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 ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)⊂ X→ X near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) ∂ A(x) exists and is continuous in x ∈ D ( A ) x∈ D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X X is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔ μ → +∞ \| U_μ -V_μ \| , where U μ U_μ and V μ V_μ are the Yosida approximations of U U and V 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 ∂ A ∂ 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.