Weak dichotomies and asymptotic integration of nonlinear differential systems. (English) Zbl 0918.34013

Naulin, Raúl

Nonlinear Stud. 5, No. 2, 201-218 (1998).

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.

Reviewer: Stepan Kostadinov (Plovdiv)

MSC:

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

Keywords:

nonlinear differential systems; weak dichotomies; asymptotic integration