Sentence examples for using the critical point from inspiring English sources

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.

In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1).

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].

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.

We also use the critical point theory and the variational method.

For the proof of Theorem 1.1, we approach the variational method and use the critical point theory for indefinite functional.

Since we use the critical point theory to investigate problem (1.1), the appropriate fractional Sobolev space is necessary.

Also, in Dancer and Du [9], the authors use the critical point theory and a sub-sup solution method on smooth critical point theory.

We approach the variational method and use the critical point theory which is the Linking Theorem for the strongly indefinite corresponding functional.