Fixed points and functional equations connected with derivations on Banach algebras

Let 1≦m≦4 be a fixed integer and let f:X→Y be a mapping with X, Y two vector spaces. The functional equation (1.1) is said to be additive if m=1, quadratic if m=2, cubic if m=3 and quartic if m=4, respectively. For convenience, a solution of (1.1) will be called an m-mapping. Let $\mathcal{A}$ , $\mathcal{B}$ be two algebras. An m-mapping $f\colon \mathcal{A} \to \mathcal{B}$ will be called m-derivation if the equality f(xy)=x ^m f(y)+f(x)y ^m is fulfilled for all $x,y\in \mathcal{A}$ . In this paper, we use a fixed point method to prove the stability and hyperstability of m-derivations on Banach algebras.     

Adsorption of diazinon and fenitothion on MCM-41 and MCM-48 mesoporous silicas from non-polar solvent

The adsorption of two common organophosphorus pesticides, diethoxy-[(2-isopropyl-6-methyl-4-pyrimidinyl)oxy]-thioxophosphorane (diazinon) and dimethoxy-(3-methyl-4-nitrophenoxy)-thioxophosphorane (fenitrothion), by MCM-41 and MCM-48 mesoporous silicas at room temperature was investigated. UVvis and IR spectroscopy, small-angle X-ray diffraction, and the specific surface area analysis (S _BET) were used to study the adsorption behavior of diazinon and fenitrothion. The results show that the MCM-41 and MCM-48 mesoporous silicas adsorb diazinon more efficiently than fenitrothion. The extraction of adsorbed materials from the adsorbents with polar solvents and subsequent analysis by ³¹P NMR showed that the adsorption of diazinon and fenitrothion on mesoporous silicas is destructive and non-destructive, respectively. Nitrogen adsorption measurements showed that the specific surface area of both silicas decreases after the adsorption of pesticides, and the larger effect is observed for diazinon. The article is published in the original.