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SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV<sup>2</sup><sub>α</sub>(I)
Aziz, Wadie,Guerrero, Jose Atilio,Merentes, Nelson Korean Mathematical Society 2015 대한수학회보 Vol.52 No.2
The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.
SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BVα2 (I)
Wadie Aziz,Jos´e Atilio Guerrero,Nelson Merentes 대한수학회 2015 대한수학회보 Vol.52 No.2
The space BVα2 (I) of all the real functions defined on interval I = [a, b] ⊂ R, which are of bounded second α-variation (in the sense De la Vall´e Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function h : I × R → R, and prove, in particular, that if H maps BVα2 (I) into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.
Aziz Muhammad Safwan,Islam Shumaila,Rosli Muhammad Izz,Basar Norazah,Krishnan Ganesan,Duralim Maisarah,Harun Sulaiman Wadi 한국물리학회 2022 Current Applied Physics Vol.39 No.-
Nonlinear optical properties of Coumarin-based compounds i.e., 3-acetyl-7-iethylaminocoumarin (C3) and 7-Ethylamino-4-trifluoromethylcoumarin (C500) is reported by Z-scan technique (Nd: YAG Continuous-wave 532 nm). The conventional C500 compound exhibited NLR properties in the order of n2 = 10–9, whereas the synthesized C3 compound manifested both NLR and NLA properties around n2 = 10–9 and β = 10–4, respectively. The results are obtained from different laser power PC500 = 0.10–0.18 W, while PC3 = 0.20–0.28 W. Both compounds provided excellent NLO attributes at consistent and stable outputs of open and closed-aperture signals, resulted in eminence of NLR and NLA values. The C3 compound is manifested reverse saturable absorption, optical limiting properties, and threshold at 0.3814 W. The exclusivity of NLR in C500 makes it suitable to develop as an optical switcher while synthesized C3 compound is more suitable as an optical limiter for safety.