• Korean Journal of Mathematics
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• v.30 no.4
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• pp.571-592
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• 2022

By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.29-42
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• 2018

In this paper, we investigate the stability of two functional equations f(ax+by + cz) - abf(x + y) - bcf(y + z) - acf(x + z) + bcf(y) - a(a - b - c)f(x) - b(b - a)f(-y) - c(c - a - b)f(z) = 0, f(ax+by + cz) + abf(x - y) + bcf(y - z) + acf(x - z) - a(a + b + c)f(x) - b(a + b + c)f(y) - c(a + b + c)f(z) = 0 by applying the direct method in the sense of Hyers and Ulam.

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

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.313-326
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• 2015

In this paper, we present a fixed point method to prove generalized Hyers-Ulam stability of the systems of quadratic-cubic functional equations with constant coefficients in modular spaces.

• Nonlinear Functional Analysis and Applications
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• v.26 no.3
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• pp.497-512
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• 2021

In this paper, we study the analytical and approximate solutions for a fractional quadratic integral equation involving Katugampola fractional integral operator. The existence and uniqueness results obtained in the given arrangement are not only new but also yield some new particular results corresponding to special values of the parameters 𝜌 and ϑ. The main results are obtained by using Banach fixed point theorem, Picard Method, and Adomian decomposition method. An illustrative example is given to justify the main results.

• 제어로봇시스템학회:학술대회논문집
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• 1998.10a
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• pp.373-378
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• 1998

A method to construct a memoryless feedback law for systems with multiple time-delays in the states is proposed. As a plant model, a differential-difference equation with multiple delayed terms is introduced, A stabilizability condition by memoryless feedback is presented. A feedback gain is calculated with a solution of a finite dimensional Riccati equation. It is shown that the resulting closed loop system is asymptotically stable, and moreover, it is a linear quadratic regulator for some cost functional. An alternative stabilizability condition which is easier to check is given.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.263-273
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• 2013

A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation $$\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y$$, where M is a given mean and $f$, F, $g$, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.413-424
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• 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
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• v.47 no.5
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• pp.987-996
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• 2010

In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.1
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• pp.69-77
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• 2006

The generalized Hyers-Ulam stability problems of the mixed type functional equation $$f\({\sum_{i=1}^{4}xi\)+\sum_{1{\leq}i<j{\leq}4}f(x_i+x_j)=\sum_{i=1}^{4}f(x_i)+\sum_{1{\leq}i<j<k{\leq}4}f(x_i+X_j+x_k)$$ is treated under the approximately even(or odd) condition and the behavior of the quadratic mappings and the additive mappings is investigated.