Sentence examples for maximal principle for from inspiring English sources

We show that there is a maximal principle for solutions of certain nonlinear equations of the formΔu(x)=F x, u(x)).

Then Balakrishnan proved a maximal principle for the optimal control and which can imply the bang bang property (see [4]), Friedman discussed the time optimal control problem on Banach spaces (see [5]), Fattorini proved that the maximal principle in 1974 for some special Banach spaces.

From the maximum principle for parabolic equations [16], it is not hard to verify that for, where is the maximal existence time of the solution.

We obtain the following criteria according to the maximal principle and they are a sufficient condition for convergence to an accurate solution.

By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- as well as positive solutions for fourth-order singular -Laplacian differential equations with integral boundary conditions.

Lemma 1.3 (maximal principle).

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

end{cases} (2.16) It follows from the maximal principle that (u^_{k}< r).

subject to the state Eq. (1.2) by using Pontryagin's maximal principle.

To prove the main result, we need the following maximal principle.

The grid spacings of difference scheme are obtained by analyzing numerical stability and convergence based on the maximal principle.