Rectangular b-metric space and contraction principles

Volume 8, Issue 6, pp 1005--1013 http://dx.doi.org/10.22436/jnsa.008.06.11 Publication Date: November 10, 2015

Authors

R. George - Department of Mathematics and Computer Science, St. Thomas College, Bhilai, Chhattisgarh, India.
S. Radenović - Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia.
K. P. Reshma - Department of Mathematics, Government VYT PG Autonomous College, Durg, Chhattisgarh, India.
S. Shukla - Department of Applied Mathematics, S.V.I.T.S. Indore (M.P.), India.


Abstract

The concept of rectangular b-metric space is introduced as a generalization of metric space, rectangular metric space and b-metric space. An analogue of Banach contraction principle and Kannan's fixed point theorem is proved in this space. Our result generalizes many known results in fixed point theory.


Keywords


References

[1] T. Abdeljawad, D. Turkoglu, Locally convex valued rectangular metric spaces and Kannan's fixed point theorem, arXiv, 2011 (2011), 11 pages.
[2] H. Aydi, M. F. Bota, E. Karapinar, S. Moradi, A common fixed point for weak \(\phi\)-contractions on b-metric spaces, Fixed Point Theory, 13 (2012), 337-346.
[3] A. Azam, M. Arshad, Kannan Fixed Point Theorems on generalised metric spaces, J. Nonlinear Sci. Appl., 1 (2008), 45-48.
[4] A. Azam, M. Arshad, I. Beg, Banach contraction principle on cone rectangular metric spaces, Appl. Anal. Discrete Math., 3 (2009), 236-241.
[5] I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., Unianowsk Gos. Ped. Inst., 30 (1989), 26-37.
[6] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Inter. J. Mod. Math., 4 (2009), 285-301.
[7] M. Boriceanu, M. Bota, A. Petrusel, Mutivalued fractals in b-metric spaces, Cen. Eur. J. Math., 8 (2010), 367-377.
[8] M. Bota, A. Molnar, V. Csaba, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory, 12 (2011), 21-28.
[9] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalised metric spaces, Publ. Math. Debrecen, 57 (2000), 31-37.
[10] C. N. Chen, Common fixed point theorem in complete generalized metric spaces, J. Appl. Math., 2012 (2012), 14 pages.
[11] S. Czerwik, Contraction mappings in b-metric spaces, Acta. Math. Inform. Univ. Ostraviensis, 1 (1993), 5-11.
[12] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 46 (1998), 263-276.
[13] S. Czerwik, K. Dlutek, S. L. Singh, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci. , 11 (1997), 87-94.
[14] S. Czerwik, K. Dlutek, S. L. Singh, Round-off stability of iteration procedures for set valued operators in b-metric spaces, J. Natur. Phys. Sci., 15(2001), 1-8.
[15] P. Das, A fixed point theorem on a class of generalized metric spaces, Korean J. Math. Sci., 9 (2002), 29-33.
[16] P. Das, A fixed point theorem in generalized metric spaces, Soochow J. Math., 33 (2007), 33-39.
[17] P. Das, B. K. Lahri, Fixed point of a Ljubomir Ciric's quasi-contraction mapping in a generalized metric space, Publ. Math. Debrecen, 61(2002), 589-594.
[18] P. Das, B. K. Lahri, Fixed Point of contractive mappings in generalised metric space, Math. Slovaca, 59 (2009), 499-504.
[19] I. M. Erhan, E. Karapinar, T. Sekulic, Fixed Points of (psi, phi) contractions on generalised metric spaces, Fixed Point Theory Appl., 2012 (2012), 12 pages.
[20] R. George, B. Fisher, Some generalised results of fixed points in cone b-metric spaces , Math. Moravic., 17 (2013), 39-50.
[21] G. S. Jeong, B. E. Rhoades, Maps for which\( F(T) = F(T^n)\), Fixed Point Theory Appl., 6 (2007), 71-105.
[22] M. Jleli, B. Samet, The Kannan's fixed point theorem in cone rectangular metric space, J. Nonlinear Sci. Appl., 2 (2009), 161-167.
[23] H. Lakzian, B. Samet, Fixed Points for (\(\psi,\phi\))-weakly contractive mapping in generalised metric spaces, Appl. Math. Lett., 25 (2012), 902-906.
[24] S. G. Mathews, Partial Metric Topology, Papers on general topology appl., Ann. New York Acad. Sci., 728 (1994), 183-197.
[25] D. Mihet, On Kannan fixed point result in generalised metric spaces, J. Nonlinear Sci. Appl., 2 (2009), 92-96.
[26] I. R. Sarma, J. M. Rao, S. S. Rao, Contractions over generalised metric spaces, J. Nonlinear Sci. Appl., 2 (2009), 180-182.