Positive periodic solutions of second-order difference equations with weak singularities

Abstract

We study the existence of positive periodic solutions of the second-order difference equationSchauder's fixed point theorem, where a, c : ℤ ℝ + are T -periodic functions, f C(ℤ × (0, ∞), ℝ) is T -periodic with respect to t and singular at u = 0. Mathematics Subject Classifications: 34B15.

Periodic Solutions of Second Order Nonlinear Difference Equations with Singularϕ-Laplacian Operator

“…An interesting question is which techniques and theorems regarding the continuous differential equations can be adapted for difference equations (see Kelly and Peterson [9], Agarwal [10], and Bereanu and Mawhin [11,12]). The purpose of this paper is to show that some known existence and multiplicity results of periodic solution for singular perturbations of the singular -Laplacian operator also hold for the corresponding difference equation and develop some results for the singular difference equation boundary value problems; see [12][13][14][15][16][17].…”

“…This problem has been studied by Ma and Lu [15][16][17] where the existence of positive solution needs a necessary condition < 0 or 0 < < 4sin 2 ( /2 ) (see [14]). For other results concerning the existence of solutions for singular nonlinear difference equation boundary value problems, see, for example, [9,10,18].…”

“…and note that = + − − . Motivated by the above results from [12,[14][15][16][17], we consider the discrete periodic problem…”

Reverse-Order Lower and Upper Functions for Periodic Problems of Second-Order Singular Difference Equations

“…Recently, there are many papers to study the existence of positive periodic solutions for second-order difference equations, see [2][3][4][5][6][7] and their references therein. However, there are few techniques for studying the existence of positive solutions of difference equations with singularity, and thus the results in the field are very rare, see [8][9][10][11][12][13]. The existence of positive periodic solutions for continuous case has been studied by Torres, see [14,15].…”