In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations $$ u_{i}^{(m_{i})}(t)=\ell _{i}(u_{i+1}) (t)+ q_{i}(t) \quad (i= \overline{1, n}) \text{ for } t\in I:=[a, b] $$ and $$ u_{i}^{(m_{i})} (t)=F_{i}(u) (t)+q_{0i}(t) \quad (i = \overline{1, n}) \text{ for } t\in I $$ under the periodic boundary conditions $$ u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij} \quad (i=\overline{1, n},j= \overline{0, m_{i}-1}), $$ where $u_{n+1}=u_{1} $, $m_{i}\geq 1$, $n\geq 2 $, $c_{ij}\in R$, $q_{i},q_{0i}\in L(I; R)$, $\ell _{i}:C^{0}_{1}(I; R)\to L(I; R)$ are monotone operators and $F_{i}$ are the local Caratheodory’s class operators. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved.

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高阶泛函微分方程系统周期问题的Lasota-Opial型条件

在本文中，我们研究了高阶泛函微分方程系统的可解性和唯一可解性的问题$$ u_ {i} ^ {（m_ {i}）}（t）= \ ell _ {i}（u_ { i + 1}）（t）+ q_ {i}（t）\ quad（i = \ overline {1，n}）\ text {for} t \ in I：= [a，b] $$和$$ u_ {i} ^ {（m_ {i}）}（t）= F_ {i}（u）（t）+ q_ {0i}（t）\ quad（i = \ overline {1，n}）\ text {对于} t \ in I $$在周期性边界条件$$ u_ {i} ^ {（j）}（b）-u_ {i} ^ {（j）}（a）= c_ {ij} \ quad （i = \ overline {1，n}，j = \ overline {0，m_ {i} -1}），$$其中$ u_ {n + 1} = u_ {1} $，$ m_ {i} \ geq 1 $，$ n \ geq 2 $，$ c_ {ij} \ in R $，$ q_ {i}，q_ {0i} \ in L（I; R）$，$ \ ell _ {i}：C ^ {0} _ {1}（I; R）\至L（I; R）$是单调运算符，而$ F_ {i} $是本地Caratheodory的类运算符。在某种意义上，本文获得了可以保证线性问题具有唯一可解性的最佳条件，