Sentence examples for applying the continuation from inspiring English sources
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.
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].
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.
In Sect. 2 we give a prior estimate and obtain the proof of Theorem 1.1 by applying the homotopy continuation method.
It entails applying the upward continuation filter to the Bouguer anomalies at progressive heights and to determine the horizontal gradient of each upward continued distance (Blakely and Simpson 1986; Everaerts and Mansy 2001; Jaffal et al. 2010; Hadhemi et al. 2016).
By applying the Manásevich-Mawhin continuation theorem, the authors proved that equation (1.2) has at least one positive T-periodic solution.
By applying the Manásevich-Mawhin continuation theorem, we establish some sufficient conditions for the existence and uniqueness of positive periodic solutions for the Liénard type ϕ-Laplacian operator equation.
