A Perron-type theorem for nonautonomous delay equations
The authors show that the asymptotic exponential behavior of the solutions to a linear delay equation \[ x'=L(t)x_{t}, \] where \(x_{t}(\theta)=x(t+\theta)\), \(\theta\in [-r,0]\) for some fixed \(r>0\), persists under small perturbations, i.e., for an appropriate choice of \(f(t,x)\) in the equation \[ x'=L(t)x_{t}+f(t,x_{t}). \] More precisely, let \(X\) be a Banach space and \(r>0\), and define \(\hat{\mathbb{D}}\) as the space of functions \(\varphi:[-r,0]\to X\) such that the one-sided limits of \(\varphi\) at \(s\) exist and \(\varphi\) is right continuous at \(s\) for every \(s\in[-r,0]\). We write \(T(t,s)\) for the evolution operator with values in \(\hat{\mathbb{D}}\). One also assumes that \(\hat{\mathbb{D}}=F_{1}(t)\oplus F_2(t) \oplus \dotsb\oplus F_{p}(t)\) for every \(t\geq0\) and \(\lambda_1<\lambda_2<\dotsb<\lambda_p\) are real numbers which satisfy {\parindent=8mm \begin{itemize}\itemH1 For each \(t,s\geq0\) and \(i=1,\dotsc,p\), one has \(T(t,s)F_{i}(s)=F_{i}(t)\); \itemH2 Given \(\varepsilon>0\), there exists \(\rho>0\) such that \[ \rho^{-1}e^{(\lambda_i-\varepsilon)(t-s)-\varepsilon s}\left\|x\right\|\leq \left\|T(t,s)x\right\|\leq \rho e^{(\lambda_i+\varepsilon)(t-s)+\varepsilon s}\left\|x\right\|,\quad t\geq s\geq0, \] for every \(i=1,\dotsc,p\) and \(x\in F_{i}(s)\); \itemH3 For each \(i=1,\dotsc,p\), the projections \(P_i(t)\) associated to the decomposition of \(\hat{\mathbb{D}}\) satisfy \[ \lim_{t\to\infty}\frac{1}{t}\log \left\|P_{i}(t)\right\|=0. \] \end{itemize}} The previous hypotheses are widely satisfied and are also motivated by ergodic theory. The main result of the work states that, under the previous assumptions, if \(x\) is a solution of \[ x_t=T(t,s)x_s+\int_{s}^{t}T(t,\tau)X_0 f(\tau,x_{\tau})\,d\tau, \] where \((X_0 u)(0)=u\) and \((X_0 u)(\theta)=0\) for every \(\theta<0\), and also \(\left\|f(t,x_t)\right\|\leq \gamma(t)\left\|x_t\right\|\) for every \(t\geq0\), and for some continuous function \(\gamma:\mathbb{R}\to\mathbb{R}\) which verifies \(\lim_{t\to\infty}\int_{t}^{t+1}e^{\delta\tau}\gamma(\tau)\,d\tau=0\) for some \(\delta>0\), then either \(x_{t}=0\) for all large enough \(t\) or there exists \(i\in\{1,\dotsc,p\}\) such that \(\lambda_{i}=\lim_{t\to\infty}\frac{1}{t}\log\left\|x_t\right\|\). In the second case, the authors prove that any solution \(x(t)\) to the equation above is essentially asymptotically tangent to the spaces \(F_{i}(t)\), with \(i\) as before. To conclude, two nontrivial consequences of the main result considering perturbations of linear equations having only negative Lyapunov exponents, and the case of linear perturbation with \(f(t,x_t)=M(t)x_t\) for some linear operator \(M(t)\), continuous for \(t\geq0\), are stated.
scientific article; zbMATH DE number 6176172