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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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