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Weak dichotomies and asymptotic integration of nonlinear differential systems. (English) Zbl 0918.34013
The linear equation \[ \dot x(t)=A(t)x(t)\tag{1} \] is called weak dichotomous if there are four positive and continuous functions \(h,p,k,q\), and a projection matrix \(P\) such that \[ \bigl|\varphi(t)P\varphi^{-1}(s)\bigr|\leq Kh(t)p(s),\quad t\geq s\geq 0, \] \[ \bigl|\varphi(t)(I-P)\varphi^{-1}(s)\bigr|\leq Kk(t)q(s),\quad s\geq t\geq 0, \] where \(\varphi(t)\) is the fundamental matrix to (1) and \(K\) is a positive constant.
Using the Schauder-Tychonoff fixed point principle the author proves an equivalence between (1) and the nonlinear equation \[ \dot x(t)=A(t)x(t)+f\bigl(t,x(t)\bigr).\tag{2} \] That means that solutions to (1) induce solutions to (2) and vice versa. The results are illustrated by some interesting examples.

34A34 Nonlinear ordinary differential equations and systems, general theory
34D05 Asymptotic properties of solutions to ordinary differential equations