Asymptotic Hyers-Ulam Stability or Superstability for Generalized Linear Equations by Unilateral Perturbations

Catherine Peppo

doi:10.68381/jca26028

Abstract

In relation to the famous problem of Ulam ``Give conditions in order for a linear mapping near an approximated linear mapping to exist'', we consider the stability or superstability of generalized linear equation

f(x+y)-f(x)-f(y)=B[\phi(x)+\phi(y)]

by left or right perturbations with some hypotheses of convexity or concavity, and – in a forthcoming paper – apply our conclusions to the generalized exponential equation

\frac{f(x+y)} {f(x)f(y)}= [\phi(x)\phi(y)]^{B}.

Author information

Contact details are reproduced from the original publication and may be historical.

Catherine Peppo

Itescia, 8 rue P. de Coubertin, 95300 Pontoise, France

cpeppo@cci-paris-idf.fr

Keywords

Hyers-Ulam stability
superstability
asymptotic stability
linear equation
exponential equation

Mathematics Subject Classification

39B62, 26A51

References

T. Aoki: On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66.https://doi.org/10.2969/jmsj/00210064 Google Scholar
Z. Gajda: On stability of additive mappings, Int. J. Math. Math. Sci. 14(3) (1991) 431–434.https://doi.org/10.1155/S016117129100056X Google Scholar
P. Gavruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436.https://doi.org/10.1006/jmaa.1994.1211 Google Scholar
P. Gavruta, L. Gavruta: A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1(2) (2010) 6 pp.Google Scholar
E. Hills, R. S. Phillips: Functional Analysis and Semi-Groups, Colloquium Publications 31, American Mathematical Society, Providence (2000).Google Scholar
D. H. Hyers: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27(1941) 222–224.https://doi.org/10.1073/pnas.27.4.222 Google Scholar
Th. M. Rassias: On the stability of the linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300.https://doi.org/10.1090/S0002-9939-1978-0507327-1 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.

[1] T. Aoki: On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66.https://doi.org/10.2969/jmsj/00210064 Google Scholar

[2] Z. Gajda: On stability of additive mappings, Int. J. Math. Math. Sci. 14(3) (1991) 431–434.https://doi.org/10.1155/S016117129100056X Google Scholar

[3] P. Gavruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436.https://doi.org/10.1006/jmaa.1994.1211 Google Scholar

[4] P. Gavruta, L. Gavruta: A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1(2) (2010) 6 pp.Google Scholar

[5] E. Hills, R. S. Phillips: Functional Analysis and Semi-Groups, Colloquium Publications 31, American Mathematical Society, Providence (2000).Google Scholar

[6] D. H. Hyers: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27(1941) 222–224.https://doi.org/10.1073/pnas.27.4.222 Google Scholar

[7] Th. M. Rassias: On the stability of the linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300.https://doi.org/10.1090/S0002-9939-1978-0507327-1 Google Scholar