YOSIDA DISTANCE AND EXISTENCE OF INVARIANT MANIFOLDS IN THE INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS

We consider the existence of invariant manifolds to evolution equations u ' ( t ) = Au ( t ), A : D ( A ) C X-+ X near its equilibrium A (0) = 0 under the assumption that its proto-derivative partial derivative A ( x ) exists and is continuous in x E D ( A ) in the sense of Yosida distance. . Yosida distance between two (unbounded) linear operators U and V in a Banach space X is defined as dY(U, Y ( U, V ) := lim sup mu ->+ infinity IIU mu U mu- V mu II, where U mu mu and V mu mu are the Yosida approximations of U and V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of partial derivative A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.