Stability criterion for a family of nonlocal difference schemes

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ. A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ ∈ (−cosh π,−1) and the original differential problem is stable, while the stability conditions for the finite-difference schemes substantially depend on γ. We obtain estimates for the energy norm of the solution of the finite-difference problem via the same norm of the initial data and prove the equivalence of the energy norm and the grid L ₂-norm.