Two-layered difference schemes for solving multipoint problems

In a Hilbert space one considers the two-layered difference scheme with the multipoint condition Here $B$ and $A(t)$ are symmetric and positive-definite operators from $H$ into $H$. Under the assumption of the operator inequalities ($\tau$ is the step of the net) $\tau$ for some constants $\varepsilon>0$, $q>0$ (3) one establishes estimates for the solution of problem (1), (2) in terms of (). On the basis of these estimates one investigates linear and non-linear schemes with weights. The obtained results are applied to establish the order of convergence of the difference methods for solving periodic boundary-value problems for second-order quasilinear parabolic equations.