Yosida Distance and Existence of Invariant Manifolds in the Infinite-Dimensional Dynamical Systems

Abstract

We introduce a new concept of Yosida distance between two (unbounded) linear operators $A$ and $B$ in a Banach space $\mathbb{X}$ defined as $d_Y(A,B):={\mathrm{limsup}}_{\mu \to +\infty }\Vert A_{\mu }-B_{\mu }\Vert$, where $A_{\mu }$ and $B_{\mu }$ are the Yosida approximations of $A$ and $B$, respectively, and then study the persistence of evolution equations under small Yosida perturbation. This new concept of distance is also used to define the continuity of the proto-derivative of the operator $F$ in the equation $u^{\prime }(t)=Fu(t)$, where $F:D(F)\subset \mathbb{X}\to \mathbb{X}$ is a nonlinear operator. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of $F$ is continuous. The Yosida distance approach to perturbation theory allows us to free the requirement on the domains of the perturbation operators. Finally, the obtained results seem to be new.