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Hyperstability of a Linear Functional Equation on Restricted Domains

  • Jaeyoung Chung
  • John Michael Rassias
  • Bogeun Lee
  • Chang-Kwon ChoiEmail author
Chapter

Abstract

Let X, Y  be real Banach spaces, f : X → Y  and \(\mathcal H\) be a subset of X such that \({\mathcal H}^c\) is of the first category. Using the Baire category theorem we prove the Ulam–Hyers stability of the linear functional equation

$$\displaystyle f(ax+by+\alpha )=Af(x)+Bf(y)+C $$
for all \(x, y\in \mathcal H\), such that ∥x∥ + ∥y∥≥ d with d > 0, where a, b, A, B are nonzero real numbers and α ∈ X is fixed. As a consequence we solve the hyperstability problem associated to
$$\displaystyle \|f(ax+by+\alpha )-Af(x)-Bf(y)-C\|\le \delta \psi (x,y) $$
for all \(x, y \in \mathcal K\), where \(\mathcal K\) is a subset of \(\mathbb {R}\) with Lebesgue measure zero and ψ(x, y) = |x|p + |y|q, p, q < 0; or ψ(x, y) = |x|p|y|q, p + q < 0; or ψ(x, y) = |x|p|y|q, pq < 0.

Keywords

Baire category theorem First category Restricted domain Second category Lebesgue measure zero Linear functional equation Ulam–Hyers stability Hyperstability 

2010 Mathematics Subject Classification

39B82 

Notes

Acknowledgements

This research was completed with the help of Professor Jaeyoung Chung. After finishing this work, Professor Jaeyoung Chung tragically passed away. Pray for the bliss of dead.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jaeyoung Chung
    • 1
  • John Michael Rassias
    • 2
  • Bogeun Lee
    • 3
  • Chang-Kwon Choi
    • 4
    Email author
  1. 1.Department of MathematicsKunsan National UniversityGunsanRepublic of Korea
  2. 2.National and Kapodistrian University of AthensPedagogical Department E. E., Section of Mathematics and InformaticsAthensGreece
  3. 3.Department of Mathematics and Institute of Pure and Applied MathematicsChonbuk National UniversityJeonjuRepublic of Korea
  4. 4.Department of Mathematics and Liberal Education InstituteKunsan National UniversityGunsanRepublic of Korea

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