• Korean Journal of Mathematics
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• 제19권1호
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• pp.77-85
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• 2011

In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $${\sum_{i=1}^{n}}\left\|x_i-{\frac{1}{n}}{\sum_{j=1}^{n}}x_j \right\|^2={\sum_{i=1}^{n}}{\parallel}x_i{\parallel}^2-n\left\|{\frac{1}{n}}{\sum_{i=1}^{n}}x_i \right\|^2$$ holds for all $x_1$, ${\cdots}$, $x_n{\in}V$. Let V, W be real vector spaces. It is shown that if an even mapping $f:V{\rightarrow}W$ satisfies $$(0.1)\;{\sum_{i=1}^{2n}f}\(x_i-{\frac{1}{2n}}{\sum_{j=1}^{2n}}x_j\)={\sum_{i=1}^{2n}}f(x_i)-2nf\({\frac{1}{2n}}{\sum_{i=1}^{2n}}x_i\)$$ for all $x_1$, ${\cdots}$, $x_{2n}{\in}V$, then the even mapping $f:V{\rightarrow}W$ is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.

• Korean Journal of Mathematics
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• 제19권1호
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• pp.17-24
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• 2011

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0, and $$2(_{2d-2}C_{d-1}-_{2d-2}C_d)f\({\sum_{j=1}^{2d}}x_j\)+{\sum_{{\iota}(j)=0,1,{{\small\sum}_{j=1}^{2d}}{\iota}(j)=d}}\;f\({\sum_{j=1}^{2d}}(-1)^{{\iota}(j)}x_j\)=2(_{2d-1}C_d+_{2d-2}C_{d-1}-_{2d-2}C_d){\sum_{j=1}^{2d}}f(x_j)$$ for all $x_1$, ${\cdots}$, $x_{2d}{\in}X$, then the even mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.

• 충청수학회지
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• 제19권1호
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• pp.69-77
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• 2006

The generalized Hyers-Ulam stability problems of the mixed type functional equation $$f\({\sum_{i=1}^{4}xi\)+\sum_{1{\leq}i<j{\leq}4}f(x_i+x_j)=\sum_{i=1}^{4}f(x_i)+\sum_{1{\leq}i<j<k{\leq}4}f(x_i+X_j+x_k)$$ is treated under the approximately even(or odd) condition and the behavior of the quadratic mappings and the additive mappings is investigated.

• 대한수학회보
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• 제51권5호
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• pp.1511-1525
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• 2014

In this paper, we prove the generalized Hyers-Ulam stability results controlled by considering approximately mappings satisfying conditions much weaker than Hyers and Rassias conditions for radical quadratic and radical quartic functional equations in quasi-${\beta}$-normed spaces.