Sentence examples for using variational methods and from inspiring English sources
By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.
By using variational methods and elementary analysis, the conditions for the existence of positive and negative solutions to a nonlinear system are given.
By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation.
By using variational methods and critical point theory, we obtain that such a system possesses at least one, two periodic solutions generated by impulses under different conditions, respectively.
We study the existence of distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.
By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity.
They proved that IBVP (1.4) has at least one solution and infinitely many solutions by using variational method and critical point theorem.
Recently, in [16], using variational method and critical point theory, Nieto and O'Regan studied the existence of solutions of the following equation: (1.3).
To improve the system performance, we establish some conditions the optimal impulse instants should satisfy using variational method, and design gradient-based algorithms to search for optimal impulse instants.
In 2011, Zhang and Wang [15] used variational methods and Morse theory to study the multiplicity of periodic solutions for (P) with double resonance between two consecutive eigenvalues at infinity.
In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects.
