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Let $X, Y$ be real normed vector spaces. We exhibit all the solutions $f:X\to Y$ 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 $\Omega\cap\{(x, y)\in X^2:\|x\|+\|y\|\ge d\}$, 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 $f:\mathbb R\to Y$.
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참고문헌 (23)
참고문헌 신청
A. Gilanyi / 1999 / Hyers-Ulam stability of monomial functional equations on a general domain / Proc. Natl. Acad. Sci. USA 96 (19) : 10588 ~ 10590
A. Najati / 2010 / Approximately quadratic mappings on restricted domains / J. Inequal. Appl 2010 : 10 ~
B. Batko / 2008 / Stability of an alternative functional equation / J. Math. Anal. Appl 339 (1) : 303 ~ 311
D. G. Bourgin / 1951 / Classes of transformations and bordering transformations / Bull. Amer. Math. Soc 57 : 223 ~ 237
D. H. Hyers / 1941 / On the stability of the linear functional equations / Proc. Nat. Acad. Sci. USA 27 : 222 ~ 224
A. Najati / 2010 / Approximately quadratic mappings on restricted domains / J. Inequal. Appl 2010 : 10 ~
B. Batko / 2008 / Stability of an alternative functional equation / J. Math. Anal. Appl 339 (1) : 303 ~ 311
D. G. Bourgin / 1951 / Classes of transformations and bordering transformations / Bull. Amer. Math. Soc 57 : 223 ~ 237
D. H. Hyers / 1941 / On the stability of the linear functional equations / Proc. Nat. Acad. Sci. USA 27 : 222 ~ 224
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