In an arbitrary Banach space, the authors consider a nonlocal boundary value problem for the difference equation \[ \frac{u_k - u_{k-1}}{\tau} + A u_k = \varphi_k, \;1 \leq k \leq N, \;N\tau = 1, \;u_0 = u_{[\lambda/\tau]} + \varphi, \tag{1} \] where \(A\) is a strongly positive operator. Stability and coercive stability of (1) in various Banach spaces are studied. As applications, difference schemes of boundary-value problems for parabolic equations are considered.