Sentence examples for use the degree theory from inspiring English sources

We use the degree theory and a priori estimates to obtain the existence of solutions for the second-order Dirichlet boundary value problems.

To solve the above problem (E), we thus use the degree theory for densely defined ( S + ) L -perturbations of maximal monotone operators introduced by Kartsatos and Quarcoo in [18].

Let U r be the ball with radius r in L 2 0, T; V*) and z ∈ U r. To prove (3.11), we will also use the degree theory for the equation z = λ ( G − A S ) w + w, 0 ≤ λ ≤ 1 (3.12). in open ball U d where the constant d satisfies ( r + ω 3 + N 2 ∣ x 0 ∣ + M g T ) ( 1 − N 2 ) − 1 < d (3.13).

In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions of C as well as Browder's degree.

and by using the degree theory, the existence of a positive solution of (1.3) is established.

Using the degree theory, we show that the Borsuk theorem is still valid for our class.

Using the degree theory, we show that the Borsuk theorem holds true for this class.

Now we can show the solvability of the given boundary value problem involving the p-Laplacian by using the degree theory.

By using the degree theory, we will show that T 1 has at least three fixed points which coincide with the solutions of (22 - 2).

By using the degree theory, we will show that (T_{2}) has at least three fixed points which are solutions of (3.18 - 1.2).

The authors [4] also proved that when c < λ 1, Λ 1 < b < Λ 2 and s < 0, (1.2) has at least three nontrivial solutions by using the degree theory.