On well‐posedness of the nonlocal boundary value problem for parabolic difference equations

Abstract

We consider the nonlocal boundary value problem for difference equations ( u k − u k −1 )/ τ + A u k = φ k , 1 ≤ k ≤ N , N τ = 1, and u 0 = u [ λ / τ ] + φ , 0 < λ ≤ 1, in an arbitrary Banach space E with the strongly positive operator A . The well‐posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed‐type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.