Abstract

In a recent paper [Asymptotic Hyers-Ulam stability or superstability for generalized linear equations by unilateral perturbations, J. Convex Analysis 26 (2019) 543–562] we considered in relation to the famous problem of Ulam ``Give conditions in order for a linear mapping near an approximated linear mapping to exist" the stability or superstability of a generalized linear equation ∣∣f(x+y)−f(x)−f(y)∣∣=B[ϕ(x)+ϕ(y)]{||{f(x+y)-f(x)-f(y)}||=B[\phi(x)+\phi(y)]} by perturbations on the convexity side, named right perturbations. In this paper we continue this research for perturbations on the concavity side with some hypotheses of concavity.

Author information

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

Keywords

  • Hyers-Ulam stability
  • superstability
  • asymptotic stability
  • linear equation
  • affine equation
  • exponential equation

Mathematics Subject Classification

39B62, 26A51

References

  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/00210064Google Scholar
  2. Z. Gajda: On stability of additive mappings, Int. J. Math. Math. Sci. 14(3) (1991) 431–434.https://doi.org/10.1155/S016117129100056XGoogle 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.1994.1211Google Scholar
  4. P. Gavruta, L. Gavruta: A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1(2) (2010) 11–18.Google Scholar
  5. E. Hille, R. S. Phillips: Functional Analysis and Semi-Groups, Colloquium Publications 31, American Mathematical Society, Providence (1996).https://doi.org/10.1090/coll/031Google 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.27.4.222Google Scholar
  7. C. Peppo: Asymptotic Hyers-Ulam stability or superstability for generalized linear equations by unilateral perturbations, J. Convex Analysis 26(2) (2019) 543–562.https://doi.org/10.68381/jca26028Google Scholar
  8. 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/S0002-9939-1978-0507327-1Google Scholar

Suggested citation

C. Peppo. “Asymptotic Hyers-Ulam Stability or Superstability by Unilateral Perturbations on the Concavity Side for Generalized Linear Equations.” Journal of Convex Analysis 28 (2021), No. 1, 143–156. https://doi.org/10.68381/jca28011

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.