On hyperstability of Cauchy functional equation in (2,\(\gamma\))-Banach spaces
Volume 23, Issue 4, pp 354--363
http://dx.doi.org/10.22436/jmcs.023.04.08
Publication Date: November 26, 2020
Submission Date: October 14, 2020
Revision Date: October 15, 2020
Accteptance Date: October 18, 2020
-
1043
Downloads
-
1993
Views
Authors
El-Sayed El-Hady
- Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia.
- asic Science Department, Faculty of Computers and Informatics, Suez Canal University, Ismailia, 41522, Egypt.
Abstract
In this paper, we generalize the recent hyperstability results obtained by Brzd{\k{e}}k and concerning the Cauchy functional equation \[f(x_1+x_2)=f(x_1)+f(x_2).\] The obtained results are in (2,\(\gamma\))-Banach spaces. The main tool used in the analysis is some fixed point theorem.
Share and Cite
ISRP Style
El-Sayed El-Hady, On hyperstability of Cauchy functional equation in (2,\(\gamma\))-Banach spaces, Journal of Mathematics and Computer Science, 23 (2021), no. 4, 354--363
AMA Style
El-Hady El-Sayed, On hyperstability of Cauchy functional equation in (2,\(\gamma\))-Banach spaces. J Math Comput SCI-JM. (2021); 23(4):354--363
Chicago/Turabian Style
El-Hady, El-Sayed. "On hyperstability of Cauchy functional equation in (2,\(\gamma\))-Banach spaces." Journal of Mathematics and Computer Science, 23, no. 4 (2021): 354--363
Keywords
- Hyperstability
- Cauchy equation
- additive function
- restricted domain
- invex set
MSC
References
-
[1]
J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989)
-
[2]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64--66
-
[3]
D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16 (1949), 385--397
-
[4]
D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223--237
-
[5]
J. Brzdȩk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., 141 (2013), 58--67
-
[6]
J. Brzdȩk, A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc., 89 (2014), 33--40
-
[7]
J. Brzdȩk, L. Cadariu, K. Ciepliński, Fixed point theory and the Ulam stability, J. Funct. Spaces, 2014 (2014), 16 pages
-
[8]
J. Brzdȩk, K. Ciepliński, Hyperstability and superstability, Abstr. Appl. Anal., 2013 (2013), 13 pages
-
[9]
J. Brzdȩk, K. Ciepliński, On a fixed point theorem in $2$-Banach spaces and some of its applications, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 377--390
-
[10]
J. Brzdȩk, K. Ciepliński, Z. Leśniak, On Ulam's type stability of the linear equation and related issues, Discrete Dyn. Nat. Soc., 2014 (2014), 14 pages
-
[11]
J. Brzdęk, E. El-hady, On approximately additive mappings in $2$-Banach spaces, Bull. Aust. Math. Soc., 101 (2020), 299--310
-
[12]
J. Brzdȩk, D. Popa, I. Raşa, Hyers-Ulam stability with respect to gauges, J. Math. Anal. Appl., 453 (2017), 620--628
-
[13]
J. Brzdȩk, D. Popa, I. Raşa, B. Xu, Ulam Stability of Operators, Academic Press, London (2018)
-
[14]
S.-C. Chung, W.-G. Park, Hyers-Ulam stability of functional equations in $2$-Banach spaces, Int. J. Math. Anal. (Ruse), 6 (2012), 951--961
-
[15]
S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59--64
-
[16]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., River Edge (2002)
-
[17]
E. El-hady, On stability of the functional equation of $p$-Wright affine functions in $(2,\alpha)$-Banach spaces, J. Egyptian Math. Soc., 27 (2019), 9 pages
-
[18]
R. W. Freese, Y. J. Cho, Geometry of Linear $2$-normed Spaces, Nova Science Publishers, Hauppauge (2001)
-
[19]
S. Gähler, Lineare $2$-normierte Räume, Math. Nachr., 28 (1964), 1--43
-
[20]
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431--434
-
[21]
J. Gao, On the stability of the linear mapping in $2$-normed spaces, Nonlinear Funct. Anal. Appl., 14 (2009), 801--807
-
[22]
E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar., 124 (2009), 179--188
-
[23]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
-
[24]
D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, Birkauser, Boston (1998)
-
[25]
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York (2011)
-
[26]
S.-M. Jung, D. Popa, M. T. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim., 59 (2014), 165--171
-
[27]
G. Maksa, Z. Páles, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 17 (2001), 107--112
-
[28]
Z. Moszner, Stability has many names, Aequationes Math., 90 (2016), 983--999
-
[29]
K. Y. Naif Sayar, A. Bergam, Some hyperstability results for a Cauchy-Jensen type functional equation in $2$-Banach spaces, Proyecciones, 39 (2020), 73--89
-
[30]
P. Nakmahachalasint, Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of additive functional equation in several variables, Int. J. Math. Math. Sci., 2007 (2007), 6 pages
-
[31]
P. Nakmahachalasint, On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations, Int. J. Math. Math. Sci., 2007 (2007), 10 pages
-
[32]
W.-G. Park, Approximate additive mappings in $2$-Banach spaces and related topics, J. Math. Anal. Appl., 376 (2011), 193--202
-
[33]
T. Phochai, S. Saejung, Hyperstability of generalised linear functional equations in several variables, Bull. Aust. Math. Soc., 102 (2020), 293--302
-
[34]
T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297--300
-
[35]
J. Senasukh, S. Saejung, On the hyperstability of the drygas functional equation on a restricted domain, Bull. Aust. Math. Soc., 102 (2020), 126--137
-
[36]
D. Zhang, On hyperstability of generalised linear functional equations in several variables, Bull. Aust. Math. Soc., 92 (2015), 259--267