Hark-Mahn Kim

The generalized Hyers–Ulam–Rassias stability of a cubic functional equation☆

Abstract In this paper, we obtain the general solution and the generalized Hyers–Ulam stability for a cubic functional equation f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+12f(x) .

EXTENDED HYERS-ULAM STABILITY FOR A CAUCHY-JENSEN MAPPINGS

In 1940, Ulam proposed the famous Ulam stability problem. In 1941, Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In 2003–2006, the last author of this paper investigated the Hyers–Ulam stability of additive and Jensen type mappings. In this paper, we improve results obtained in 2003 and 2005 for Jensen type mappings and establish new theorems about the Ulam stability of additive and alternative additive mappings. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.

Approximate homomorphisms and derivations between C∗-ternary algebras

In 1940, Ulam proposed the famous Ulam stability problem. In this paper we introduce a general Cauchy–Jensen functional equation and prove the generalized Ulam stability of C∗-ternary homomorphisms and C∗-ternary derivations in C∗-ternary algebras for the general Cauchy–Jensen equation.

ON THE HYERS-ULAM STABILITY OF A GENERALIZED QUADRATIC AND ADDITIVE FUNCTIONAL EQUATION

In this paper, we obtain the general solution of a gen-eralized quadratic and additive type functional equation f(x + ay) + af(x - y) = f(x - ay) + af(x + y) for any integer a with a -1. 0, 1 in the class of functions between real vector spaces and investigate the generalized Hyers- Ulam stability problem for the equation.

Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings

We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

Stability of Functional Inequalities with Cauchy-Jensen Additive Mappings

We investigate the generalized Hyers-Ulam stability of the functional inequalities associated with Cauchy-Jensen additive mappings. As a result, we obtain that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists a Cauchy-Jensen additive mapping near the mapping.

On Functional Inequalities Originating from Module Jordan Left Derivations

We first examine the generalized Hyers-Ulam stability of functional inequality associated with module Jordan left derivation (resp., module Jordan derivation). Secondly, we study the functional inequality with linear Jordan left derivation (resp., linear Jordan derivation) mapping into the Jacobson radical.

Fuzzy approximation of Euler-Lagrange quadratic mappings

AbstractIn this article, we consider the Hyers-Ulam stability of the Euler-Lagrange quadratic functional equation f(kx+ly)+f(kx−ly)=kl[f(x+y)+f(x−y)]+2(k−l)[kf(x)−lf(y)] in fuzzy Banach spaces, where k, l are nonzero rational numbers with k≠l.

A result concerning the stability of some difference equations and its applications

In this paper, we investigate the Hyers-Ulam stability problem for the difference equation f(x +p, y +q)- φ(x, y)f(x, y)- ψ(x, y)= 0.

Approximate linear derivations and functional inequalities with applications

Abstract In this paper, we prove that any approximate linear derivation on a semisimple Banach algebra is continuous. We deal with the functional inequalities associated with additive mappings and some stability theorems are proved. Based on these facts, we obtain some results for the functional inequalities corresponding to the additive mappings and the equation f ( x y ) = x f ( y ) + f ( x ) y .