015).

I. Altun, G. Durmaz, G. Minak, S. Romaguera, Multivalued Almost F -contractions on complete metric spaces, Filomat, 30 (2016), 441–448.

V. Berinde, Generalized contractions in quasimetric spaces, In: Seminar on Fixed Point Theory, (1993), 3–9.

M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), 772–782.

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

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.

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

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.

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.

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

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

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

S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 263–276.

S. Czerwik, K. Dlutek, SL. Singh, Round-off stability of iteration procedures for set-valued operators in b-metric spaces, J. Natur. Phys. Sci., 11 (2007), 87–94.

M. Cosentino, M. Jleli, B. Samet, C. Vetro, Solvability of integrodifferential problems via fixed point theory in b-metric spaces, Fixed Point Theory and Applications, 70 (2015).

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

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

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

Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), 103–112.

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

M.E. Gordji, H. Baghani, H. Khodaei, M. Ramezani, A generalization of Nadler’s fixed point theorem, The Journal of Nonlinear Science and Applications, 3 (2010), 148–151.

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

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

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

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

S.K. Mohanta, Fixed Points in C ∗ -algebra valued b-metric spaces endowed with a Graph, Mathematica Slovaca, 68 (2018), 639–654.

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. Lin. Top. Alg., 6 (2017), 301–321.

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

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

J. Tiammee, S. Suantal, Coincidence point theorems for graph-preserving multi-valued mappings, Fixed Point Theory and Applications, 70 (2014).

D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Applications, 94 (2012).