An optimal condition for the uniqueness of a periodic solution for linear functional differential systems

S. Mukhigulashvili, Academy of Sciences of the Czech Republic, Brno, Czech Republic
I. Grytsay, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

E. J. Qualitative Theory of Diff. Equ., No. 59. (2009), pp. 1-12.

Abstract: Unimprovable effective efficient conditions are established for the unique solvability of the periodic problem
$$
\begin{aligned}
u'_i (t)&=\sum\limits_{j=2}^{i+1} \ell_{i,j}(u_{j})(t) + q_i(t) \qquad \text{for} \quad 1 \leq i\leq n-1,\\
u'_n (t)&=\sum\limits_{j=1}^{n} \ell_{n,j}(u_{j} )(t) + q_n(t),\\
u_j (0)& = u_j (\omega) \qquad \text{for} \quad 1 \leq j\leq n,
\end{aligned}
$$
where $\omega >0$, $\ell_{ij}:C([0,\omega])\to L([0,\omega])$ are linear bounded operators, and
$q_i \in L([0,\omega])$.

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