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HYERS{ULAM STABILITY OF FUNCTIONAL INEQUALITIES ASSOCIATED WITH CAUCHY MAPPINGS
Kim, Hark-Mahn,Oh, Jeong-Ha 충청수학회 2007 충청수학회지 Vol.20 No.4
In this paper, we investigate the generalized Hyers-Ulam stability of the functional inequality $$||af(x)+bf(y)+cf(z)||{\leq}||f(ax+by+cz))||+{\phi}(x,y,z)$$ associated with Cauchy additive mappings. As a result, we obtain that if a mapping satisfies the functional inequality with perturbing term which satisfies certain conditions then there exists a Cauchy additive mapping near the mapping.
REMARKS ON THE PAPER: ORTHOGONALLY ADDITIVE AND ORTHOGONALLY QUADRATIC FUNCTIONAL EQUATION
Kim, Hark-Mahn,Jun, Kil-Woung,Kim, Ahyoung Chungcheong Mathematical Society 2013 충청수학회지 Vol.26 No.2
The main goal of this paper is to present the additional stability results of the following orthogonally additive and orthogonally quadratic functional equation $$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)=\frac{3}{2}f(x)-\frac{1}{2}f(-x)+f(y)+f(-y)+f(z)+f(-z)$$ for all $x,y,z$ with $x{\bot}y$, which has been introduced in the paper [11], in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.
Stability of the Functional Equations related to a Multiplicative Derivation
Hark-Mahn Kim,Ick-Soon Chang 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.11 No.1-2
In this paper, using an idea from the direct method of Hyers and Ulam,we investigate the situations so that the Hyers-Ulam-Rassias stability of the func-tional equation g(x2) = 2 xg(x) is satised.AMS Mathematics Subject Classication : 39B52, 39B72.Key words and phrases : Hyers-Ulam stability, multiplicative derivation1. IntroductionIn 1940, S. M. Ulam [14] gave a wide ranging talk before the mathematicsclub of the University of Wisconsin in which he discussed a number of importantunsolved problems. Among those was the question concerning the stability ofgroup homomorphisms:LetG1 be a group and letG2 be a metric group with the metricd(·,·). Given0, does there exist a 0 such that if a functionh :G1 → G2 satises theinequality d(h(xy),h(x)h(y)) for allx,y∈G1, then there exists a homomor-phism H :G1 → G2 withd(h(x),H(x)) for allx ∈G1?In other words, we are looking for situations when the homomorphisms are
ON THE STABILITY OF A MODIFIED JENSEN TYPE CUBIC MAPPING
Kim, Hark-Mahn,Ko, Hoon,Son, Jiae Chungcheong Mathematical Society 2008 충청수학회지 Vol.21 No.1
In this paper we introduce a Jensen type cubic functional equation <TEX>$$f\(\frac{3x+y}{2}\)+f\(\frac{x+3y}{2}\)\\=12f\(\frac{x+y}{2}\)+2f(x)+2f(y),$$</TEX> and then investigate the generalized Hyers-Ulam stability problem for the equation.
Stability of Approximate Quadratic Mappings
Kim, Hark-Mahn,Kim, Minyoung,Lee, Juri Hindawi Publishing Corporation 2010 Journal of inequalities and applications Vol.2010 No.1
<P>We investigate the general solution of the quadratic functional equation f(2x+y)+3f(2x-y)=4f(x-y)+12f(x), in the class of all functions between quasi-β-normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.</P>
STABILITY OF THE FUNCTIONAL EQUATIONS RELATED TO A MULTIPLICATIVE DERIVATION
Kim, Hark-Mahn,Chang, Ick-Soon 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.11 No.1
In this paper, using an idea from the direct method of Hyers and Ulam, we investigate the situations so that the Hyers-Ulam-Rassias stability of the functional equation $g(x^2)\;=\;2xg(x)$ is satisfied.