Inner Product Spaces and Quadratic Functional Equations

被引：0

作者：

Park, Choonkil ^{[1
]}

Park, Won-Gil ^{[2
]}

Rassias, Themistocles M. ^{[3
]}

机构：

[1] Hanyang Univ, Dept Math, Seoul, South Korea

[2] Mokwon Univ, Daejeon, South Korea

[3] Natl Tech Univ Athens, Athens, Greece

来源：

COMPUTATIONAL ANALYSIS, AMAT 2015 | 2016年 / 155卷

关键词：

Inner product space; Quadratic mapping; Quadratic Functional equation; IIyers-Ulam stability; ULAM STABILITY; BANACH-SPACES; MAPPINGS;

D O I：

10.1007/978-3-319-28443-9_10

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we prove that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer n >= 2 n parallel to Sigma(n)(i=1)x(i)parallel to(2) + Sigma(n)(i=1)parallel to nx(i) - Sigma(n)(j=1)x(j)parallel to(2) = n(2)Sigma(n)(i=1)parallel to x(i)parallel to(2) holds for all x(1),..., x(n) is an element of V. Let V, W be real vector spaces. It is shown that if a mapping f : V -> W satisfies nf(Sigma(n)(i=1)x(i)) + Sigma(n)(i=1)f(nx(i) - Sigma(n)(j=1)x(j)) = n(2)Sigma(n)(i=1)f(x(i)), (n > 2) or nf(Sigma(n)(i=1)x(i)) + Sigma(n)(i=1)f(nx(i) - Sigma(n)(j=1)x(j)) = n(2) + n/2 Sigma(n)(i=1)f(x(i)) + n(2) - n/2 Sigma(n)(i=1)f(-x(i)), (n >= 2) for all x(1),..., x(n) is an element of V, then the mapping f : V -> W is Cauchy additive-quadratic. Furthermore, we prove the Hyers-Ulam stability of the above quadratic functional equations in Banach spaces.

引用

页码：137 / 151

页数：15

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