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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.

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