Some applications with new admissibility contractions in \(b\)-metric spaces

Volume 10, Issue 8, pp 4162--4174

Publication Date: 2017-08-11


Ljiljana Paunović - University of Priˇstina-Kosovska Mitrovica, Teacher Education School in Prizren-Leposavi´c, 38218 Leposavi´c, Serbia.
Preeti Kaushik - Department of Mathematics, DCRUST, Murthal, Sonepat 131039, India.
Sanjay Kumar - Department of Mathematics, DCRUST, Murthal, Sonepat 131039, India.


The work presented in this paper extends the idea of \(\alpha-\beta\)-contractive mappings in the framework of \(b\)-metric spaces. Fixed points are investigated for such kind of mappings. An example is given to show the superiority of our results. As applications we discuss Ulam-Hyres stability, well-posedness and limit shadowing of fixed point problem.


\(\alpha-\beta(b)\)-admissible mappings, fixed point, \(b\)-metric space, stability.


[1] I. A. Bakhtin, The contraction mapping principle in quasi-metric spaces, J. Funct. Anal., 30 (1989), 26–37.
[2] M. F. Bota, E. Karapnar, O. Mlesnite, Ulam-Hyers stability results for fixed point problems via \(\alpha-\psi\)-contractive mapping in (b)-metric space, Abstr. Appl. Anal., 2013 (2013), 6 pages.
[3] M. F. Bota-Boriceanu, A. Petrusel, Ulam-Hyers stability for operatorial equations, An. tiin. Univ. Al. I. Cuza Iai. Mat., 57 (2011), 65–74.
[4] J. Brzdek, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal., 74 (2011), 6728–6732.
[5] J. Brzdek, K. Cieplinski, A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl., 400 (2013), 68–75.
[6] L. Cadariu, L. Gavruta, P. Gavruta, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal., 2012 (2012), 10 page.
[7] S. Chandok, M. Jovanović, S. Radenović, Ordered b-metric spaces and Geraghty type contractive mappings, Military Technical Courier, 65 (2017), 331–345.
[8] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.
[9] F. S. de Blassi, J. Myjak, Sur la porosite des contractions sans point fixed, C. R. Acad. Sci. Paris Sr. I Math., 308 (1989), 51–54.
[10] D. Dukić, Z. Kadelburg, S. Radenović, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstr. Appl. Anal., 2011 (2011), 13 pages.
[11] M. Eshaghi Gordji, M. Ramezani, Y. J. Cho, S. Pirbavafa, A generalization of Geraghty’s theorem in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012 (2012), 9 pages.
[12] A. Felhi, S. Sahmim, H. Aydi, Ulam-Hyers stability and well-posedness of fixed point problems for \(\alpha-\lambda\)-contractions on quasi b-metric spaces, Fixed Point Theory Appl., 2016 (2016), 20 pages.
[13] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222–224.
[15] M. Jovanovic, Z. Kadelburg, S. Radenovic, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010 (2010), 15 pages.
[16] M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., 2010 (2010), 7 pages.
[17] B. K. Lahiri, P. Das, Well-posedness and porosity of certain classes of operators, Demonstratio Mathematica., 38 (2005), 170–176.
[18] V. L. Lažar, Ulam-Hyers stability for partial differential inclusions, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 19 pages.
[19] Z. Mustafa, H. Huang, S. Radenović, Some remarks on the paper, Some fixed point generalizations are not real generalizations, J. Adv. Math. Stud., 9 (2016), 110–116.
[20] V. Popa, Well posedness of fixed point problem in orbitally complete metric spaces, Stud. Cercet. tiin. Ser. Mat. Univ. Bacu., 16 (2006), 18–20.
[21] V. Popa, Well posedness of fixed point problem in compact metric spaces, Bul. Univ. Petrol-Gaze, Ploiesti, Sec. Mat. Inform. Fiz., 60 (2008), 1–4.
[22] G. S. Rad, S. Radenović, D. Dolićanin-Dekić, A shorter and simple approach to study fixed point results via b-simulation functions, Iran. J. Math. Sci. Inform., (Accepted).
[23] S. Radenović, S. Chandok, W. Shatanawi, Some cyclic fixed point results for contractive mappings, University Though, Publication in Nature Sciences, 6 (2016), 38–40.
[24] S. Radenovic´, T. Došenović, V. Osturk, Ć. Dolićanin, A note on the paper, Integral equations with new admisibility types in b-metric spaces, J. Fixed Point Theory Appl., 2017 (2017), 9 pages.
[25] S. Reich, A. J. Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci., 3 (2001), 393–401.
[26] J. R. Roshan, V. Parvaneh, Z. Kadelburg, Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces, J. Nonlinear Sci. Appl., 7 (2014), 229–245.
[27] A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305–320.
[28] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for - -contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165.
[29] R. J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl., 2014 (2014), 23 pages.
[30] W. Sintunavarat, Generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for \(\alpha-\beta\)- contraction mapping in metric spaces, The Sci. World J., 2014 (2014), 7 pages.
[31] W. Sintunavarat, S. Plubtieng, P. Katchang, Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation, Fixed Point Theory and Appl., 2013 (2013), 13 pages.
[32] F. A. Tise, I. C. Tise, Ulam-Hyers-Rassias stability for set integral equations, Fixed Point Theory, 13 (2012), 659–668.
[33] S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, (1964).


XML export