Sentence examples for apply the existence from inspiring English sources

In addition, we apply the existence theorem to a specially doubly degenerate case.

For instance, in the following example, we apply the existence result to an operator Φ having exponential growth.

We will develop and apply the existence theory for first-order differential equations (see [16, 17]) to study certain second-order differential equations associated with nonlinear maximal monotone operators in Hilbert spaces.

Proof Rewrite (1) as an equivalent system of the form y ′ = − G − 1 ( u ), u ′ = − p ( t ) F ( y ), where G − 1 is the inverse of G. Then we apply the existence result [[9], Theorem 1] to obtain DS ≠ ∅.

Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems.

Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.

end{aligned} Applying the existence and uniqueness theorem [25, 26] to (5.7), we obtain that u' x,t -w' x,t -w' xad text{a.e. on }Q_{T},thence the entire sequence begin{aligned} u_{k}(cdot,t)-w_{k}(cdot,t)rightarrow0 end{aligned} (5.8) uniformly on any compact subset of (mathbb{R}^{N}) as (krightarrowinfty ).

In this subsection, we consider the extension of our results to degenerate elliptic equations and in particular apply the classical existence results, Theorems 4.1 and 4.2, to yield the existence of (C^{1,1}) admissible solutions for the oblique boundary value problems.

We apply the following existence principle which follows from [11 13] to prove the solvability of problem (1.12), (1.2).

Thus, we can take ( ρ, u, w, H ) ( x, T ∗ ) as the initial data and apply the local existence theorem to extend the local strong solutions beyond T ∗.

Now, we apply the local existence theorem taking the initial time and obtain that these solutions coincide on some interval, which give us a contradiction with the fact that is the maximal time of coincidence.