On the behavior of solutions of some periodic differential equations

This paper investigates the conditions that ensure the existence of unique periodic solutions for ordinary differential equations of the form $u^{\prime }=A\left(t\right)u+f\left(t\right)$, where both $A\left(t\right)$ and $f\left(t\right)$ are T-periodic. Particular emphasis is placed on understanding the behavior of bounded solutions on $R$. Moreover, we examine cases where the equation's coefficients are nonnegative.