Document Type : Research Paper

Authors

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

Abstract

In this paper, we give some properties of $m_b$-metric topology and prove Cantor's intersection theorem in $m_b$-metric spaces. Moreover, we introduce some new
classes of $H_b^+ $-contractions for a pair of multi-valued and single-valued mappings and discuss their coincidence points. Some examples are provided to justify the validity of our main results.

Keywords

[1] S.M.A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Convergence of an iterative scheme for multifunctions, J. Fixed Point Theory Appl., 12 (2012), pp. 239-246.
[2] S.M.A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Fixed points of hemi-convex multifunctions, Topo. Metd. Nonlinear Anal., 37 (2011), pp. 383-389.
[3] S.M.A. Aleomraninejad and N. Shahzad, On fixed point generalizations of Suzuki's method, Appl. Math. Lett., 24 (2011), pp. 1037-1040.
[4] S.M.A. Aleomraninejad and N. Shahzad, Some fixed point results on a metric space with a graph, Topo. Appl., 159 (2012), pp. 659-663.
[5] M. Asadi, E. Karapinar and P. Salimi, New extension of $p$-metric spaces with some fixed-point results on $M$-metric spaces, J. Inequal. Appl., 2014 (2014), pp. 1-9.
[6] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal.,Gos. Ped. Inst. Unianowsk, 30 (1989), pp. 26-37.
[7] I. Beg and A. Azam, Fixed points of multivalued locally contractive mappings, Boll. Unione Matematica Italiana, 7 (1990), pp. 227-233.
[8] I. Beg and A.R. Butt, Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Analysis: Theory, Methods and Appl., 71 (2009), pp. 3699-3704.
[9] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostrav, 1 (1993), pp. 5-11.
[10]B. Damjanovic, B. Samet and C. Vetro, Common fixed point theorems for multi-valued maps, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), pp. 818-824.
[11] R.H. Haghi and Sh. Rezapour, Fixed points of multifunctions on regular cone metric spaces, Expositiones Mathematicae, 28 (2010), pp. 71-77.
[12] R.H. Haghi, Sh. Rezapour and N. Shahzad, Be caraful on partial metric fixed point results, Topo. Appl., 160 (2013), pp. 450-454.
[13] R.H. Haghi, Sh. Rezapour and N. Shahzad, On fixed points of quasi-contraction type multifunctions, Appl. Math. Lett., 25(5) (2012), pp. 843-846.
[14] R.H. Haghi, Sh. Rezapour and N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Analysis: Theory, Methods and Appl., 74(5) (2011), pp. 1799-1803.
[15] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), pp. 1468-1476.
[16] E. Karapinar, A note on common fixed point theorems in partial metric spaces, Miskolc Math. Notes, 12 (2011), pp. 185-191.
[17] E. Karapinar, Generalizations of Caristi Kirk's theorem on partial metric spaces, Fixed Point Theory Appl., 2011 (2011), pp. 1-7.
[18] E. Karapinar and S. Romaguera, Nonunique fixed point theorems in partial metric spaces, Filomat, 27 (2013), pp. 1305-1314.
[19] Z. Ma and L. Jiang, $C^*$-algebra-valued $b$-metric spaces and related fixed point theorems, Fixed Point Theory Appl., 2015 (2015), pp. 1-12.
[20] Z. Ma, L. Jiang and H. Sun, $C^*$-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl., 2014 (2014), pp. 1-11.
[21] S. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), pp. 183-197.
[22] S.K. Mohanta and S. Mohanta, A common fixed point theorem in $G$-metric spaces, Cubo, A Mathematical Journal, 14 (2012), pp. 85-101.
[23] S.K. Mohanta and S. Mohanta, Some fixed point results for mappings in $G$-metric spaces, Demonstratio Mathematica, 47 (2014), pp. 179-191.
[24] S.K. Mohanta and S. Patra, Coincidence points and common fixed points for hybrid pair of mappings in $b$-metric spaces endowed with a graph, J. Linear. Topological. Algebra., 6 (2017), pp. 301-321.
[25] Z. Mustafa and B. Sims, Fixed point theorems for contractive mappings in complete $G$-metric spaces, Fixed Point Theory Appl., 2009 (2009), pp. 1-10.
[26] S.B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), pp. 475-488.
[27] Sh. Rezapour and R.H. Haghi, Two results about fixed point of multifunctions, Bull. Iranian Math. Soc., 36 (2010), pp. 279-287.
[28] Sh. Rezapour, R.H. Haghi and N. Shahzad, Some notes on fixed points of quasi-contraction maps, Appl. Math. Lett., 23 (2010), pp. 498-502.
[29] H. Sahin, I. Altun and D. Turkoglu, Fixed point results for mixed multivalued mappings of Feng-Liu type on $M_b$-metric spaces, Mathematical Methods in Engineering, Nonlinear Systems and Complexity, 23 (2019), pp. 67-80.