Shizuo Miyajima

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.

Generators of positive

Publisher Summary This chapter focuses on the generators of positive C o -semigroups. It presents an exposition of some earlier results in a slightly improved form. The problem to be treated is concerned with the generators of positive C o -semigroups on Banach lattices. Abstract Katos inequality for the generators of positive semigroups and Katos inequality for formal differential operators are presented in the chapter. The generators of such semigroups satisfy weak abstract Katos inequality, and if a differential operator generates a positive C o- semigroup on L p (Ω) then it is an (possibly degenerate) elliptic second order differential operator. An improvement of a result of Baoswan Wong–Dzung is also presented in the chapter.

C_0

We consider a natural correspondence between a family of inequalities and a closed convex set. As an application, we give new types of power mean inequalities and the Hölder-type inequalities.