Dynamics of a class of ODEs more general than almost periodic
Given a Banach space \(X\), the authors investigate the function space \[ {\mathcal F}:=\{f:{\mathbb R}\to X\mid f\text{ is uniformly continuous with \(f({\mathbb R})\) being precompact}\} \] and provide interesting applications to solutions of nonautonomous ODEs in Banach spaces with respect to solutions in \({\mathcal F}\), the saddle-point structure around them, as well as pullback attraction properties. First, it is established that \({\mathcal F}\) is a closed subset of the bounded uniformly continuous functions. Moreover, provided \(A(t):D\to X\) with dense domain \(D\) generates an evolution operator, it is shown that \({\mathcal F}\) is an admissible space. This means, under exponential dichotomy, equicontinuity and compactness assumption on \(\dot x=A(t)x\), that the inhomogeneous problem \(\dot x=A(t)x+f(t)\) has a unique solution in \({\mathcal F}\) for every perturbation \(f\in{\mathcal F}\). After an insertion on sequential sets for \({\mathcal F}\) and dichotomy roughness, the authors verify that the above admissibility property also holds for weakly nonlinear problems \[ \qquad\dot x=A(t)x+f(t,x,\epsilon),\tag{\(*\)} \] provided \(f\) is Lipschitz in \(x\) and satisfies certain compactness properties. Around the resulting solution \(x^\ast(\epsilon)\) in \({\mathcal F}\), there exists a saddle-point structure whose stable and unstable integral manifolds are constructed using a Lyapunov-Perron approach. Applications to coupled heat and beam equations are given. Finally, under the assumption of a uniformly asymptotically stable linear part \(\dot x=A(t)x\), it is shown that \(x^\ast(\epsilon)\) is the pullback attractor of the weakly nonlinear problem \((\ast)\). The reviewer points out that for continuously Fréchet-differentiable \(f\) an alternative persistence proof for the solution \(x^\ast(\epsilon)\) and its saddle-point structure on the basis of the implicit mapping theorem might be possible [cf. the reviewer, Commun. Pure Appl. Anal. 10, No.~3, 937--961 (2011), \url{doi:10.3934/cpaa.2011.10.937}].
scientific article; zbMATH DE number 5865543
scientific article