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GENERALIZED QUASI–BANACH SPACES AND QUASI–(2; p)–NORMED SPACES
Choonkil Park 충청수학회 2006 충청수학회지 Vol.19 No.2
In this paper, the notion of a generalized quasi-normed space is introduced and its completion is investigated. We introduce quasi-2-normed spaces and quasi-(2; p)-normed spaces, and investigate the properties of quasi-2-normed spaces and quasi-(2; p)-normed spaces.
HYERS-ULAM-RASSIAS STABILITY OF ISOMORPHISMS IN C<SUP>*</SUP>-ALGEBRAS
Park, Choonkil 충청수학회 2006 충청수학회지 Vol.19 No.2
This paper is a survey on the Hyers-Ulam-Rassias stability of the Jensen functional equation in $C^*$-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Its content is divided into the following sections: 1. Introduction and preliminaries. 2. Approximate isomorphisms in $C^*$-algebras. 3. Approximate isomorphisms in Lie $C^*$-algebras. 4. Approximate isomorphisms in $JC^*$-algebras. 5. Stability of derivations on a $C^*$-algebra. 6. Stability of derivations on a Lie $C^*$-algebra. 7. Stability of derivations on a $JC^*$-algebra.
APPROXIMATE RING HOMOMORPHISMS OVER p-ADIC FIELDS
Park, Choonkil,Jun, Kil-Woung,Lu, Gang 충청수학회 2006 충청수학회지 Vol.19 No.3
In this paper, we prove the generalized Hyers-Ulam stability of ring homomorphisms over the p-adic field $\mathbb{Q}_p$ associated with the Cauchy functional equation f(x+y) = f(x)+f(y) and the Cauchy-Jensen functional equation $2f(\frac{x+y}{2}+z)=f(x)+f(y)+2f(z)$.
QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES: A FIXED POINT APPROACH
PARK, CHOONKIL,SEO, JEONG PIL The Kangwon-Kyungki Mathematical Society 2015 한국수학논문집 Vol.23 No.2
In this paper, we solve the following quadratic $\rho$-functional inequalities ${\parallel}f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-x(y)-f(z){\parallel}\;(0.1)\\{\leq}{\parallel}{\rho}(f(x+y+z+)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}<\frac{1}{8}$, and ${\parallel}f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z){\parallel}\;(0.2)\\{\leq}{\parallel}{\rho}(f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}$ < 4. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.
GENERALIZED (θ, ø)-DERIVATIONS ON BANACH ALGEBRAS
Park, Choonkil,Shin, Dong Yun The Kangwon-Kyungki Mathematical Society 2014 한국수학논문집 Vol.22 No.1
We introduce the concept of generalized (${\theta}$, ${\phi}$)-derivations on Banach algebras, and prove the Cauchy-Rassias stability of generalized (${\theta}$, ${\phi}$)-derivations on Banach algebras.