Sentence examples for by using the saddle from inspiring English sources

Next, we obtain the second main result by using the saddle point theorem.

Xu and Tang [23] proved the existence of periodic solutions for problem (1) by using the saddle point theorem.

Next, we will prove Theorem 1.2 by using Ekeland's variational principle and the mountain pass theorem and Theorem 1.3 by using the saddle point theorem.

Recently, the authors in [19] considered the existence of periodic and subharmonic solutions for the nonlinear difference equations with p-Laplace operator by using the saddle point theorem.

In [8], under a generalized Landesman-Lazer type condition, Ke and Tang obtained the existence of at least two solutions for a class of asymptotically linear noncooperative elliptic systems by using the saddle point theorem.

Especially, in [4], under the asymptotic noncrossing conditions and the nonquadraticity conditions, Costa proved that there was at least a weak solution for the semilinear elliptic systems by using the saddle point theorem and they proved there was at least a nontrivial solution for the semilinear elliptic systems under the crossing condition by using the linking theorem.

Secondly, we prove the well-posedness of these weak problems by using the classical saddle point theory.

By using the generalized saddle point theorem due to Liu [5], we can prove the existence of multiple periodic solutions for the second order Josephson-type differential systems for a new and large range of the nonlinear term.

This results in using the saddle points for merging on a 'level-by-level' basis.

A variational approach is applied using the saddle point theorem.

By using the variational minimizing method and the saddle point theorem, the periodic solutions for non-autonomous second-order discrete Hamiltonian systems are considered.