Sentence examples for upper solutions for from inspiring English sources

We define lower and upper solutions for problem (3.1).

In this paper, we extended the concept of lower and upper solutions for (1.1).

We begin by introducing the concepts of lower and upper solutions for problem (2.1).

It is elementary matter to check that and,, are lower and upper solutions for the problem.

In Section 2 we develop a method to construct lower and upper solutions for (1.1).

Suppose that, hold, are strict lower solutions, are strict upper solutions,,, for some, and?? satisfies Nagumo condition with respect to.

By this, an upper solution for (1.1) is obtained.

We now construct an upper solution for this problem.

An upper solution for (1.1) satisfies the reversed inequalities.

A function is an upper solution for (3.2) if and (3.4).

So, α and β are, respectively, a lower and an upper solution for problem (20).