Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

Abstract:

We consider the existence of invariant manifolds to evolution equations $u’(t)=Au(t)$, $A:D(A)\subset \mathbb {X}\to \mathbb {X}$ near its equilibrium $A(0)=0$ under the assumption that its proto-derivative $\partial A(x)$ exists and is continuous in $x\in D(A)$ in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators $U$ and $V$ in a Banach space $\mathbb {X}$ is defined as $d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \|$, where $U_\mu$ and $V_\mu$ 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 $\partial A$ 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.