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Quadratic functional equations in a set of Lebesgue measure zero

Authors:	Jaeyoung Chung John Michael Rassias

Institution:	1. Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea;2. National and Capodistrian University of Athens, Pedagogical Department E. E., Section of Mathematics and Informatics, Greece

Abstract:	Let R $R$ be the set of real numbers, Y a Banach space and f:R→Y $f : R \to Y$ . We prove the Hyers–Ulam stability theorem for the quadratic functional inequality ‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤? $‖ f (x + y) + f (x - y) - 2 f (x) - 2 f (y) ‖ \leq ?$ for all (x,y)∈Ω $(x, y) \in Ω$ , where Ω⊂R² $Ω \subset R^{2}$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.

Keywords:	Baire category theorem First category Lebesgue measure Quadratic functional equation
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