Sentence examples for upper function from inspiring English sources

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).

Proof According to Lemma 3.4, there exist k 0 ∈ N and an upper function β to the problems (3.2), (1.3) for k ≥ k 0 satisfying (3.8).

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

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

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

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.