Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero

On Homoclinic Solutions for First-Order Superquadratic Hamiltonian Systems with Spectrum Point Zero

“…Lemma 3 (see [16], Lemma 2.1). − ,2 ⊂ 1 loc and is embedded compactly in ∞ loc and continuously in for all ≥ 2.…”

“…During the last decades, many authors are devoted to the study of homoclinic orbits for Hamiltonian systems via modern variational methods. For example, see [1][2][3][4] for the second-order systems and [5][6][7][8][9][10][11][12][13][14][15][16] for the first-order systems. To be precise, in 1999, Ding and Willem [15] studied the existence of homoclinic solutions for a class of the first-order periodic Hamiltonian systems (HS) with spectrum point zero under the well-known (AR) condition as follows:…”

Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales

“…Although considerable attention has been dedicated to homoclinic orbits for continuous or discrete Hamiltonian systems, see [7,8,16,17,20,25,[35][36][37] and the references therein. To the best of our knowledge, there is few work on homoclinic orbits for Hamiltonian systems on time scales.…”

An index theory with applications to homoclinic Orbits of Hamiltonian systems and Dirac equations

“…Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from the origin. The existence and multiplicity of homoclinic orbits for the first order system were studied extensively by means of critical point theory, and many results were obtained under the assumption that H(t, z) depends periodically on t (see [4,10,13,16,22,33,45,46,48,49,50,51,52,56] and the references therein). As authors known, the periodicity is used to protect some kind of compactness such as the (PS) condition.…”