Sentence examples for apply the continuation from inspiring English sources

In this section, we shall apply the continuation theorem of Mawhin's coincidence degree theory to establish the global existence of at least one positive periodic solution.

To apply the continuation theorem, we investigate the operator equation begin{aligned} &x^{Delta}(t)=lambda biggl[a(t -b(t -bp bigl(x(t) bigr)-frac {c(t)exp (y(t))}{alpha(t)+bigl(t) ex t(x(t))+m(t)exp (y(t))} bigr -frac^{Delta}(t)=lambigr -frac-d(t)+frac{c(t)exp (x(t))}{alpha(t)+beta(t) exp (x(t))+m(t)exp (y(t))}{alpha].

By applying the continuation theorems, Gaines and Mawhin proved that the Rayleigh equation can support periodic solutions.

By applying the continuation theorem and some analytic techniques, we shall establish several new criteria for the existence of positive periodic solutions for the considered problem.

Applying the continuation theorem, we prove the main results on the solvability of the given semilinear wave equation, with the aid of spectral theory for densely defined closed linear operators.

It is adopted to apply the upward continuation technique as a tool of regional/residual separation in order to investigate the orientation of subsurface structures.

We now apply the Leray-Schauder continuation theorem to the operator equation u = K N u.

In [15], Lu, Li and Su applied the homotopy continuation method to give a constructive proof of Mawhin's continuation theorem in [20], which can be used to obtain the periodic solutions by track along a (C^{1}) trajectory.

In [15], Lu, Li and Su applied the homotopy continuation method to give a constructive proof of Mawhin's continuation theorem in [20], and thus established a global method for finding periodic solutions of ordinary differential equations (systems).

Next, applying the Manśevich Mawhin continuation theorem, we prove the following theorems.

The sufficient conditions for the existence of solutions of coupled fractional differential equations are obtained by applying the Mawhin continuation theorem.