Sentence examples for methods some new from inspiring English sources

Based on variational methods, some new results are obtained in regard to the existence of solutions.

To enable these methods, some new types of designs are defined.

Under suitable hypotheses and by using variational methods, some new criteria to guarantee that the fractional differential system has at least two nontrivial and nonnegative solutions are obtained.

Under some natural assumptions, by employing variational methods, some new results for the existence of at least two nontrivial and nonnegative solutions of the system ((P_{lambda })) are obtained.

Based on the fact that the (PS) condition can be replaced by the weaker Cerami condition for some deformation theorems which are the footstone for minimax methods, some new p-superlinear growth conditions were established in order to weaken ((AR)).

In addition, the moisture and rutting resistance of OGFC mixture could be evaluated by the common or modified test methods, but some new methods should be developed to properly evaluate these properties.

Based on Lyapunov second method, some new absolute stability conditions were proposed.

Then, by using the variational method, some new existence results are obtained.

Different from the free-weighting matrix method, some new sharper summation inequalities are developed via auxiliary functions.

Combining with the Halanay inequality and the Lyapunov functional method, some new sufficient conditions are derived for the model to be globally exponentially stable.

Based on a piecewise analysis method and the Lyapunov functional method, some new delay-dependent synchronization criteria are derived in the form of LMIs by introducing free-weighting matrices.