Sentence examples for by constructing upper and from inspiring English sources

By constructing upper and lower solutions, the existence of positive solutions for the problem is established.

By constructing upper and lower solutions of (2.2), the authors obtained the asymptotic behavior of the solution to (2.1).

In this section, inspired by [14, 16, 26], we first study the existence of a coexistence solution to problem (1.3) by constructing upper and lower solutions.

A necessary and sufficient condition for the existence of as well as positive solutions is given by constructing upper and lower solutions and with the maximum principle.

Moreover, the dynamic properties of a non-autonomous logistic model with Dirichlet boundary conditions are obtained by constructing upper and lower solutions.

We will establish the following theorem to deal with this case by constructing upper and lower solutions to solve this problem.

Some sufficient conditions for constructing upper and lower solutions are given in each section for applications.

By constructing upper-lower solutions, we establish the existence of the traveling wavefronts when the wave speed is large enough.

Moreover, by constructing upper-lower solutions, we find that there exist traveling wavefronts which connect the zero equilibrium and positive equilibrium when the wave speed is large enough and the prey intrinsic growth rate and the death rate of the predator are relatively big.

By constructing available upper and lower solutions and combining the Schauder's fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence of as well as positive solutions for a class of singular boundary value problems on time scales.

The existence of traveling wave solutions is proved by constructing generalized upper and lower solutions.