Abstract
We prove the existence of one‐signed periodic solutions of second‐order nonlinear difference equation on a finite discrete segment with periodic boundary conditions by combining some properties of Green′s function with the fixed‐point theorem in cones.
Highlights
Let R be the set of real numbers, Z be the integers set, T, a, b ∈ Z with T > 2, a > b, and a, b Z {a, a 1, . . . , b}.In recent years, the existence and multiplicity of positive solutions of periodic boundary value problems for difference equations have been studied extensively, see 1–5 and the references therein
We say that a solution y of 1.4 has a generalized zero at t0 provided that y t0 0 if t0 0 and if t0 > 0 either y t0 0 or y t0 − 1 y t0 < 0
We consider the existence of one-signed solutions of 1.7
Summary
Let R be the set of real numbers, Z be the integers set, T, a, b ∈ Z with T > 2, a > b, and a, b Z {a, a 1, . . . , b}.In recent years, the existence and multiplicity of positive solutions of periodic boundary value problems for difference equations have been studied extensively, see 1–5 and the references therein. 1.6 and obtains the sign properties of Green’s function of 1.4 , 1.5 . We say that a solution y of 1.4 has a generalized zero at t0 provided that y t0 0 if t0 0 and if t0 > 0 either y t0 0 or y t0 − 1 y t0 < 0. Assume that the distance between two consecutive generalized zeros of a nontrivial solution of 1.4 is greater than T .
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