Sentence examples for using the comparison theorem from inspiring English sources

Using the comparison theorem, we can derive the first inequality.

Using the comparison theorem, similar to the proof of the first assertion, we directly obtain (3.14).

The persistence and extinction of the system are discussed by using the comparison theorem of differential equations.

In Section 3, by using the comparison theorem of the impulsive differential equations, we give the permanence of system (1.2).

Using the comparison theorem, liminf_{trightarrow+infty} x t geqfrac{k alpha _{1}-c_{1}varepsilon)}{alpha_{1}}.

From system (3.4) and Lemma 2.4, we easily obtain (Z(t)leq0) using the comparison theorem of impulsive differential equations.

Similar to Theorems 3 and 6 of paper [7] and Theorems 3 and 5 of paper [8], existence results of problems (1.1 - 1.4 1.1 - 1.4btained by using the canparison results Theorems 1.1, 1.2, 2.1, and 2.2 in this paper.

For example, using the Sturmian comparison theorem (see [14], Theorem 1.2.4), we can proceed for perturbed Euler type half-linear equations as follows.

But for elliptic gradient estimates for f-Laplacian under the ∞-Bakry Émery Ricci curvature, in order to using the weighted comparison theorem, the assumption (|nabla f|leqtheta) is forced commonly.

There are many papers on the persistence in mean and extinction for stochastic models, and a common means to do this is by using the stochastic comparison theorem and the sample Lyapunov exponent; see [7, 20].

end{aligned} By the assumption (H1) and (3.3), considering the auxiliary equation (3.4) and using the standard comparison theorem and (a) of Lemma 2.2, we can obtain limsup_{trightarrowinfty}mathrm{E}bigl[x^{p}(t bigr]le limsup_{trightarrowinfty}y^{ast}(t):=H(p).