An equivalence of results in $C^*$-algebra valued b-metric and b-metric spaces

Nguyen Van Dung; Vo Thi Le Hang; Diana Dolicanin-Djekic

doi:10.4995/agt.2017.5185

Authors

Nguyen Van Dung Ton Duc Thang University
Vo Thi Le Hang Dong Thap University
Diana Dolicanin-Djekic University of Pristina-Kosovska Mitrovica

DOI:

https://doi.org/10.4995/agt.2017.5185

Keywords:

metric, b-metric, $C^*$-algebra-valued metric, $C^*$-algebra-valued b-metric, fixed point.

Abstract

We construct a $b$-metric from a given $C^*$-algebra-valued $b$-metric and prove some equivalences between them. Then we show that not only fixed point results but also topological properties on $C^*$-algebra-valued $b$-metric spaces may be deduced from certain results in $b$-metric spaces. In particular, every $C^*$-algebra-valued $b$-metric space is metrizable.

Downloads

Download data is not yet available.

References

M. Abbas, G. V. R. Babu and G. N. Alemayehu, On common fixed points of weakly compatible mappings satisfying "generalized condition (B)", Filomat 25, no. 2 (2011), 9-19.

https://doi.org/10.2298/FIL1102009A

A. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64, no. 4 (2014), 941-960.

https://doi.org/10.2478/s12175-014-0250-6

H. Aimar, B. Iaffei and L. Nitti, On the Macías-Segovia metrization of quasi-metric spaces, Rev. Un. Mat. Argentina 41 (1998), 67-75.

H. H. Alsulami, R. P. Agarwal, E. Karapinar and F. Khojasteh, A short note on $C^*$-valued contraction mappings, J. Inequal. Appl. 2016:50 (2016), 1-3.

https://doi.org/10.1186/s13660-016-0992-5

T. V. An, N. V. Dung, Z. Kadelburg and S. Radenovic, Various generalizations of metric spaces and fixed point theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109 (2015), 175-198.

https://doi.org/10.1007/s13398-014-0173-7

T. V. An, L. Q. Tuyen and N. V. Dung,Stone-type theorem on $b$-metric} spaces and applications, Topology Appl. 185/186 (2015), 50-64.

https://doi.org/10.1016/j.topol.2015.02.005

S. Batul and T. Kamran, $C^*$-valued contractive type mappings, Fixed Point Theory Appl. 2015:142 (2015), 1-9.

https://doi.org/10.1186/s13663-015-0393-3

J. Caristi and W. A. Kirk, Geometric fixed point theory and inwardness conditions, In: The geometry of metric and linear spaces, vol. 490, pp. 74-83, Springer Berlin Heidelberg, 1975.

https://doi.org/10.1007/BFb0081133

S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1, no. 1 (1993), 5-11.

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

N. V. Dung, T. V. An, and V. T. L. Hang, Remarks on Frink's metrization technique and applications, Fixed Point Theory, to appear.

N. V. Dung and V. T. L. Hang, On relaxations of contraction constants and Caristi's theorem in b-metric spaces, J. Fixed Point Theory Appl. 18, no. 2 (2016), 267-284.

https://doi.org/10.1007/s11784-015-0273-9

I. Gelfand and M. Naimark, On the embedding of normed rings into the ring of operators in Hilbert space, Mat. Sb. 12 (1943), 197-213.

J. Jachymski, Common fixed point theorems for some families of maps, Indian J. Pure Appl. Math. 25, no. 9 (1994), 925-937.

M. Jovanovic, Z. Kadelburg and S. Radenovic, Common fixed point results in metric-type spaces, Fixed Point Theory Appl. 2010 (2010), 1-15.

https://doi.org/10.1155/2010/978121

G. Jungck, Common fixed points for noncontinuous nonself mappings on a nonmetric space, Far East J. Math. Sci. 4, no. 2 (1996), 199-212.

M. A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. 73, no. 9 (2010), 3123-3129.

https://doi.org/10.1016/j.na.2010.06.084

W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, Cham, 2014.

Z. Ma and L. Jiang, $C^*$-algebra-valued b-metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2015, 2015:222, 12 pp.

Z. Ma, L. Jiang and H. Sun, $C^*$-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2014, 2014:206, 11 pp.

H. K. Nashine and Z. Kadelburg, Cyclic generalized $varphi$-contractions in b-metric spaces and an application to integral equations, Filomat 28, no. 10 (2014), 2047- 2057.

https://doi.org/10.2298/FIL1410047N

X. Qiaoling, J. Lining and M. Zhenhua, Common fixed point theorems in $C^*$-algebra-valued metric spaces, arXiv:1506.05545v2 (2015).

D. E. Shehwar, S. Batul, T. Kamran and A. Ghiura, Caristi's fixed point theorem on $C^*$-algebra valued metric spaces, J. Nonlinear Sci. Appl. 9 (2016), 584-588.

D. E. Shehwar and T. Kamran, $C^*$-valued G-contractions and fixed points, J. Inequal. Appl. 2015, 2015:304, 8 pp.