Existence and roughness of nonuniform \((h,k,\mu,\nu)\)-trichotomy for nonautonomous differential equations
The property of exponential trichotomy for a nonautonomous system of linear differential equations \[ \dot{x}=A(t)x \tag{1} \] on a Banach space $\mathcal{X}$ has been introduced by R. J. Sacker and G. Sell for the special case $\mathcal{X}=\mathbb{R}^{n}$. This property has been generalized in several ways. In particular, this article introduces the property of nonuniform $(h,k,\mu,\nu)$-trichotomy on $\mathbb{R}^{+}$ in the following way: \par There exists projections $P(t),Q(t),R(t): \mathcal{X}\to \mathcal{X}$ such that \[ P(t)+Q(t)+R(t)=\mathrm{Id} \] and the evolution operator $T(t,s)$ associated to (1) satisfies the following properties: \begin{itemize} \item[(i)] For any $t,s\in \mathbb{R}^{+}$ it follows that: \[ \begin{aligned} T(t,s)P(s)&=P(t)T(t,s)\\ T(t,s)Q(s)&=Q(t)T(t,s)\\ T(t,s)R(s)&=R(t)T(t,s). \end{aligned} \] \item[(ii)] There exist constants $a\leq 0 < b$, $c\leq 0<d$, $\varepsilon \geq 0$ and $K_{1}\geq 0$ such that, for any $t\geq s \geq 0$, \[ \begin{aligned} |T(t,s)P(s)|&\leq K_{1}\left(\frac{h_{1}(t)}{h_{1}(s)}\right)^{a}\mu_{1}^{\varepsilon}(s),\\ |T(t,s)R(s)|&\leq K_{1}\left(\frac{h_{2}(t)}{h_{2}(s)}\right)^{b}\mu_{2}^{\varepsilon}(s), \end{aligned} \] and, for any $0\leq t \leq s$, \[ \begin{aligned} |T(t,s)Q(s)|&\leq K_{1}\left(\frac{k_{1}(s)}{k_{1}(t)}\right)^{c}\nu_{1}^{\varepsilon}(s),\\ |T(t,s)R(s)|&\leq K_{1}\left(\frac{k_{2}(s)}{k_{2}(t)}\right)^{d}\nu_{2}^{\varepsilon}(s), \end{aligned} \] where the functions $h_{i},k_{i},\mu_{i},\nu_{i}$ ($i=1,2$) are in the set \[ \triangle=\left\{ u: \mathbb{R}^{+}\to [1,+\infty) \mid u\text{ is increasing}, u(0)=1\text{ and }\lim_{t\to +\infty}u(t)=+\infty\right\}. \] \end{itemize} In order to introduce the first result, the authors focus its attention on finite dimensional systems (1), where the linear operator $A(t)$ is of the type \[ A(t)=\left[\begin{matrix} A_{1}(t) & 0 & 0 \\ 0 & A_{2}(t) & 0 \\ 0 & 0 & A_{3}(t) \end{matrix}\right], \tag{2} \] which leads to the subsystems \[ x_{1}'=A_{1}(t)x_{1}, \quad y_{1}'=A_{2}(t)y_{1}, \quad z_{1}'=A_{3}(t)z_{1} \tag{3} \] and its corresponding adjoints \[ x_{2}'=-A_{1}^{*}(t)x_{2}, \quad y_{2}'=-A_{2}^{*}(t)y_{2}, \quad z_{2}'=-A_{3}^{*}(t)z_{2}. \tag{4} \] The first result of the article (Theorem 2.3) provides a set of sufficient conditions ensuring that a linear system with structure given by (2) has a nonuniform $(h,k,\mu,\nu)$-trichotomy on $\mathbb{R}^{+}$. These conditions are described in terms of a weighted Lyapunov exponents (called $(H,K,L)$-Lyapunov exponents) for the subsystems (3)--(4). The resulting functions $h_{i},k_{i},\mu_{i}$ and $\nu_{i}\in \triangle$ are also described in terms of the weights $(H,K,L)$ mentioned above. \par The proof use the fact that the exponents $a$,$b$,$c$ and $d$ of some dichotomies and have a strong relation with the characteristic exponents. This idea is extended for this trichotomy in an interesting way. \par As any linear system is kinematically similar to a system of type (2), the authors say that the nonuniform $(h,k,\mu,\nu)$-trichotomy is widely present in nonautonomous linear systems. \par The second result of the article (Theorem 2.4) is a \textit{roughness type} result for the perturbed system \[ x'=[A(t)+B(t)]x, \tag{5} \] that is, under the assumption that the system (1) has a nonuniform $(h,k,\mu,\nu)$-trichotomy on $\mathbb{R}^{+}$, the authors provide a set of smallness conditions for $B(t)$ combined with technical assumptions for $K_{1}$ and the functions $h_{i},k_{i},\mu_{i},\nu_{i}$ in order to ensure that the perturbed system (5) also has a nonuniform $(h,k,\mu,\nu)$-trichotomy on $\mathbb{R}^{+}$. \par The proof of this result is fashioned along some roughness results for nonuniform $(h,k,\mu,\nu)$-dichotomies. Nevertheless, technical difficulties arise in this new context.
scientific article; zbMATH DE number 6999283
scientific article