On well-posedness of the nonlocal boundary value problem for parabolic difference equations. (English) Zbl 1077.39015
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.
Reviewer: Victor I. Tkachenko (Kyïv)
MSC:
39A12 | Discrete version of topics in analysis |
39A10 | Additive difference equations |
47B39 | Linear difference operators |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
34G10 | Linear differential equations in abstract spaces |