On the exponential dichotomy on \(\mathbb R\) of linear differential equations in \(\mathbb R^n\) (Q2754812)

The exponential dichotomy of a real \(n\)-dimensional system NEWLINE\[NEWLINE\dot x=A(t)x\tag{1}NEWLINE\]NEWLINE with measurable and integrally bounded matrix \(A(t)\) (this means that \(\sup_{s\in \mathbb{R}}\int_{s}^{s+1}A(t) dt<\infty \)) is studied. The author presents a complete and rigorous proof of the following result. NEWLINENEWLINENEWLINESystem (1) with integrally bounded matrix \(A(t)+A^*(t)\) has the property of exponential dichotomy on the whole axis iff one of the following statements is true:NEWLINENEWLINENEWLINE(a) for any bounded measurable function \(f:\mathbb{R} \mapsto \mathbb{R}^n\), the inhomogeneous system \(\dot x=A(t)x+f(t)\) has a unique bounded on solution \(\mathbb{R}\);NEWLINENEWLINENEWLINE(b) there exists an indefinite matrix-valued function \(W(t)\in C(\mathbb{R} \mapsto \mathbb{R}^{n\times n})\) which is bounded, nondegenerate, almost everywhere differentiable on \(\mathbb{R}\), and for some \(\alpha >0\) satisfies the inequality NEWLINE\[NEWLINEd\langle W(t)x(t),x(t)\rangle /dt\leq-\alpha \|x(t)\|^2\quad \forall t\in \mathbb{R}, NEWLINE\]NEWLINE where \(x(t)\) is an arbitrary solution to system (1).NEWLINENEWLINENEWLINEIt should be noted that one can find an analogous theorem in the book by \textit{Yu. L. Daletskij} and \textit{M. G. Krein} [Stability of solutions of differential equations in Banach space. Moscow: Nauka (1970; Zbl 0233.34001)]. Unfortunately, their formulation of statement (b) is inaccurate. It does not contain the requirement of nondegeneracy of \(W(t)\).