On the Convergence of Difference Approximations to Nonlinear Contraction Semigroups in Hilbert Spaces

Abstract

Convergence properties of the difference schemes (S) \[ {h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0,\], for evolution equations (E) \[ \frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)} \] are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation $x\prime = \lambda x$ for $\lambda \in {\text {C}} - K$, where K is a compact subset of the right half-plane, then (S) is convergent as $h \downarrow 0$, with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis $t \geqslant 0$, if the solution $u(t)$ tends strongly to a constant as $t \to \infty$. We also show that under weaker stability conditions one can construct conditionally convergent methods.