• 대한수학회보
• /
• 제56권3호
• /
• pp.801-813
• /
• 2019

In this paper, using the Baire category theorem we investigate the Hyers-Ulam stability problem of pexiderized Jensen functional equation $$2f(\frac{x+y}{2})-g(x)-h(y)=0$$ and pexiderized Jensen type functional equations $$f(x+y)+g(x-y)-2h(x)=0,\\f(x+y)-g(x-y)-2h(y)=0$$ on a set of Lebesgue measure zero. As a consequence, we obtain asymptotic behaviors of the functional equations.

• 대한수학회보
• /
• 제58권2호
• /
• pp.269-275
• /
• 2021

Let ℝ be the set of real numbers, f, g : ℝ → ℝ and �� ≥ 0. In this paper, we consider the stability of partially pexiderized exponential-radical functional equation $$f({\sqrt[n]{x^N+y^N}})=f(x)g(y)$$ for all x, y ∈ ℝ, i.e., we investigate the functional inequality $$\|f({\sqrt[n]{x^N+y^N}})-f(x)g(y)\|{\leq}{\epsilon}$$ for all x, y ∈ ℝ.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제19권2호
• /
• pp.193-198
• /
• 2012

We show that every unbounded approximate Pexiderized exponential type function has the exponential type. That is, we obtain the superstability of the Pexiderized exponential type functional equation $$f(x+y)=e(x,y)g(x)h(y)$$. From this result, we have the superstability of the exponential functional equation $$f(x+y)=f(x)f(y)$$.

• Korean Journal of Mathematics
• /
• 제16권1호
• /
• pp.103-114
• /
• 2008

In this paper, we investigate the superstability problem for the pexiderized cosine functional equations f(x+y) +f(x−y) = 2g(x)h(y), f(x + y) + g(x − y) = 2f(x)g(y), f(x + y) + g(x − y) = 2g(x)f(y). Consequently, we have generalized the results of stability for the cosine($d^{\prime}Alembert$) and the Wilson functional equations by J. Baker, $P.\;G{\check{a}}vruta$, R. Badora and R. Ger, and G.H. Kim.

• 충청수학회지
• /
• 제20권2호
• /
• pp.117-137
• /
• 2007

We establish the Hyers-Ulam-Rassias stability of the Pexiderized mixed type quadratic equation $f_1(x+y+z)+f_2(x-y)+f_3(x-z)-f_4(x-y-z)-f_5(x+y)-f_6(x+z)=0$ in the spirit of D. H. Hyers, S. M. Ulam and Th. M. Rassias.

• 호남수학학술지
• /
• 제33권1호
• /
• pp.73-84
• /
• 2011

Let f, g, h, k : $\mathbb{R}^n{\rightarrow}\mathbb{C}$ be locally integrable functions. We deal with the $L^{\infty}$-version of the Hyers-Ulam stability of the quadratic functional inequality and the Pexiderized quadratic functional inequality $${\parallel}f(x + y) + f(x - y) -2f(x) - 2f(y){\parallel}_{L^{\infty}(\mathbb{R}^n)}\leq\varepsilon$$ $${\parallel}f(x + y) + g(x - y) -2h(x) - 2f(y){\parallel}_{L^{\infty}(\mathbb{R}^n)}\leq\varepsilon$$ based on the concept of linear functionals on the space of smooth functions with compact support.

• 대한수학회보
• /
• 제44권3호
• /
• pp.569-587
• /
• 2007

In this paper, we establish the generalized Hyers-Ulam-Rassias stability of the Pexider type quadratic equation $f_1(x+y+z)+f_2(x-y)+f_3(x-z)-f_4(x-y-z)-f_5(x+y)-f_6(x+z)=0$ and its general solution.

• 대한수학회논문집
• /
• 제31권1호
• /
• pp.101-113
• /
• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• 대한수학회보
• /
• 제46권1호
• /
• pp.45-60
• /
• 2009

In this paper, we investigate homomorphisms in proper $JCQ^*$-triples and derivations on proper $JCQ^*$-triples associated to the following Pexiderized functional equation $$f(x+y+z)=f_0(x)+f_1(y)+f_2(z)$$. This is applied to investigate homomorphisms and derivations in proper $JCQ^*$-triples.

• Journal of applied mathematics & informatics
• /
• 제29권5_6호
• /
• pp.1421-1433
• /
• 2011

In this paper, using a fixed point approach, the generalized Hyeres-Ulam stability of the following quadratic functional equation $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=3(f(x)+f(y)+f(z))$ will be studied, where f is a function from abelian group G into a quasi-${\beta}$-normed space and ${\sigma}$ is an involution on the group G. Next, we consider its pexiderized equation of the form $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=g(x)+g(y)+g(z)$ and its generalized Hyeres-Ulam stability.