We will modify and extend W to an appropriate W̃ and show for the associated modified functional I the existence of a sequence of homoclinic solutions converging to zero in (L^{infty}) norm, and therefore we obtain infinitely many homoclinic solutions for the original problem.
In order to prove our results via variational approach, inspired by the methods of [27, 28], first, we need to modify and extend f to an appropriate f̃ and to show for the associated modified functional the existence of solutions.
Some properties of the associated modified equations are proven and then extended to the semi-discrete case.
Based on the flow resistivity, the frequency-dependent acoustic absorption coefficient of the needled nonwovens was predicted using the empirical model of Delany and Bazley (1970) and its associated modified models.
By making use of a recently introduced block tridiagonal partitioning of the collocation matrix, which yields faster rates of convergence, and its associated modified red-black ordering of unknowns and equations, we succeeded to increase the scalability of the problem.
For example, in Fagaceae it is derived from a highly modified and reduced branch system with its associated modified leaves or bracts, at least in the genus Quercus; and in Betulaceae the husks are thought to be made up of several modified bracts.
However, the metabolic effects of pills are extremely variable; for example estrogens impair insulin action dose-dependently and the associated progestins may modify these effects [ 14].
Since TRMOD based decision boundaries are derived from data and are comparable to eDISH limits for liver chemistry data, it has been suggested to name the associated plot "modified eDISH", "mDISH".
Finally, the immune response and the associated resistance can be modified by the type of antigen that is recognized and by such factors as age, nutrition, and the number of infections.
As described earlier, the cytoskeleton is made of three different classes of polymers and a large number of associated modifying factors (Section 2.2.3).
