Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods

Abstract

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales T of the formWe construct a variational framework of the above-mentioned problem, and some new results on the existence of a homoclinic orbit or an unbounded sequence of homoclinic orbits are obtained by using the mountain pass lemma and the symmetric mountain pass lemma, respectively. The interesting thing is that the variational method and the critical point theory are used in this paper. It is notable that in our study any periodicity assumptions on p(t), q(t) and f (t, u) are not required.

Homoclinic solutions for n-dimensional p-Laplacian neutral differential systems with a time-varying delay

“…In the past few decades, the existence of homoclinic solutions for second-order differential equations has been widely investigated by using critical point theory, the methods of bifurcation theory, or Mawhin's continuation theorem (see [1][2][3][4][5][6][7][8]). However, the corresponding results on the existence of homoclinic solutions to a neutral differential equation are relatively infrequent.…”