• The Pure and Applied Mathematics
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• v.19 no.4
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• pp.409-421
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• 2012

In this paper, using an efficient change of variables we refine the Hyers-Ulam stability of the alternative Jensen functional equations of J. M. Rassias and M. J. Rassias and obtain much better bounds and remove some unnecessary conditions imposed in the previous result. Also, viewing the fundamentals of what our method works, we establish an abstract version of the result and consider the functional equations defined in restricted domains of a group and prove their stabilities.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.283-295
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• 2002

In this Paper we show the Solution of the following Jensen functional equation with three variables and prove the stability of this equations in the spirit of Hyers, Ulam, Rassias and Gavruta: (equation omitted).

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.149-161
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• 2011

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras and of derivations on Banach algebras for the following Jensen type functional equation: $$f({\frac{x+y}{2}})+f({\frac{x-y}{2}})=f(x)$$.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.499-507
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• 2006

In this paper, we obtain the general solution and the stability of the hi-Jensen functional equation $$4f(\frac {x+y} 2,\;\frac {z+w} 2)=f(x,\;z)+f(x,\;w)+f(y,\;z)+f(y,\;w)$$.

• 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.22 no.3
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• pp.383-398
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• 2009

In this paper, we study the generalized Hyers-Ulam stability of a bi-Jensen functional equation $$4f(\frac{x+y}{2},\;\frac{z+w}{2})=f(x,\;z)+f(x,w)+f(y,\;z)+f(y,w)$$. Moreover, we establish stability results on the punctured domain.

• Kyungpook Mathematical Journal
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• v.54 no.3
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• pp.401-411
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• 2014

In this paper, we prove the generalized Hyers-Ulam stability of the following Jensen type functional equation $$f(\frac{x-y}{n}+z)+f(\frac{y-z}{n}+x)+f(\frac{z-x}{n}+y)=f(x)+f(y)+f(z)$$ in p-Banach spaces for any fixed nonzero integer n.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.705-720
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• 2008

In this paper, we obtain the Hyers-Ulam-Rassias stability of a bi-Jensen functional equation $4f(\frac {x+y}{2},\;\frac {z+w}{2})=f(x,z)+f(x,w)+f(y,z)+f(y,w)$.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.159-170
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• 2012

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