Sentence examples for by variational methods in from inspiring English sources

For, and for,, such transformed problem was solved by variational methods in [7, 8], respectively.

When the elastic beam equation does not contain parameter λ, the existence of multiple positive solutions and unique positive solution was presented in [17] by variational methods and in [18] by a fixed point theorem, respectively; monotone positive solutions were obtained by using the monotone iteration method in [19].

For the case of bounded domain, some interesting studies by variational methods can be found in [6 14] for equation (1.4) with several growth conditions on f.

Critical point theory has been widely used to study the existence of periodic solutions and solutions for nonlinear difference boundary value problems since the first result was established by using variational methods in 2003 (see [2]).

In [23], in the asymptotically periodic case, a nontrivial solution is obtained by variational methods.

In [21], the existence criteria of radial solutions are established under different conditions by variational methods.

Many excellent results were obtained by variational methods [1 11].

In [22], the authors also studied the existence of infinitely many nontrivial energy solutions by variational methods.

In [17] and [18], the authors also dealt with some one-dimensional impulsive problems with damped term by variational methods.

A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ on Rd which can be expressed as the moment measures of suitable convex functions u, i.e. are of the form (∇u)#e−u for u Rd→R∪ and finds the corresponding u by a variational method in the class of convex functions.

It seems difficult to consider the case by variational method.