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A note on the existence of $$\Psi$$-bounded solutions for a system of differential equations on $$\mathbb{R}$$. (English) Zbl 1175.34042
The author considers the inhomogeneous linear equation
$x'= A(t)x+f(t),\tag{1}$
where $$A(t)$$ is a continuous $$d\times d$$-real matrix function on $$\mathbb{R}$$ with values in $$\mathbb{R}^d$$. The author calls the solution $$\varphi:\mathbb{R}\to\mathbb{R}^d$$ $$\Psi$$-bounded if the function $$\Psi_\varphi$$ $$(\Psi=\text{diag}[\psi_1,\dots,\psi_d], \psi_i:\mathbb{R}\to(0,\infty))$$ is bounded on $$\mathbb{R}$$. By means of (exponential) dichotomy similar conditions the author finds necessary and sufficient conditions for the existence of $$\psi$$-bounded solutions of (1).
##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems
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