Sentence examples for by using critical point from inspiring English sources

In this section, we will prove our main results by using critical point theorem.

In this paper, we consider (1.1) by using critical point theory and variational methods.

Under some suitable assumptions, the existence of solutions is proved by using critical point theory.

In [16], the authors obtained the multiplicity results for periodic solutions to (1.1) by using critical point theory.

By using critical point theory, we obtain some existence theorems of solutions for the nonlinear impulsive problem.

We also refer to [6 9] for more results on the difference BVP by using critical point theory.

By using the critical point theory, the existence and multiple results are obtained.

There are also authors who studied the Duffing-type equations by using the critical point theory (see [12, 13]).

In section 3, we prove Theorem 1.1 by using the critical point theory and variation of linking method.

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems.

Recently, there are some new results on periodic solutions of nonlinear difference equations by using the critical point theory in the literature; see [1 3].