Sentence examples for maximum principles obtained from inspiring English sources

Existence and uniqueness results were established by the new maximum principles obtained by estimating the fractional derivative of a function at its extreme points.

But the maximum principle obtained by these method contains unknown parameters.

In this paper, by constructing completely different auxiliary functions from those in [26], [27] and technically using maximum principles, we obtain the conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global.

In this paper, by constructing completely different auxiliary functions with those in [21 23] and technically using maximum principles, we obtain some existence theorems of a global solution, an upper estimate of the global solution, the existence theorems of a blow-up solution, an upper bound of 'blow-up time', and an upper estimates of 'blow-up rate'.

The maximum principle is obtained.

Also, Pontryagin's maximum principle is obtained for our problem.

Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle.

The second-order variational equation for the BSDE (3) and the maximum principle are obtained.

They showed that the model (1.4) admits a maximum principle and obtained a necessary and sufficient condition that the solution of the regularized QG equations (1.4) develops a singularity in finite time and proves that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times.

By maximum principle, we obtain that.

By the strong maximum principle, we obtain that on.