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NOTES ON ORTHOGONAL-COMPLETE METRIC SPACES

Part of: Spaces with richer structures Nonlinear operators and their properties

Published online by Cambridge University Press:  11 May 2021

NGUYEN VAN DUNG Open the ORCID record for NGUYEN VAN DUNG [Opens in a new window]
NGUYEN VAN DUNG*
Affiliation:
Department of Mathematics and Information Technology Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap Province, Vietnam
*
e-mail: [email protected]
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Abstract

We prove that the restriction of a given orthogonal-complete metric space to the closure of the orbit induced by the origin point with respect to an orthogonal-preserving and orthogonal-continuous map is a complete metric space. Then we show that many existence results on fixed points in orthogonal-complete metric spaces can be proved by using the corresponding existence results in complete metric spaces.

Keywords

complete metric spacefixed pointorthogonal-completeorthogonal set

MSC classification

Primary: 47H10: Fixed-point theorems
Secondary: 54E99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 154 - 160
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

An, T. V., Dung, N. V. and Hang, V. T. L., ‘A new approach to fixed point theorems on $G$ -metric spaces’, Topology Appl. 160 (2013), 1486–1493.CrossRefGoogle Scholar
An, T. V., Dung, N. V., Kadelburg, Z. and Radenović, S., ‘Various generalizations of metric spaces and fixed point theorems’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109(1) (2015), 175–198.CrossRefGoogle Scholar
Baghani, H., Gordji, M. E. and Ramezani, M., ‘Orthogonal sets: the axiom of choice and proof of a fixed point theorem’, J. Fixed Point Theory Appl. 18(3) (2016), 465–477.CrossRefGoogle Scholar
Bahraini, A., Askari, G., Gordji, M. E. and Gholami, R., ‘Stability and hyperstability of orthogonally $\ast$ - $m$ -homomorphisms in orthogonally Lie ${C}^{\ast }$ -algebras: a fixed point approach’, J. Fixed Point Theory Appl. 20(2) (2018), 1–12.Google Scholar
Diaz, J. B. and Margolis, B., ‘A fixed point theorem of the alternative, for contractions on a generalized complete metric space’, Bull. Amer. Math. Soc. 74 (1968), 305–309.CrossRefGoogle Scholar
Dung, N. V., An, T. V. and Hang, V. T. L., ‘Remarks on Frink’s metrization technique and applications’, Fixed Point Theory 20(1) (2019), 157–176.CrossRefGoogle Scholar
Gordji, M. E., Habibi, H. and Sahabi, M. B., ‘Orthogonal sets; orthogonal contractions’, Asian-Eur. J. Math. 12(03) (2019), 1–10.CrossRefGoogle Scholar
Gordji, M. E., Ramezani, M., De La Sen, M. and Cho, Y. J., ‘On orthogonal sets and Banach fixed point theorem’, Fixed Point Theory 18(2) (2017), 569–578.CrossRefGoogle Scholar
Haghi, R. H., Rezapour, S. and Shahzad, N., ‘Some fixed point generalizations are not real generalizations’, Nonlinear Anal. 74 (2011), 1799–1803.CrossRefGoogle Scholar
Hazarika, B., ‘Applications of fixed point theorems and general convergence in orthogonal metric spaces’, in: Advances in Summability and Approximation Theory (eds. Mohiuddine, S. and Acar, T.) (Springer, Singapore, 2018), 23–51.CrossRefGoogle Scholar
Khamsi, M. A., ‘Generalized metric spaces: a survey’, J. Fixed Point Theory Appl. 17(3) (2016), 455–475.CrossRefGoogle Scholar
Meir, A. and Keeler, E., A theorem on contraction mappings’, J. Math. Anal. Appl. 28 (1969), 326–329.CrossRefGoogle Scholar
Pata, V., ‘A fixed point theorem in metric spaces’, J. Fixed Point Theory Appl. 10 (2011), 299–305.CrossRefGoogle Scholar
Rabbani, M. and Eshaghi, M., ‘Introducing of an orthogonally relation for stability of ternary cubic homomorphisms and derivations on ${C}^{\ast }$ -ternary algebras’, Filomat 32(4) (2018), 1439–1445.CrossRefGoogle Scholar
Ramezani, M. and Baghani, H., ‘Some new stability results of a Cauchy–Jensen equation in incomplete normed spaces’, J. Math. Anal. Appl. 495(2) (2021), article ID 124752.CrossRefGoogle Scholar
Sawangsup, K., Sintunavarat, W. and Cho, Y. J., ‘Fixed point theorems for orthogonal $F$ -contraction mappings on $O$ -complete metric spaces’, J. Fixed Point Theory Appl. 22(1) (2020), 10.CrossRefGoogle Scholar
Senapati, T., ‘Weak orthogonal metric spaces and fixed point results’, Preprint, 2018, arXiv:1806.09928, 12 pages.Google Scholar
Touail, Y. and El Moutawakil, D., ‘ ${\perp}_{\psi_F}$ -contractions and some fixed point results on generalized orthogonal sets’, Rend. Circ. Mat. Palermo (2) (2020). https://doi.org/10.1007/s12215-020-00569-4.CrossRefGoogle Scholar
Wardowski, D., ‘Fixed points of a new type of contractive mappings in complete metric spaces’, Fixed Point Theory Appl. 2012 (2012), 94, 6 pages.CrossRefGoogle Scholar