Sentence examples for an initial function from inspiring English sources

Choose an initial function (u_{0}(x)).

where φ i ( t ) are coordinates of an initial function φ ( t ) used in (2).

For this purpose, sequences (5.2) and (5.4) should be computed simultaneously, with (5.4) serving as an initial function for (5.2) at the next step, and vice versa.

uniformly stable if for any there exists a such that if and is an initial function on with then for all ; uniformly asymptotically stable if it is uniformly stable, and if there exists a such that, for any there exists an with the property that, if and is an initial function on with then for all.

(i) uniformly stable if for any there exists a such that if and is an initial function on with then for all ;   (ii) uniformly asymptotically stable if it is uniformly stable, and if there exists a such that, for any there exists an with the property that, if and is an initial function on with then for all.  .

Here, we claim that δ 1 ( ε, t 0 ) ≤ ε 1. Suppose this is not the case; then there exists an initial function φ ∈ C such that ∥ φ ∥ < δ 1 ( ε, t 0 ) and | D | = ε 1.

It is easy to see that, for any given and a given initial function, there is a unique solution of (1.1), denoted by such that (1.2).

For the behavior of a solution, generated by a nonmonotone initial function there are two possibilities: either is eventually monotone and, consequently, asymptotically convergent, or is eventually nonmonotone.

Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function.

So it is important to find a good initial function.

We will assume that a monotone initial function is given.