• Kyungpook Mathematical Journal
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• 제47권2호
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• pp.239-251
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• 2007

We show how the minimization can be used to solve the quadratic matrix equation and then compare two different types of conjugate gradient method which are Polak and Ribi$\acute{e}$re version and Fletcher and Reeves version. Finally, some results of the global and local convergence are shown.

• 충청수학회지
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• 제14권2호
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• pp.1-7
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• 2001

In this paper we prove the stability of some generaliged quadratic functional equation $$a^2f(\frac{x+y}{a})+a^2f(\frac{x-y}{a})=2f(x)+2f(y)$$.

• East Asian mathematical journal
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• 제35권3호
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• pp.331-340
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• 2019

In this paper, we investigate the stability for the functional equation $f(x+ky)-kf(x+y)+kf(x-y)-f(x-ky)-(k^3-k)f(y)+(k^3-k)f(-y)=0$ in the sense of M. S. Moslehian and Th. M. Rassias.

• 호남수학학술지
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• 제26권1호
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• pp.45-59
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• 2004

In this paper we obtain the general solution the functional equation $a^2f(\frac{x-2y}{a})+f(x)+2f(y)=2a^2f(\frac{x-y}{a})+f(2y).$ The type of the solution of this equation is Q(x)+A(x)+C, where Q(x), A(x) and C are quadratic, additive and constant, respectively. Also we prove the stability of this equation in the spirit of Hyers, Ulam, Rassias and $G\check{a}vruta$.

• 대한수학회보
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• 제42권1호
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• pp.57-73
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• 2005

In this paper we solve a generalized quadratic Jensen type functional equation $m^2 f (\frac{x+y+z}{m}) + f(x) + f(y) + f(z) =n^2 [f(\frac{x+y}{n}) +f(\frac{y+z}{n}) +f(\frac{z+x}{n})]$ and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and Gavruta.

• 대한수학회지
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• 제50권1호
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• pp.173-187
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• 2013

This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

• 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.

• Journal of applied mathematics & informatics
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• 제15권1_2호
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• pp.435-446
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• 2004

In this paper, we obtain the general solution of a quadratic functional equation $b^2f(\frac{x+y+z}{b})+f(x-y)+f(x-z)=\;a^2[f(\frac{x-y-z}{a})+f(\frac{x+y}{a})+f(\frac{x+z}{a})]$ and prove the stability of this equation.

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.371-380
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• 2002

In this paper we obtain the general solution of a quadratic Jensen type functional equation : (equation omitted) and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and Gavruta.

• 호남수학학술지
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• 제30권2호
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• pp.219-225
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• 2008

In this paper, we obtain the general solution and the stability of the 2-dimensional vector variable quadratic functional equation f( x + y, z - w) + f(x - y, z + w) = 2f(x, z ) + 2f(y, ${\omega}$). The quadratic form f( x, y) = $ax^2$ + $by^2$ without cross product terms is a solution of the above functional equation.