Sentence examples for by using the proposition from inspiring English sources

Proposition 2.2 can be easily shown by using the Proposition 2.1 (see [6]).

By using the Proposition 2 we obtain the following result: (overline{x}_{2}) is a saddle point if (q-p<1), (overline{x}_{2}) is a repeller if (q-p>1).

We use the propositions 1 to 4 in Section 3 to reduce the searching range effectively, since all the QPBTOCAS and the APBTOCAS seem only one if they can get from one sequence by using the propositions 1 to 4. Therefore, by means of an exhaustive computer search for the possible class of PBTOCAS, we get some QPBTOCAS within length 25 in Table 1 and APBTOCAS within length 26 listed in Table 2.

The Hermit-Hadamard type inequalities have been obtained for ICR functions by using the following proposition in [7].

The Hermite-Hadamard type inequalities are shown for IPH functions by using the following proposition which is very important for IPH functions.

However, it is not necessary to solve each (mathrm{P2} upsilon)) associated with each (upsilonin B^{theta}) for searching the solution of problem (P), that is, by using the following proposition we can obtain an improvement of the algorithm.

for all x ∈ X, where q is a constant such that q ∈ [ 0, 1 ). Then T has a unique fixed point. Samet et al. [37] proved that Theorem 2.4-Theorem 2.7 are the consequences of Theorem 2.9 by using the following proposition.

By using the qualitative result (see the proposition), the existence of the relaxation oscillation of this kind of predator-prey system with distributed delay can be determined quickly.

By using the same techniques as Proposition 2.2, we get mathbb{E} sup_{0 leqthetaleq tau} bigl(bigl|(CV) (theta) - (C G_{l}) (theta bigr|^{2}bigr) leq (2tau+ 8) mathbb{E} int_{0}^{tau} h(theta) gbigl(bigl|Vbigl(f(theta) bigr) - G_{l}bigl(f(theta )bigr bigr|^{2}bigr),dtheta.

By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.

where and Moreover, by using the same argument as in Proposition 3.5, we have for every and Thus, (3.35).