Hyperstability of a Linear Functional Equation on Restricted Domains

Jaeyoung Chung (), John Michael Rassias (), Bogeun Lee () and Chang-Kwon Choi ()
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Jaeyoung Chung: Kunsan National University, Department of Mathematics
John Michael Rassias: Pedagogical Department E. E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens
Bogeun Lee: Chonbuk National University, Department of Mathematics and Institute of Pure and Applied Mathematics
Chang-Kwon Choi: Kunsan National University, Department of Mathematics and Liberal Education Institute

A chapter in Frontiers in Functional Equations and Analytic Inequalities, 2019, pp 27-42 from Springer

Abstract: Abstract Let X, Y be real Banach spaces, f : X → Y and ℋ $$\mathcal H$$ be a subset of X such that ℋ c $${\mathcal H}^c$$ is of the first category. Using the Baire category theorem we prove the Ulam–Hyers stability of the linear functional equation f ( a x + b y + α ) = A f ( x ) + B f ( y ) + C $$\displaystyle f(ax+by+\alpha )=Af(x)+Bf(y)+C $$ for all x , y ∈ ℋ $$x, y\in \mathcal H$$ , 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 ∥ f ( a x + b y + α ) − A f ( x ) − B f ( y ) − C ∥ ≤ δ ψ ( x , y ) $$\displaystyle \|f(ax+by+\alpha )-Af(x)-Bf(y)-C\|\le \delta \psi (x,y) $$ for all x , y ∈ K $$x, y \in \mathcal K$$ , where K $$\mathcal K$$ is a subset of ℝ $$\mathbb {R}$$ with Lebesgue measure zero and ψ(x, y) = |x|p + |y|q, p, q

Keywords: Baire category theorem; First category; Restricted domain; Second category; Lebesgue measure zero; Linear functional equation; Ulam–Hyers stability; Hyperstability; 39B82 (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-28950-8_2

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DOI: 10.1007/978-3-030-28950-8_2

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