Exponential dichotomy roughness and structural stability for evolution families without bounded growth and decay
An evolution family \(U=\{u(t,s)\}_{t\geq s}\) of linear and bounded operators is called exponentially dichotomous iff there exist constants \(N_i\), \(\nu_i>0\), \(i=1,2\) and a projection valued function \(\rho: \Omega\to{\mathcal L}(E)\) \(({\mathcal L} (E)\) is the space of all linear bounded operators actend in the Banach space \(E)\) Strongly continuous, verifying the following relations \(P(t)U(t,s)= U(t,s)P(s)\) \(U_Q(t,s):Q(s)E \to Q(t)\) \(E\) is invertible \(\|U_p(t,s)\| \leq N_1\ell^{-\nu_1(t-s)}\) \(\|U_Q(t,s)^{-1}\|\leq N_2\ell^{-\nu_2(t-s)}\), for all \(t\geq s\), where \(Q(t)=I-P(t)\) and. If \(P(t)U(t,s)=U(t,s)P(s)\), for all \(t\geq s\) then \(U_p(t,s)=U(t,s)\) \(P(s)\) and \(U_Q(t,s)=U(t,s)Q(s)\). The author prove a roughness theorem for an exponentially dichotomous family \(U=\{U(t, s)\}_{t\geq s}\). The perturbed evolution family \(V=\{V(t,s)\}_{t\geq s}\) is given by following equality \[ V(t,s)=U(t,s)+\int_s^tU(t,\tau)B(\tau) V (t,s)d \tau. \] Two evolution families \(U=\{U(t,s)\}_{t\geq s}\) and \(V= \{V(t,s)\}_{t\geq s}\) are called topological equivalent iff there exists a continuous map \(h:\mathbb{R} \times E\) satisfying the following conditions -- For each \(t\in\mathbb{R}\), the function \(h_t(x)=h(t,x)\) is an homeomorphism -- \(h_tU(t,\tau)= V(t,\tau) h_\tau\) for each \(t,\tau \in\mathbb{R}\). -- There exist two increasing functions \(L_i: \mathbb{R}_+\to \mathbb{R}_+\), \(i=1,2\) with \(L_i(0)=0\), continuous at 0, for which \(\|h_t(x) \|\leq L_1(\|x\|)\) and \(\|h_t^{-1}(x)\|\leq L_2(\|x\|)\), \(x\in E\), \(t\in \mathbb{R}\). The evolution family \(U=\{U(t,s)\}_{t\geq s}\) is called structurally stable iff for small \(B(t)\) the perturbed family \(V=\{V(t, s)\}_{t\geq s}\) is topological equivalent to \(U=\{U(t,s)\}_{t\geq s}\). The author prove the important fact, that an exponentially dichotomic evolution family is structurally stable.
scientific article; zbMATH DE number 5569389
scientific article