Stability of Functional Equations in Dislocated Quasi-Metric Spaces

Hejmej, Beata

doi:10.2478/amsil-2018-0005

Abstract

We present a result on the generalized Hyers-Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.

References

[1] Amini-Harandi A., Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012, 2012:204, 10 pp.10.1186/1687-1812-2012-204
Search in Google Scholar Back to article
[2] Bahyrycz A., Piszczek M., Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), 353-365.10.1007/s10474-013-0347-3
Search in Google Scholar Back to article
[3] Brillouët-Belluot N., Brzdek J., Cieplinski K., On some recent developments in Ulam’s type stability, Abstr. Appl. Anal. 2012, Art. ID 716936, 41 pp.10.1155/2012/716936
Search in Google Scholar Back to article
[4] Brzdek J., On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Aust. J. Math. Anal. Appl. 6 (2009), Art. 4, 10 pp.
Search in Google Scholar Back to article
[5] Hyers D.H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.10.1073/pnas.27.4.222107831016578012
Open DOI Search in Google Scholar Back to article
[6] Hyers D.H., Isac G., Rassias T.M., Stability of Functional Equations in Several Variables, Birkhäuser Boston, Boston, 1998.10.1007/978-1-4612-1790-9
Open DOI Search in Google Scholar Back to article
[7] Piszczek M., Remark on hyperstability of the general linear equation, Aequationes Math. 88 (2014), 163-168.10.1007/s00010-013-0214-x
Search in Google Scholar Back to article
[8] Piszczek M., Hyperstability of the general linear functional equation, Bull. Korean Math. Soc. 52 (2015), 1827-1838.10.4134/BKMS.2015.52.6.1827
Search in Google Scholar Back to article
[9] Rahman M.U., Sarwar M., Some new fixed point theorems in dislocated quasi-metric spaces, Palest. J. Math. 5 (2016), 171-176.10.18576/amisl/050102
Search in Google Scholar Back to article
[10] Sarwar M., Rahman M.U., Ali G., Some fixed point results in dislocated quasi-metric (dq-metric) spaces, J. Inequal. Appl. 2014, 2014:278, 11 pp.10.1186/1029-242X-2014-278
Search in Google Scholar Back to article
[11] Ulam S.M., A Collection of Mathematical Problems, Interscience Publishers, New York-London, 1960. Reprinted as: Problems in Modern Mathematics, John Wiley & Sons, New York, 1964.
Search in Google Scholar Back to article

Stability of Functional Equations in Dislocated Quasi-Metric Spaces

Abstract

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