A MEASURE ZERO STABILITY OF A FUNCTIONAL EQUATION ASSOCIATED WITH INNER PRODUCT SPACE
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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 $f:\mathbb R\to Y$.
Abstract
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 $f:\mathbb R\to Y$.
참고문헌 (23)
- J. Aczel, "Lectures on Functional Equations and Their Applications" (1966), Academic Press
- J. Aczel, "Functional Equations in Several Variables" (1989), Cambridge University Press
- T. Aoki, "On the stability of the linear transformation in Banach spaces", J. Math. Soc. Japan, 2, pp.64~66 (1950)
- B. Batko, "Stability of an alternative functional equation", J. Math. Anal. Appl, 339(1), pp.303~311 (2008)
- D. G. Bourgin, "Classes of transformations and bordering transformations", Bull. Amer. Math. Soc, 57, pp.223~237 (1951)
- J. Brzdek, "On the quotient stability of a family of functional equations", Nonlinear Anal, 71(10), pp.4396~4404 (2009)
- J. Brzdek, "A conditional exponential functional equation and its stability", Nonlinear Anal, 72(6), pp.2929~2934 (2010)
- J. Chung, "Quadratic functional equations in a set of Lebesgue measure zero", J. Math. Anal. Appl, 419(2), pp.1065~1075 (2014)
- A. Gilanyi, "Hyers-Ulam stability of monomial functional equations on a general domain", Proc. Natl. Acad. Sci. USA, 96(19), pp.10588~10590 (1999)
- D. H. Hyers, "On the stability of the linear functional equations", Proc. Nat. Acad. Sci. USA, 27, pp.222~224 (1941)
- P. Jordan, "On inner products in linear metric spaces", Ann. of Math, 36(3), pp.719~723 (1935)
- S.-M. Jung, "On the Hyers-Ulam stability of the functional equations that have the qua-dratic property", J. Math. Anal. Appl, 222(1), pp.126~137 (1998)
- S.-M. Jung, "Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analisis" (2011), Springer
- S.-M. Jung, "Stability of a generalized Jensen equation on restricted domains", J. Math. Inequal, 4(2), pp.191~206 (2010)
- P. Kannappan, "Quadratic functional equation and inner product spaces", Results Math, 27(3-4), pp.368~372 (1995)
- A. Najati, "Approximately quadratic mappings on restricted domains", J. Inequal. Appl, 2010, pp.10~ (2010)
- J. C. Oxtoby, "Measure and Category" (1980), Springer
- J. M. Rassias, "On the Ulam stability of mixed type mappings on restricted domains", J. Math. Anal. Appl, 281(2), pp.747~762 (2002)
- J. M. Rassias, "Asymptotic behavior of mixed type functional equations", Aust. Math. Anal. Appl, 1(1), pp.1~21 (2004)
- Th. M. Rassias, "On the stability of the linear mapping in Banach spaces", Proc. Amer. Math. Soc, 72(2), pp.297~300 (1978)
- J. Sikorska, "On two conditional Pexider functional equations and their stabilities", Non-linear Analy, 70(7), pp.2673~2684 (2009)
- F. Skof, "Sull’ approssimazione delle applicazioni localmente δ-additive", Atti della Ac-cademia delle Scienze di Torino, 117, pp.377~389 (1983)
- S. M. Ulam, "Problems in Modern Mathematics. Vol. VI, Science Edn" (1940), Wiley