Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity

Existence and multiple solutions for second-order p-Laplacian difference equations

“…In this paper, we investigate the solutions of boundary value problems (1.1) with (1.2) for a second-order p-Laplacian difference equation [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. By using the critical point theory [26][27][28], the existence and multiple results are obtained.…”

Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

“…Therefore, the study of homoclinic orbits [10][11][12][13][14][15][16][17][18][19][20][21] of difference equation is meaningful.…”

“…By using the critical point theory, Guo and Yu [23] established sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations. Compared to first-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, for example, [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references contained therein). Peil and Peterson [26] in 1994 studied the asymptotic behavior of solutions of 2nth-order difference equation…”

“…However, to the best of our knowledge, the results on homoclinic solutions of higher-order nonlinear difference equations are scarce in the literature [10,11,15,20,24,27]. Furthermore, since (1) contains both advance and retardation, there are very few manuscripts dealing with this subject.…”

“…The proof is based on the notable Mountain Pass Lemma in combination with variational technique. The motivation for the present work stems from the recent papers [1,11,20].…”