Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics

Multiple Heteroclinic solutions of bilateral difference systems with Laplacian operators

“…The methods used in [22] are based upon the growth argument and the upper and lower solutions methods. In [23], motivated by some models arising in hydrodynamics, Rachunek and Rachunkoa studied the second-order non-autonomous difference equation…”

“…We note that the difference equations discussed in [19,22,23] are those ones defined on N ¼ f0; 1; 2; Á Á Ág. The existence of homoclinic solutions for second-order discrete Hamiltonian systems have been studied in [24,26] by using fountain theorem.…”

The Solvability of the Discrete Boundary Value Problem on the Half-Line

“…In various physical areas, such as hydrodynamics or the unsteady flow of gas through a semi-infinite porous media, studying radially symmetric solutions leads to the Sturm-Liouville equation with boundary value conditions of the form x (0) = 0, x(∞) = C, C ∈ (0, 1); see for example [1,2]. Let us remind the reader of the classical Sturm-Liouville boundary value problem on the half-line:…”

“…where x(∞) = lim n→∞ x(n), d ∈ R, α, β ∈ R, α 2 + β 2 > 0. We want to construct sufficient conditions for the existence of a solution to (2) in dependence on the parameters α, β. First, we divide our consideration into two cases, when problem (2) is without resonance, which means that β = α ∑ ∞ l=0 1 a(l) , and with resonance.…”

Unbounded solutions of second order discrete BVPs on infinite intervals

“…We refer here to some works using the upper and lower solutions method, e.g., see [3,10,11,13,14,15,18,20,21,24,25,28,29] for finite interval problems, and [1,5,7,27] for infinite interval problems. Discrete infinite interval problems have also been studied by several other methods in [2,4,6,8,9,12,16,17,19,21,22,26].…”