Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations

Abstract

We consider a 2nth-order nonlinear difference equation containing both many advances and retardations with p-Laplacian. Using the critical point theory, we obtain some new explicit criteria for the existence and multiplicity of periodic and subharmonic solutions. Our results generalize and improve some known related ones.

Positive Solutions for a Class of Discrete Mixed Boundary Value Problems with thep,q-Laplacian Operator

“…So, we have ‖u − n ‖ ⟶ 0 as n ⟶ +∞. It means that there exists an M > 0 such that u − n ≤ M. From (10) we know that…”

“…In recent years, with the development of mechanical engineering, control system, computer science, and economics, the existence of solutions of difference equations has attracted wide attention (see [1][2][3][4][5][6]). For example, applying Ricceri variational principle to obtain the existence of multiple solutions [7][8][9], taking the invariant sets of descending flow to prove the existence of sign-changing solutions [10], making the linking theorem to get the existence and multiplicity of periodic solutions [11], and using critical point theory to obtain the existence of homoclinic solutions [12][13][14][15] and heteroclinic solutions [16].…”

Multiple Solutions for Boundary Value Problems of p-Laplacian Difference Equations Containing Both Advance and Retardation

“…Relevant examples and mathematical models can be found in [1][2][3]. By means of critical point theory, many researchers devote themselves to the study of difference equations and achieve many excellent results, for example, the results on boundary value problems [4][5][6][7][8][9][10][11][12][13][14][15][16], periodic solutions [17][18][19][20][21][22], and homoclinic solutions [23][24][25][26][27][28][29][30][31] had been obtained.…”

“…where a(k) and b(k) are real valued for each k ∈ Z and ω ∈ R. Mei and Zhou [21] considered the existence of the periodic and subharmonic solutions of a 2nth order p-Laplacian difference equation containing both advances and retardations:…”

“…ere are more about the results of the difference equations containing both advance and retardation which can be seen in [13,19,22,24].…”