The Hartman-Grobman theorem for reversible systems on Banach spaces
The object of the paper is to present a new version of the well-known Hartman-Grobman Theorem for reversible systems on a Banach space. Let \(A:D\subset B\to B\) be the infinitesimal generator of a \(C_0\)-semigroup \(T(t)\), here \(B\) is a Banach space. Let \(A\) be reversible with respect to \(S\) (i.e., \(D\) is invariant under \(S\), and \(ASx=-SAx\) for any \(x\in D\)). Let \[ W_s=\{x\in B:T(t)x\to 0,\text{ as }t\to\infty\} \] and \[ W_u=\{x\in B:T(t)x\to 0,\text{ as }t\to-\infty\}. \] Suppose that \(B=W_s\oplus W_u\), and there exists a projection \(P:B\to W_s\), such that \(SP=(I-P)S\) and \[ |T(t)Px|\leq Ke^{-\alpha t}|x| \] for any \(t\geq 0\) and any \(x\in B\); \(K\geq 1\), \(\alpha\geq 0\). Finally, let \(C_*^0\) denote the space of all bounded and uniformly continuous functions from \(B\) into \(B\); \(L_\mu\) denote the subspace of \(C_*^0\) of uniformly Lipschitz continuous functions with Lipschitz constant \(\mu\). Then the authors prove that there exists \(\mu_0>0\) such that, for any \(f\in L_{\mu_0}\) satisfying \((T(1)+f)\circ S=S\circ(T(1)+f)^{-1}\), there is a unique homeomorphism \(h=h(f)\) with \(h(0)=I\), \(h-I\in C_*^0\), and such that \(h\circ(T(1)+f)=T(1)\circ h\). Moreover, the set \(\{x\in B:Sx=x\}\) is invariant under \(h\). The authors give some applications of the obtained results to partial differential equations.
scientific article; zbMATH DE number 1016996
scientific article