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Date Published:
14/06/2021
Online Published:
14/06/2021
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Views PDF: 29
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Natural Sciences Issue
How to Cite
Pham, T. M. T., Vo, T. L. H., & Nguyen, V. D. (2021). On the hyperstability of the Drygas functional equations. Dong Thap University Journal of Science, 10(3), 21-30. https://doi.org/10.52714/dthu.10.3.2021.864
On the hyperstability of the Drygas functional equations
Main Article Content
Abstract
In this paper we study the hyperstability of the Drygas functional equation of the form f(x + y) + f(x - y) = 2 f(x) + f(y) + f(-y) in quasi-normed spaces, where f is a map from X into Y and x, y, x + y, x - y X. The obtained results are the extensions of the results of Aiemsomboon and Sintunavarat (2016) on the Drygas functional equation in normed spaces.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Functional equation, hyperstability, quasi-norm
References
Aiemsomboon, L., & Sintunavarat, W. (2016a). Two new generalised hyperstability results for the Drygas functional equation. Bull. Aust. Math. Soc., 12 pages.
Aiemsomboon, L., & Sintunavarat, W. (2016b). On generalized hyperstability of a general linear equation. Acta Math. Hungar. 149(2), 413-422.
Aiemsomboon, L., & Sintunavarat, W. (2017). A note on the generalised hyperstability of the general linear equation. Bull. Aust. Math. Soc., 96(2), 263-273.
Bourgin, D. G. (1949). Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J., 16, 385-397.
Brzdek, J. (2013). Stability of additivity and fixed point methods. Fixed Point Theory Appl., 2013, Article ID 285, 9 pages.
Brzdek, J. (2015). Remarks on stability of some inhomogeneous functinal equations. Aequationes Math., 89, 83-96.
Brzdek, J., Chudziak, J., & Pales, Zs. (2011). Fixed point approach to stability of functional equations. Nonlinear Anal., 74, 6728-6732.
Brzdek, J., & Cieplinski, K. (2013). Hyperstability and superstability. Abstr. Appl. Anal, 2013, Article ID 401756, 13 pages.
Czerwik, S. (1998). Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena, 46, 263-276.
Drygas, H. (1987). Quasi-inner products and their applications, in: Advances in Multivariate Statistical Analysis (ed. K. Gupta) (Springer, Netherlands, 13-30.
Dung, N. V., & Hang, V. T. L. (2018). The generalized hyperstability of general linear equations in quasi-Banach spaces, J. Math. Anal. Appl., 462, 131-147.
Ebanks, B. R., Kannappan, P. l., & Sahoo P. K. (1992). A common generalization of functional equations characterizing normed and quassi-inner-product spaces. Canad. Math. Bull., 35(3), 321-327.
Faiziev, V. A., & Sahoo, P. K. (2007). On the stability of Drygas functional equation on groups. Banach J. Math. Anal., 1(1), 43-55.
Faiziev, V. A., & Sahoo, P. K. (2007). Stability of Drygas functional equation on T (3, ). Int. J. Math. Stat., 7, 70-81.
Jung, S. M., & Sahoo, P. K. (2002). Stability of a functional equation of Drygas. Aequationes Math., 64, 263-273.
Kalton, N. (2003). Quasi-Banach spaces, in: Johnson W.B., Lindenstrauss J. (Eds.), Handbook of the Geometry of Banach Spaces 2, Elsevier, 1099-1130.
Maksa, Gy., & Pales, Zs. (2001). Hyperstability of a class of linear functional equations. Acta Math. Acad. Paedagog. Nyhazi. (N.S), 17, 1007-112.
Paluszyński, M., & Stempak, K. (2009). On quasi- metric and metric spaces. Proc. Amer. Math. Soc., 137(12), 43074312.
Piszczek, M. (2015). Hyperstability of the general linear functional equation. Bull. Korean Math. Soc., 52, 1827-1838.
Piszczek, M., & Szczawinka, J. (2013). Hyperstability of the Drygas functional equation. J. Funct. Spaces Appl., 2013, Article ID 912718, 4 pages.
