• Nonlinear Functional Analysis and Applications
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• 제29권1호
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• pp.295-306
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• 2024

In this work, we investigate the generalized Hyers-Ulam stability of quadratic functional inequality in modular spaces satisfying ∆₂-conditions and Fatou property, and in 𝛽-homogeneous Banach spaces.

• 대한수학회보
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• 제40권2호
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• pp.253-267
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• 2003

In this paper we deal With the quadratic functional equation (equation omitted) deriving from an inequality of T. Popoviciu for convex functions. We solve this functional equation by proving that its solutions we the polynomials of degree at most two. Likewise, we investigate its stability in the spirit of Hyers, Ulam, and Rassias.

• 호남수학학술지
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• 제33권1호
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• pp.73-84
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• 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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제24권3호
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• pp.179-190
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• 2017

In this paper, we introduce and solve the following quadratic (${\rho}_1$, ${\rho}_2$)-functional inequality (0.1) $$N\left(2f({\frac{x+y}{2}})+2f({\frac{x-y}{2}})-f(x)-f(y),t\right){\leq}min\left(N({\rho}_1(f(x+y)+f(x-y)-2f(x)-2f(y)),t),\;N({\rho}_2(4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y)),t)\right)$$ in fuzzy normed spaces, where ${\rho}_1$ and ${\rho}_2$ are fixed nonzero real numbers with ${{\frac{1}{{4\left|{\rho}_1\right|}}+{{\frac{1}{{4\left|{\rho}_2\right|}}$ < 1, and f(0) = 0. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic (${\rho}_1$, ${\rho}_2$)-functional inequality (0.1) in fuzzy Banach spaces.

• 충청수학회지
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• 제20권4호
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• pp.515-523
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• 2007

In this paper, we investigate the Hyers-Ulam-Rassias stability of generalized Jensen type quadratic functional equations in Banach spaces.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제13권3호
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• pp.237-247
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• 2006

In this paper we solve the Hyers-Ulam stability problem for quadratic functional equations on restricted domains, and then obtain an asymptotic behavior of quadratic mappings on restricted domains.

• 대한수학회논문집
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• 제23권3호
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• pp.371-376
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• 2008

In this paper, it is shown that if f satisfies the following functional inequality (0.1) $${\parallel}\sum\limits_{i,j=1}^3\;f{(xi,yj)}{\parallel}{\leq}{\parallel}f(x_1+x_2+x_3,\;y_1+y_2+y_3){\parallel}$$ then f is a bi-additive mapping. We moreover prove that if f satisfies the following functional inequality (0.2) $${\parallel}2\sum\limits_{j=1}^3\;f{(x_j,\;z)}+2\sum\limits_{j=1}^3\;f{(x_j,\;w)-f(\sum\limits_{j=1}^3\;xj,\;z-w)}{\parallel}{\leq}f(\sum\limits_{j=1}^3\;xj,\;z+w){\parallel}$$ then f is an additive-quadratic mapping.

• 대한수학회논문집
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• 제19권2호
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• pp.253-266
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• 2004

In this note, by using the fixed point alternative, we investigate the modified Hyers-Ulam-Rassias stability for the following mixed type functional inequality which is either cubic or quadratic: $\parallel$8ｆ(x-3y) + 24f(x+y) + f(8y) -8〔f(x+3y) + 3f(x-y) + 2f(2y)〕$\parallel$$\leq$$\varphi$(x,y).

• 전기학회논문지
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• 제67권5호
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• pp.651-655
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• 2018

In this paper, based on an augmented Lyapunov-Krasovskii functional(LKF) with time-delay product quadratic terms, the stability result in the form of linear matrix inequality(LMI) is proposed. In getting an LMI result, the free matrix based integral inequality is used. Finally, two well-known numerical examples are given to demonstrate the usefulness of the proposed result.

• 충청수학회지
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• 제20권4호
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• pp.535-542
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• 2007

In this paper, we investigate the generalized Hyers-Ulam{Rasssias stability of the following Euler-Lagrange type quadratic functional equation $$f(ax+by+cz)+f(ax+by-cz)+f(ax-by+cz)+f(ax-by-cz)=4a^2f(x)+4b^2f(y)+4c^2f(z)$$.