On (N, Q)-periodicity for Weyl-like fractional difference equations: extending existence results under weighted-sum conditions or the H$$\ddot{\hbox {o}}$$lder continuity condition

Abstract In this paper, we continue to study the ( N , Q )-periodicity for a class of Weyl-like fractional difference equations of the form $$\begin{aligned} \triangle ^{\alpha }v(n)=Lv(n+1)+p(n, v(n)), \ n\in \mathbb {Z}. \end{aligned}$$ ▵ α v ( n ) = L v ( n + 1 ) + p ( n , v ( n ) ) , n ∈ Z . Some new existence theorems of ( N , Q )-periodic mild solutions for the equations are established under weighted-sum conditions or the H $$\ddot{\hbox {o}}$$ o ¨ lder continuity condition of order $$\epsilon \in (0,1)$$ ϵ ∈ ( 0 , 1 ) , which relax the Lipschitz-type conditions of [13]. The results are essentially a continuation and generalization of some recent results on this issue.