Homoclinic orbits of a class of second-order difference equations

Abstract

a b s t r a c tIn this paper, we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second-order differencesuperquadratic and subquadratic. Under the assumptions that L(t) is positive definite for sufficiently large |t| ∈ Z, we show that there exists at least one non-trivial homoclinic orbit of the difference equation. Further, if V (t, x) is superquadratic and even with respect to x, then it has infinitely many different non-trivial homoclinic orbits. At the end, two illustrative examples are provided.

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

“…It is easy to verify that k k and k k 0 are equivalent; see [25]. Thus, there exists C 1 , C 2 > 0 such that…”

“…Now, we introduce on X the following inner product and the norm where u = u − + u 0 + u + and v = v − + v 0 + v + with respect to the decomposition . It is easy to verify that ‖ · ‖ and ‖ · ‖ 0 are equivalent; see . Thus, there exists C 1 , C 2 > 0 such that for any u ∈ X .…”

“…In 2012, Zhang and Shi studied the existence of homoclinic solutions of under the assumptions that L ( n ) is positive definite for sufficiently large and W ( n , u ) is super quadratic or subquadratic.…”

“…3) Under the assumption that L.n/ is positive definite for all n 2 Z, most of them treated the super quadratic case [18][19][20][21][22][23], and [24] treated the subquadratic case. In 2012, Zhang and Shi [25] studied the existence of homoclinic solutions of (1.1) under the assumptions that L.n/ is positive definite for sufficiently large jnj 2 Z and W.n, u/ is super quadratic or subquadratic.…”

Multibump solutions of a class of second-order discrete Hamiltonian systems

“…the Fréchet derivative of f at x in (l 2 , ·, · 2 ). By the methods used in [8,35], we could show that f (u) is differentiable.…”