• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.163-172
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• 2007

In this paper, we prove the Hyers-Ulam-Rassias stability of a Cauchy-Jensen functional equation $$2f(x+y,\frac{z+w}{2})=f(x,z)+f(x,w)+f(y,z)+f(y,w)$$.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.173-182
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• 2007

In this paper, we prove the Cauchy-Rassias stability of derivations on quasi-Banach algebras associated to the Cauchy functional equation and the Jensen functional equation. We use the Cauchy-Rassias inequality that was first introduced by Th. M. Rassias in the paper "On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300".

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.393-399
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• 2015

W. Park [J. Math. Anal. Appl. 376 (2011) 193-202] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. But there are serious problems in the control functions given in all theorems of the paper. In this paper, we correct the statements of these results and prove the corrected theorems. Moreover, we prove the superstability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces under the original given conditions.

• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.377-386
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• 2008

In this paper, we prove the stability of a Cauchy-Jensen functional equation $$2f(x+y,\;\frac{z+w}2)$$=f(x, z)+f(x, w)+f(y, z)+f(y, w) in the sense of Th. M. Rassias.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.205-214
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• 2008

In this paper, we prove the generalized Hyers-Ulam stability of a Cauchy-Jensen functional equation $2f(x+y,\frac{z+w}{2})=f(x,z)+f(x,w)+f(y,z)+f(y,w)$ in the spirit of $P.G{\breve{a}}vruta$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.197-208
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• 2008

In this paper, we obtain the Hyers-Ulam-Rassias stability of a Cauchy-Jensen functional equation $$f(x+y,z)-f(x,z)-f(y,z)=0,\\2f(x,\frac{y+z}2)-f(x,y)-f(x,z)=0$$ in the spirit of Th. M. Rassias.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.167-178
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• 2009

In this paper, we obtain the generalized Hyers-Ulam stability of a Cauchy-Jensen functional equation f(x+y, z)-f(x, z)-f(y, z)=0, $$2f\;x,\;{\frac{y+z}{2}}-f(x,\;y)-f(x,\;z)=0$$ in the spirit of P. $G{\breve{a}}vruta$.

• Korean Journal of Mathematics
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• v.22 no.4
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• pp.659-670
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• 2014

In this paper, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach algebras and of derivations on fuzzy Banach algebras associated to the Cauchy-Jensen functional equation.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.361-370
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• 2008

We consider the Hyers-Ulam type stability of the Cauchy, Jensen, Pexider, Pexider-Jensen differences: $$(0.1){\hspace{55}}C(u):=u{\circ}A-u{\circ}P_1-u{\circ}P_2,\\(0.2){\hspace{55}}J(u):=2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2,\\(0.3){\hspace{18}}P(u,v,w):=u{\circ}A-v{\circ}P_1-w{\circ}P_2,\\(0.4)\;JP(u,v,w):=2u{\circ}\frac{A}{2}-v{\circ}P_1-w{\circ}P_2$$, with respect to bounded distributions.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.83-90
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• 2009

In this paper, we investigate isomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary 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 by Baak in [2].