Periodic and subharmonic solutions for second-order nonlinear difference equations

Existence of periodic solutions for second-order nonlinear difference equations

“…Critical point theory is also an important tool to deal with problems on differential equations [14,19]. Because of applications in many areas of difference equations [1,2,3], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions of discrete systems, see [8,9,10,17,21,23]. Particularly, Guo and Yu [8,9,10] and Shi et al [17] studied the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory.…”

“…Because of applications in many areas of difference equations [1,2,3], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions of discrete systems, see [8,9,10,17,21,23]. Particularly, Guo and Yu [8,9,10] and Shi et al [17] studied the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory. However, to the best of our knowledge, when δ = 1 the results on periodic solutions of second-order nonlinear difference equation (1.1) are very scarce in the literature (see [4]), because there are few known methods for considering the existence of periodic solutions of discrete systems.…”

Existence of periodic solutions of fourth-order nonlinear difference equations

“…Critical point theory is also an important tool to deal with problems on differential equations [9,11,12,24,29,33]. Because of applications in many areas for difference equations [1,21,25], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions on discrete systems, see [16][17][18]30,36,38]. Particularly, Guo and Yu [16][17][18] and Shi et al [30] studied the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory.…”

“…Because of applications in many areas for difference equations [1,21,25], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions on discrete systems, see [16][17][18]30,36,38]. Particularly, Guo and Yu [16][17][18] and Shi et al [30] studied the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory. Compared to one-order or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention(see, for example, [1,7,10,14,21,27,28,32,34] and the references contained therein).…”

Nonexistence and existence results for a 2 $$n$$ n th-order discrete mixed boundary value problem

“…Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [16][17][18] and Shi et al [29] have successfully proved the existence of periodic solutions of second-order nonlinear difference equations. We also refer to [33,34] for the discrete boundary value problems.…”