UPPER SEMICONTINUITY OF ATTRACTORS FOR LINEAR MULTISTEP METHODS APPROXIMATING SECTORIAL EVOLUTION-EQUATIONS

This paper sets out a theoretical framework for approximating the attr actor A Of a semigroup S(t) defined on a Banach space X by a a-step se midiscretization in time with constant steplength k. Using the one-ste p theory of Hale, Lin and Raugel, sufficient conditions are establishe d for the existence of a family of attractors (A(k)) subset of X(q), f or the discrete semigroups S-k(n) defined by the numerical method. The convergence properties of this family are also considered. Full detai ls of the theory are exemplified in the context of strictly A(alpha)-s table linear multistep approximations of an abstract dissipative secto rial evolution equation.