Search

We will consider the Banach space valued differential equation y′(t) = Ay(t) , where A is an n× n complex matrix. We give a necessary and sufficient condition in order that the equation have the Hyers-Ulam stability. As a Corollary, we prove that the Banach space valued linear differential equation with constant coefficients y(n)(t) + an−1y(n−1)(t) + · · ·+ a1y′(t) + a0y(t) = 0 has the Hyers-Ulam stability if and only if Reλ = 0 for all the solutions λ of the equation zn +an−1zn−1 + · · ·+a1z+a0 = 0 . Mathematics subject classification (2010): Primary 46B99; Secondary 39B72.

We introduce a metric in the set \({{\mathcal S}(H)}\) of all semiclosed operators in a Hilbert space H, and its topological structures are studied.

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.

Suppose A is a Banach algebra and suppose f : A → A is an approximate ring derivation in the sense of Hyers–Ulam–Rassias. This stability phenomenon was introduced for the first time in the subject of functional equations by Th.M. Rassias [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300]. If A has an approximate identity, or if A is semisimple and commutative, then we prove that f is an exact ring derivation.

We give some necessary and su-cient conditions in order that a closed operator in a Hilbert space into another have the Hyers-Ulam stability. Moreover, we prove the existence of the stability constant for a closed operator. We also determine the stability constant in terms of the lower bound.

SupposeAis a Banach algebra without order. We show that an approximate multiplierT:A→Ais an exact multiplier. We also consider an approximate multiplierTon a Banach algebra which need not be without order. If, in addition,Tis approximately additive, then we prove the Hyers-Ulam-Rassias stability ofT.

Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called $d$-preserving maps between subsets of groups and apply them in several directions. We consider $d$-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group $\mathbb T$. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.

In this paper, we present a recent development of the Hyers–Ulam stability of the Butler–Rassias functional equation \(f(x + y) - f(x)f(y) = d\sin x\sin y\). We also show our most recent results on a Butler–Rassias type functional equation with complex variables.

In this paper, we show a recent development of the Hyers–Ulam stability and Ger type stability of the first order linear differential equation \(y + py + q = 0\)for holomorphic functions on convex open sets of the complex number field and continuously differentiable functions on open intervals of the real number field.

Let p: ℝ → ℂ be a continuous function. We give a sufficient condition in order that the integral equation \(f(t) = f(0) +{ \int \nolimits \nolimits }_{0}^{\,t}p(s)f(s)\,\mathrm{d}s\) have the Hyers–Ulam stability. We also prove that if p has no zeros, then the sufficient condition is a necessary condition.

In this paper, we will consider Hyers-Ulam-Rassias stability of multipliers and ring derivations between Banach algebras. As a corollary, we will prove superstability of ring derivations and multipliers. That is, approx- imate multipliers and approximate ring derivations are exact multipliers and ring derivations.

Background/Aims: The basic principle of treatment of congestive heart failure is achieving adequate control of preload and afterload through enhancement of cardiac contractility. In severe cases, however, we have usually applied continuous hemodiafiltration (CHDF) as a type of mechanical support. In this study, we investigated hemodynamic changes caused by CHDF in patients with congestive heart failure. Methods: We treated seven patients with congestive heart failure complicated by multiple organ failure by CHDF over 72 h, during which we measured hemodynamic parameters to determine their changes. Results: Implementation of CHDF resulted in a significant decrease in pulmonary artery occluded pressure and significant increases in cardiac index and left ventricular stroke work index. In addition, 72-hour cumulative water balance was found to be –1,791 ± 2,119 ml, and systemic vascular resistance index decreased significantly. Conclusion: Hemodynamics of patients were improved with CHDF through strict control of preload and consequently tissue oxygen metabolism was improved.

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 .