• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.1
• /
• pp.43-49
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.2
• /
• pp.147-162
• /
• 2011

In this paper, we investigate the Ulam-Hyers stability of $C^{\star}$-ternary algebra 3-homomorphisms for the functional equation $$f(x_1+x_2,y_1+y_2,z_1+z_2)=\;\displaystyle\sum_{1{\leq}i,j,k{\leq}2}\;f(x_i,y_j,z_k)$$ in $C^{\star}$-ternary algebras.

• The Pure and Applied Mathematics
• /
• v.18 no.3
• /
• pp.231-244
• /
• 2011

We prove the Hyers-Ulam stability for trigonometric type functional inequalities in restricted domains with time variables. As consequences of the result we obtain asymptotic behaviors of the inequalities and stability of related functional inequalities in almost everywhere sense.

• East Asian mathematical journal
• /
• v.34 no.5
• /
• pp.693-702
• /
• 2018

In this paper, we consider the generalized Hyers-Ulam stability for the following additive-quadratic functional equation with involution $f(x+2y)-f(2x+y)+f(x+y)+f({\sigma}(x)+y)+f(x)-4f(y)-f({\sigma}(y))=0$ in non-Archimedean spaces.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.1
• /
• pp.195-209
• /
• 2010

In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in $C^*$-ternary algebras and of bi-derivations on $C^*$-ternary algebras for the following bi-additive functional equation f(x + y, z - w) + f(x - y, z + w) = 2f(x, z) - 2f(y, w). This is applied to investigate bi-isomorphisms between $C^*$-ternary algebras.

• Kyungpook Mathematical Journal
• /
• v.53 no.4
• /
• pp.639-646
• /
• 2013

We investigate the stability of orthogonally additive set-valued functional equation $$F(x+y)=F(x)+F(y)(x{\perp}y)$$ in Hausdorff topology on closed convex subsets of a Banach space.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.2
• /
• pp.189-196
• /
• 2006

The generalized Hyers-Ulam stability problems of the cubic functional equation f(x + y + z) + f(x + y - z) + 2f(x - y) + 4f(y) = f(x - y + z) + f(x - y - z) +2f(x + y) + 2f(y + z) + 2f(y - z) shall be treated under the approximately odd condition and the behavior of the cubic mappings and the additive mappings shall be investigated. The generalized Hyers-Ulam stability problem for functional equations had been posed by Th.M. Rassias and J. Tabor [7] in 1992.

• Communications of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.55-82
• /
• 2019

Given ${\sigma}:G{\rightarrow}G$ an involutive automorphism of a semigroup G, we study the solutions and stability of the following functional equations $$f(x{\sigma}(y))=f(x)g(y)+g(x)f(y),\;x,y{\in}G,\\f(x{\sigma}(y))=f(x)f(y)-g(x)g(y),\;x,y{\in}G$$ and $$f(x{\sigma}(y))=f(x)g(y)-g(x)f(y),\;x,y{\in}G$$, from the theory of trigonometric functional equations. (1) We determine the solutions when G is a semigroup generated by its squares. (2) We obtain the stability results for these equations, when G is an amenable group.

• Korean Journal of Mathematics
• /
• v.16 no.3
• /
• pp.413-424
• /
• 2008

In [21], 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 $n{\geq}2$ $$n{\parallel}\frac{1}{n}\sum\limits_{i=1}^{n}x_i{\parallel}^2+\sum\limits_{i=1}^{n}{\parallel}x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j{\parallel}^2=\sum\limits_{i=1}^{n}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\dots},x_n{\in}V$. We consider the functional equation $$nf(\frac{1}{n}\sum\limits^n_{i=1}x_i)+\sum\limits_{i=1}^{n}f(x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j)=\sum\limits_{i=1}^nf(x_i)$$ Using fixed point methods, we prove the generalized Hyers-Ulam stability of the functional equation $$(1)\;2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})=f(x)+f(y)$$.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.2
• /
• pp.339-352
• /
• 2012

We prove the stability for the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients on the probabilistic normed spaces (briefly PN spaces).