Sentence examples for using the degree theory from inspiring English sources

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 holds true for this class.

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

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).

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 assuming the existence of two pairs of unbounded upper and lower solutions, the existence of at least three solutions is obtained using the degree theories.

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].

In 2003, Facchinei and Pang [17, 18] used the degree theory to obtain a necessary and sufficient condition of variational inequality problems for continuous pseudomonotone mappings in a finite-dimensional space.

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).