Sentence examples for the use of variational methods from inspiring English sources

This dependence on the gradient prevents the use of variational methods directly on equation (1.1).

These requirements need to develop a nonstandard approach, in particular prevent the use of variational methods.

Essential features of the present work are the dependence on the gradient ∇u, which prevents the use of variational methods, and the unboundedness of the domain, which produces lack of compactness.

Recently, there is a large literature on the use of variational methods to the existence of homoclinic or heteroclinic orbits of Hamiltonian systems; see [2 7] and the references therein.

Sobolev spaces of functions on time scales, which were first introduced in [15], opened a very fruitful new approach in the study of dynamic equations on time scales: the use of variational methods in the context of boundary value problems on time scales (see [16, 17]) or in second-order Hamiltonian systems [18].

By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields.

In the 1989 paper of Rabinowitz [1], we find the first substantial use of variational methods to study heteroclinic orbits for Hamiltonian systems.

As an application, we will discuss the existence of nontrivial solutions to the p-Laplace equations with resonance on the Fučik spectrum by making use of variational methods and Morse theory.

In this connection, several existence and multiplicity results for solutions to second-order ordinary differential nonlinear equations, with the nonlinearity dependent on the derivative and Dirichlet conditions at the ends, have been investigated making use of variational methods.

The existence of new periodic solutions with a prescribed energy is established by the use of constrained variational methods.

This paper presents the use of a variational iteration method for systems of nonlinear difference equations.