Yang, D. (2004). Remarks on the stability of Drygas equation and the Pexider-quadratic equation. Aequationes Math., 64, 108-116.
Zhang, D. (2015), On hyperstability of generalised linear functional equations in several variables. Bull. Aust. Math. Soc., 92, 259-267.
Zhang, D. (2016). On Hyers-Ulam stability of generalized linear functional equation and its induced Hyers-Ulam programming problem. Aequationes Math., 90, 559-568.
Aiemsomboon, L., & Sintunavarat, W. (2016b). On generalized hyperstability of a general linear equation. Acta Math. Hungar. 149(2), 413-422.
Aiemsomboon, L., & Sintunavarat, W. (2017). A note on the generalised hyperstability of the general linear equation. Bull. Aust. Math. Soc., 96(2), 263-273.
Bourgin, D. G. (1949). Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J., 16, 385-397.
Brzdek, J. (2013). Stability of additivity and fixed point methods. Fixed Point Theory Appl., 2013, Article ID 285, 9 pages.
Brzdek, J. (2015). Remarks on stability of some inhomogeneous functinal equations. Aequationes Math., 89, 83-96.
Brzdek, J., Chudziak, J., & Pales, Zs. (2011). Fixed point approach to stability of functional equations. Nonlinear Anal., 74, 6728-6732.
Brzdek, J., & Cieplinski, K. (2013). Hyperstability and superstability. Abstr. Appl. Anal, 2013, Article ID 401756, 13 pages.
Czerwik, S. (1998). Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena, 46, 263-276.
Drygas, H. (1987). Quasi-inner products and their applications, in: Advances in Multivariate Statistical Analysis (ed. K. Gupta) (Springer, Netherlands, 13-30.
Dung, N. V., & Hang, V. T. L. (2018). The generalized hyperstability of general linear equations in quasi-Banach spaces, J. Math. Anal. Appl., 462, 131-147.
Ebanks, B. R., Kannappan, P. l., & Sahoo P. K. (1992). A common generalization of functional equations characterizing normed and quassi-inner-product spaces. Canad. Math. Bull., 35(3), 321-327.
Faiziev, V. A., & Sahoo, P. K. (2007). On the stability of Drygas functional equation on groups. Banach J. Math. Anal., 1(1), 43-55.
Faiziev, V. A., & Sahoo, P. K. (2007). Stability of Drygas functional equation on T (3, ). Int. J. Math. Stat., 7, 70-81.
Jung, S. M., & Sahoo, P. K. (2002). Stability of a functional equation of Drygas. Aequationes Math., 64, 263-273.
Kalton, N. (2003). Quasi-Banach spaces, in: Johnson W.B., Lindenstrauss J. (Eds.), Handbook of the Geometry of Banach Spaces 2, Elsevier, 1099-1130.
Maksa, Gy., & Pales, Zs. (2001). Hyperstability of a class of linear functional equations. Acta Math. Acad. Paedagog. Nyhazi. (N.S), 17, 1007-112.
Paluszyński, M., & Stempak, K. (2009). On quasi- metric and metric spaces. Proc. Amer. Math. Soc., 137(12), 43074312.
Piszczek, M. (2015). Hyperstability of the general linear functional equation. Bull. Korean Math. Soc., 52, 1827-1838.
Piszczek, M., & Szczawinka, J. (2013). Hyperstability of the Drygas functional equation. J. Funct. Spaces Appl., 2013, Article ID 912718, 4 pages.
Yang, D. (2004). Remarks on the stability of Drygas equation and the Pexider-quadratic equation. Aequationes Math., 64, 108-116.
Zhang, D. (2015), On hyperstability of generalised linear functional equations in several variables. Bull. Aust. Math. Soc., 92, 259-267.
Zhang, D. (2016). On Hyers-Ulam stability of generalized linear functional equation and its induced Hyers-Ulam programming problem. Aequationes Math., 90, 559-568.
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