THE GENERATION OF AN EVOLUTION OPERATOR IN A BANACH LATTICE

Abstract

Evolution equations of the type: (UNFORMATTED TABLE FOLLOWS)

(E) x' + A(t)x (CONT) f(t), t (ELEM) 0,T ,

x(0) = x(,o)

(TABLE ENDS)

are studied.

The underlying space is a general Banach lattice. The operators A(t)u are maximal lattice accretive (mL-accretive) in u and satisfy a weak smoothness condition in t. The function f(t) is an L('1)-function.

Recent results of Evans concerning general Banach spaces are extended to the present case. Namely, an evolution operator is generated for the problem (E) via a difference scheme. Functionals involving the Gateaux derivative of the norm play an important role in the development of theory.

Some applications are given in the theory of non-linear partial differential equations.