Sin-Ei Takahasi

A characterization of Hyers-Ulam stability of first order linear differential operators

Abstract Let X be a complex Banach space and h : R → C a continuous function. Let T h :C 1 ( R ,X)→C( R ,X) be the linear differential operator defined by Thu=u′+hu. We give a necessary and sufficient condition in order that the operator Th has the Hyers–Ulam stability.

HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy

The aim of this paper is to prove the stability in the sense of Hyers-Ulam- Rassias of the Banach space valued differentialequation y`

A refinement of various mean inequalities

Faiziev [3] obtained a refinement of the classical arithmetic mean and geometric mean inequality. Also Alzer [1] obtained a continuous version of Faiziev’s refinement and Pearid [4] gave a simple proof of the above Alzer-Faiziev inequality. Recently Takahasi and Miura [5] obtained a generalization of the Alzer-Faiziev inequality. Our main purpose of this paper is to give a new refinement of the classical arithmetic mean and geometric mean inequality (Theorem 2.1).

Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to.

ESSENTIAL NORMS AND STABILITY CONSTANTS OF WEIGHTED COMPOSITION OPERATORS ON C(X)

For a weighted composition operator on C(X), we determine its essential norm and the constant for its Hyers-Ulam stability, in terms of the set

Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem

\varphi(\{x\;\in\;X\;:\;u(x)\;\geq\;r\})

Stability of multipliers on Banach algebras

(r > 0).

A new interpretation of Jensen's inequality and geometric properties of ϕ-means

A class of commutative Banach algebras which satisfy a Bochner- Schoenberg-Eberlein-type inequality is introduced. Commutative C*-algebras, the disk algebra and the Hardy algebra on the open disk are examples.

GER-TYPE AND HYERS-ULAM STABILITIES FOR THE FIRST-ORDER LINEAR DIFFERENTIAL OPERATORS OF ENTIRE FUNCTIONS

Suppose A is a Banach algebra without order. We show that an approximate multiplier T : A → A is an exact multiplier. We also consider an approximate multiplier T on a Banach algebra which need not be without order. If, in addition, T is approximately additive, then we prove the Hyers-Ulam-Rassias stability of T .

A spectral mapping theorem for some representations of compact abelian groups

We introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function φ. Such a mean is called a φ-mean of f and written by Mφ (f). We first give a new interpretation of Jensens inequality by φ-mean. Next, as an application, we consider some geometric properties of Mφ (f), for example, refinement, strictly monotone increasing (continuous) φ-mean path, convexity, etc.Mathematics Subject Classification (2000): Primary 26E60; Secondary 26B25, 26B05.