Abstract
In this paper, the concept of \(b_v(s)\)-metric space is introduced as a generalization of metric space, rectangular metric space, b-metric space, rectangular b-metric space and v-generalized metric space. We next give proofs of the Banach and Reich contraction principles in \(b_v(s)\)-metric spaces. Using a new result, we provide short proofs which are different from of the original ones in metric spaces. The results we obtain generalize many known results in fixed point theory. We also provide a solution to an open problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bakhtin, I.A.: The contraction mapping principle in quasimetric spaces. Funct. Anal. Ulianowsk Gos. Ped. Inst. 30, 26–37 (1989)
Branciari, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 57, 31–37 (2000)
Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)
Dominguez Benavides, T., Lorenzo, J., Gatica, I.: Some generalizations of Kannan’s fixed point theorem in \(K\)-metric spaces. Fixed Point Theory 13(1), 73–83 (2012)
Dung, N.V., Hang, V.T.L.: On relaxations of contraction constants and Caristi’s theorem in \(b\)-metric spaces. J. Fixed Point Theory Appl. 18, 267–284 (2016)
George, R., Radenović, S., Reshma, K.P., Shukla, S.: Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 8, 1005–1013 (2015)
Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)
Miculescu, R., Mihail, A.: New fixed point theorems for set-valued contractions in b-metric spaces. J. Fixed Point Theory Appl. (2017). doi:10.1007/s11784-016-0400-2
Mitrović, Z.D.: On an open problem in rectangular b-metric space. J. Anal. (2017). doi:10.1007/s41478-017-0036-7
Reich, S.: Some remarks concerning contraction mappings. Canad. Math. Bull. 14, 121–124 (1971)
Suzuki, T., Alamri, B., Khan, L.A.: Some notes on fixed point theorems in \(v\)-generalized metric space. Bull. Kyushu Inst. Tech. Pure Appl. Math. 62, 15–23 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mitrović, Z.D., Radenović, S. The Banach and Reich contractions in \(\varvec{b_v(s)}\)-metric spaces. J. Fixed Point Theory Appl. 19, 3087–3095 (2017). https://doi.org/10.1007/s11784-017-0469-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-017-0469-2