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
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
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.
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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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Chung, J., Rassias, J.M., Lee, B., Choi, CK. (2019). Hyperstability of a Linear Functional Equation on Restricted Domains. In: Anastassiou, G., Rassias, J. (eds) Frontiers in Functional Equations and Analytic Inequalities. Springer, Cham. https://doi.org/10.1007/978-3-030-28950-8_2
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DOI: https://doi.org/10.1007/978-3-030-28950-8_2
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