Sentence examples for be proved using standard from inspiring English sources
The following result can be proved using standard arguments as in [26, 28].
Lemmas 1 and 2 can be proved using standard arguments as the Lemmas 1 and 2 in [13], respectively.
The well-posedness of the discrete and continuous problems can be proved using standard argument of Lax-Milgram in appropriate functional spaces; refer, for instance, to [34] for a special case of fractional Maxwell model.
Corollary 6 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 2′, i ∈ N ˜ 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 4-6 can be proved using standard arguments as Theorems 4-6.
Corollary 3 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 1′, i ∈ N 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 1-3 can be proved using standard arguments as Theorems 1-3. Remark 2 Theorems 1-3 and Corollaries 1-3 are obtained based on Mexican-hat-type feedback function (8) and piecewise constant feedback function (9).
Surface-enhanced Raman scattering property is proved using 4-mercaptobenzoic acid (4-MBA) as standard analyte and the detection concentration level is down to 10−10 M.
Although the overall system is infinite dimensional, convergence of the observer is proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces.
Since local existence results can be proved by using standard arguments, say, Galerkin method, we only deal with the a priori estimates.
The next lemma is proved by using standard Schwartz's arguments in [38].
