• 대한수학회보
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• 제47권5호
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• pp.987-996
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• 2010

In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.

• 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.

• 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.

• 충청수학회지
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• 제22권4호
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• pp.815-830
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• 2009

In this paper we establish the general solution of the following functional equation with mixed type of quadratic and additive mappings f(mx+y)+f(mx-y)+2f(x)=f(x+y)+f(x-y)+2f(mx), where $m{\geq}2$ is a positive integer, and then investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

• 충청수학회지
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• 제25권1호
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• pp.43-49
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• 2012

In this paper, we investigate the stability of a functional equation $$f(x+y+z)-f(x+y)-f(y+z)-f(x+z)+f(x)+f(y)+f(z)=0$$ by using the fixed point theory in the sense of L. $C\breve{a}dariu$ and V. Radu.

• 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.