Sentence examples for upper functions from inspiring English sources

are, respectively, lower and upper functions of (4.1) and (4.2).

We show this result utilizing the method of lower and upper functions.

Suppose that there exist lower and upper functions and of problems (1.1) and (1.2), respectively.

The following definitions of lower and upper functions are suitable for us.

Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.

In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation (1.1).

Consequently, β is an upper function to (1.1).

Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with k = k 0 satisfying (3.8).

Obviously, in view of (3.4) and the non-negativity of ρ 0, it follows that β is also an upper function to (3.2), (1.3) for k ≥ k 0. □.

Then there exist k 0 ∈ N and an upper function β to the problems (3.2), (1.3) for k ≥ k 0 satisfying (3.8).

Then there exists an upper function β to the problem (3.1), (1.3) satisfying β ( t ) ≥ x 0 for t ∈ [ 0, ω ]. (3.8).