Sentence examples for use the continuation theorem from inspiring English sources

To obtain the existence of positive ω -periodic solutions of system (2), we will use the continuation theorem.

Since ImQ is isomorphic to KerL, there exists an isomorphism J : Im Q → Ker L. In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [8].

In fact, to use the continuation theorem, it suffices to prove that there exists a positive constant 0 < ε 0 ≪ 1 such that, for any possible solution ( x 1 ( t ), x 2 ( t ) ) of (2.2), the following condition holds: | x 2 ( t ) | < 1 − ε 0. (2.4).

In order to use the continuation theorem to study the positive T-periodic solutions for equation (1.5), we consider the following system: textstylebegin{cases}u^{(m)}(t)= [A^{-1}varphi_{q} v)](t), v^{(m)}(t)=-f(u(t))u'(t)-g(t,u(t-tA^{-1}varphi_{q} ves} (2.1) where (q>1) is a constant with (1/p+1/q=1).

(1.2) Using the continuation theorem of coincidence degree theory, they obtained the existence of periodic solutions for (1.2).

By using the continuation theorem and some analysis techniques, some new results on the existence of periodic solutions are obtained.

By using the continuation theorem due to Ge, we obtain some existence results for such boundary value problems.

By using the continuation theorem of coincidence degree theory, we obtain a new result on the existence of solutions for the considered problem.

By using the continuation theorem of Manásevich and Mawhin, a new result on the existence of positive periodic solution is obtained.

Gao et al. [8, 9] considered the existence of periodic solutions for two kinds of Rayleigh type p-Laplacian equations by using the continuation theorem.

Under certain nonlinear growth conditions of the nonlinearity, we obtain a new result on the existence of solutions by using the continuation theorem of coincidence degree theory.