摘要：In this work we consider a discrete nonlocal problem of the following type: { Δ ( q Δ x ) ( n ) + f ( n + 1 , x ( n + 1 ) ) = 0 , x ( ∞ ) = g ( x ) , $$\textstyle\begin{cases} \Delta ( q \Delta \mathbf {x} )(n) + f (n+1, \mathbf {x}(n+1) ) =0, \\ \mathbf {x}(\infty) = g(\mathbf {x}), \end{cases} $$ in the context of an arbitrary Banach space ( X , ∥ ⋅ ∥ X ) $(X,\lVert\cdot\rVert _{X})$ , and we give sufficient conditions that ensure the existence of solutions to it. In order to present our result, we shall need to study conditions that ensure the existence of solutions with a nonlocal asymptotic behavior for the following equation: x ( n ) = g ( x ) − ∑ k = n + 1 ∞ ∑ j = n k − 1 1 q ( j ) f ( k , x ( k ) ) . $$\mathbf {x}(n) = g(\mathbf {x}) - \sum_{k=n+1}^{\infty}\sum_{j=n}^{k-1} \frac){q(j)} f \bigl(k, \mathbf {x}(k) \bigr). $$
关键词：nonlocal problem ; asymptotic behavior ; second order difference equation ; Leray-Schauder type fixed point theorem
