• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.261-271
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• 2011

We consider the Hyers-Ulam-Rassias stability problem ${\parallel}2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the Schwartz distributions u, which is a distributional version of the Hyers-Ulam-Rassias stability problem of the Jensen functional equation ${\mid}2f(\frac{x+y}{2})-f(x)-F(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the function f : ${\mathbb{R}}^n{\rightarrow}{\mathbb{C}}$.

• The Pure and Applied Mathematics
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• v.18 no.2
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• pp.113-128
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• 2011

Let f : ${\mathbb{R}}{\rightarrow}{\mathbb{C}}$. We consider the Hyers-Ulam stability of Jensen type functional inequality $$|f(px+qy)-Pf(x)-Qf(y)|{\leq}{\epsilon}$$ in the half planes {(x, y) : $kx+sy{\geq}d$} for fixed d, k, $s{\in}{\mathbb{R}}$ with $k{\neq}0$ or $s{\neq}0$. As consequences of the results we obtain the asymptotic behaviors of f satisfying $$|f(px+qy)-Pf(x)-Qf(y)|{\rightarrow}0$$ as $kx+sy{\rightarrow}{\infty}$.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.371-380
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• 2002

In this paper we obtain the general solution of a quadratic Jensen type functional equation : (equation omitted) and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and Gavruta.

• Honam Mathematical Journal
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• v.27 no.3
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• pp.421-429
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• 2005

Generalizing the results in [7] that considers several functional equations in the spaces of the Schwartz tempered distributions and the Fourier hyperfunctions we consider Pexider type functional equations in the space of distributions.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.129-138
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• 2008

In this paper we introduce a Jensen type cubic functional equation $$f\(\frac{3x+y}{2}\)+f\(\frac{x+3y}{2}\)\\=12f\(\frac{x+y}{2}\)+2f(x)+2f(y),$$ and then investigate the generalized Hyers-Ulam stability problem for the equation.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.245-261
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• 2006

In this paper, we prove the generalized Hyers-Ulam stability of ring homomorphisms over the p-adic field $\mathbb{Q}_p$ associated with the Cauchy functional equation f(x+y) = f(x)+f(y) and the Cauchy-Jensen functional equation $2f(\frac{x+y}{2}+z)=f(x)+f(y)+2f(z)$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.335-348
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• 2010

In this paper, we study the Hyers-Ulam-Rassias stability of a bi-Pexider functional equation $$f(x+y,z)-f_1(x,z)-f_2(y,z)=0$$, $$f(x,y+z)-f_3(x,y)-f_4(x,z)=0$$. Moreover, we establish stability results on the punctured domain.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.111-118
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• 2008

Using the Hyers-Ulam-Rassias stability method, we investigate isomorphisms in quasi-Banach algebras and derivations on quasi-Banach algebras associated with the Cauchy-Jensen functional equation $$2f(\frac{x+y}{2}+z)$$=f(x)+f(y)+2f(z), which was introduced and investigated in [2, 17]. The concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Furthermore, isometries and isometric isomorphisms in quasi-Banach algebras are studied.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.159-175
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• 2006

This paper is a survey on the Hyers-Ulam-Rassias stability of the Jensen functional equation in $C^*$-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Its content is divided into the following sections: 1. Introduction and preliminaries. 2. Approximate isomorphisms in $C^*$-algebras. 3. Approximate isomorphisms in Lie $C^*$-algebras. 4. Approximate isomorphisms in $JC^*$-algebras. 5. Stability of derivations on a $C^*$-algebra. 6. Stability of derivations on a Lie $C^*$-algebra. 7. Stability of derivations on a $JC^*$-algebra.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.53-61
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• 2003

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra, and prove the generalized Hyers-Ulam-Rassias stability of the Jensen's functional equations in a Hilbert module over a unital $C^{*}$-algebra.