In this paper, we prove the generalized Hyers-Ulam-Rassias stability of mappings of commutative semigroups into Banach spaces. In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative norms.

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In this paper we prove the generalized Hyers-Ulam-Rassias stability of extended derivations on unital Banach algebras associated to a generalized Jensen equation.

We say a function f 0 is an approximate solution of (E) if E 1 (f 0 ) and E 2 (f 0 ) are close in some sense. The stability problem is whether or not there is an exact solution of (E) near f 0 .In this paper, the stability of derivations on Hilbert C * -modules is inves-1

In this paper, we introduce a new type of parallelism for bounded linear operators on a Hilbert space H , ·, · based on numerical radius. More precisely, we consider operators T and S which satisfy ω(T + λS) = ω(T ) + ω(S) for some complex unit λ. We show that T ω S if and only if there exists a sequence of unit vectors {xn} in H such that lim n→∞ T xn, xn Sxn, xn = ω(T )ω(S).We then apply it to give some applications. z, y x for all z ∈ H . Also, we shall use [x] to denote the linear space spanned by vector x ∈ H .