A MEASURE ZERO STABILITY OF A FUNCTIONAL EQUATION ASSOCIATED WITH INNER PRODUCT SPACE

作者：Chung Jaeyoung^*; Rassias John Michael

来源：Journal of the Korean Mathematical Society, 2017, 54(2): 697-711.

DOI：10.4134/JKMS.j160272

• Baire category theorem
• first category
• Lebesgue measure
• quadratic functional equation
• (r, s)-quasi-quadratic functional equation
• second category
• hyers-Ulam stability

Let X, Y be real normed vector spaces. We exhibit all the solutions f : X -> Y of the functional equation f(rx + sy) + rsf(x - y) = rf(x) + sf(y) for all x, y is an element of 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 boolean AND {(x, y) is an element of X-2: parallel to x parallel to + parallel to y parallel to >= d}, where Omega is a rotation of H x H subset of 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 : R -> Y .

• 出版日期2017-3