• 대한수학회보
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• 제56권3호
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• pp.801-813
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• 2019

In this paper, using the Baire category theorem we investigate the Hyers-Ulam stability problem of pexiderized Jensen functional equation $$2f(\frac{x+y}{2})-g(x)-h(y)=0$$ and pexiderized Jensen type functional equations $$f(x+y)+g(x-y)-2h(x)=0,\\f(x+y)-g(x-y)-2h(y)=0$$ on a set of Lebesgue measure zero. As a consequence, we obtain asymptotic behaviors of the functional equations.

• 대한수학회지
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• 제54권2호
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• pp.697-711
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• 2017

Let X, Y be real normed vector spaces. We exhibit all the solutions $f:X{\rightarrow}Y$ of the functional equation f(rx + sy) + rsf(x - y) = rf(x) + sf(y) for all $x,y{\in}X$, where r, s are nonzero real numbers satisfying r + s = 1. In particular, if Y is a Banach space, we investigate the Hyers-Ulam stability problem of the equation. We also investigate the Hyers-Ulam stability problem on a restricted domain of the following form ${\Omega}{\cap}\{(x,y){\in}X^2:{\parallel}x{\parallel}+{\parallel}y{\parallel}{\geq}d\}$, where ${\Omega}$ is a rotation of $H{\times}H{\subset}X^2$ and $H^c$ is of the first category. As a consequence, we obtain a measure zero Hyers-Ulam stability of the above equation when $f:\mathbb{R}{\rightarrow}Y$.