Ick-Soon Chang

Stability of a functional equation deriving from cubic and quadratic functions

Abstract In this paper we establish the general solution of the functional equation 6 f ( x + y )−6 f ( x − y )+4 f (3 y )=3 f ( x +2 y )−3 f ( x −2 y )+9 f (2 y ) and investigate the Hyers–Ulam–Rassias stability of this equation.

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.

Approximation of Generalized Left Derivations

We need to take account of the superstability for generalized left derivations (resp., generalized derivations) associated with a Jensen-type functional equation, and we also deal with problems for the Jacobson radical ranges of left derivations (resp., derivations).

Higher Ring Derivation and Intuitionistic Fuzzy Stability

We take account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.

DERIVATIONS ON PRIME AND SEMI-PRIME RINGS

In this paper we will show that if there exist deriva- tions D, G on a n!-torsion free semi-prime ring R such that the mapping D 2 + G is n-commuting on R, then D and G are both commuting on R. And we shall give the algebraic conditions on a ring that a Jordan derivation is zero. Throughout this paper R will be represent an associative ring with center Z(R). The commutator xy i yx (resp. the Jordan product xy + yx) will be denoted by (x;y) (resp. hx;yi). We make extensive use of the basic identities (xy;z) = (x;z)y + x(y;z), (x;yz) = (x;y)z + y(x;z). Let rad(A) denote the (Jacobson) radical of an algebra A. Recall that R is prime if aRb = (0) implies that either a = 0 or b = 0, and is semi-prime if aRa = (0) implies a = 0. An additive mapping D from R to R is called a derivation if D(xy) = D(x)y + xD(y) holds for all x;y 2 R. A derivation D is inner if there exists a 2 R such that D(x) = (a;x) holds for all x 2 R. And also, an additive mapping D from R to R is called a Jordan derivation if D(x 2 ) = D(x)x + xD(x) holds for all x 2 R. An additive mapping F from R to R is said to be a commuting (resp. centralizing) if (F(x);x) = 0 (resp. (F(x);x) 2 Z(R)) holds for all x 2 R. More generally, for a positive integer n, we define a mapping F to be n-commuting if (F(x);x n ) = 0 for all x 2 R. The underlying idea of our research is Posners second theorem (7, Theorem 2) which is the beginning of the study concerning centralizing and commuting mappings, which states that the existence of a nonzero

Functional Inequalities Associated with Additive Mappings

The functional inequality is investigated, where is a group divisible by and are mappings, and is a Banach space. The main result of the paper states that the assumptions above together with (1) and (2) , or , imply that is additive. In addition, some stability theorems are proved.

ON STABILITY OF THE FUNCTIONAL EQUATIONS HAVING RELATION WITH A MULTIPLICATIVE DERIVATION

In this paper we study the Hyers-Ulam-Rassias stability of the functional equations related to a multiplicative derivation.

ON (α,β)-SKEW-COMMUTING AND (α,β)-SKEW-CENTRALIZING MAPS IN RINGS WITH LEFT IDENTITY

Let R be a ring with left identity. Let G : be a symmetric biadditive mapping and g the trace of G. Let be an endomorphism and an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is ()-skew-commuting on R, then we have G = 0. (ii) If g is ()-skew-centralizing on R, then g is ()-commuting on R. (iii) Let . Let R be (n+1)!-torsion-free. If g is n-()-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-()-commuting on R, then g is ()-commuting on R.

On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equation in an intuitionistic fuzzy normed algebra with unit.

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 .