Stability of functional equations in dislocated quasimetric spaces



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.


Keywords

stability of functional equations; square symmetric groupoid; dislocated quasi-metric space; semigroup; Cauchy equation

1. Amini-Harandi A., Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012, 2012:204, 10 pp.
2. Bahyrycz A., Piszczek M., Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), 353–365.
3. Brillouët-Belluot N., Brzdęk J., Ciepliński K., On some recent developments in Ulam’s type stability, Abstr. Appl. Anal. 2012, Art. ID 716936, 41 pp.
4. Brzdęk 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.
5. Hyers D.H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
6. Hyers D.H., Isac G., Rassias T.M., Stability of Functional Equations in Several Variables, Birkhäuser Boston, Boston, 1998.
7. Piszczek M., Remark on hyperstability of the general linear equation, Aequationes Math. 88 (2014), 163–168.
8. Piszczek M., Hyperstability of the general linear functional equation, Bull. Korean Math. Soc. 52 (2015), 1827–1838.
9. Rahman M.U., Sarwar M., Some new fixed point theorems in dislocated quasi-metric spaces, Palest. J. Math. 5 (2016), 171–176.
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.
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.
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Published : 2018-05-14


HejmejB. (2018). Stability of functional equations in dislocated quasimetric spaces. Annales Mathematicae Silesianae, 32, 215-225. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13921

Beata Hejmej  bhejmej1f@gmail.com
Instytut Matematyki, Uniwersytet Pedagogiczny im. Komisji Edukacji Narodowej w Krakowie  Poland



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