Abstract
AbstractIn the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations $$ u_{i}^{(m_{i})}(t)=\ell _{i}(u_{i+1}) (t)+ q_{i}(t) \quad (i= \overline{1, n}) \text{ for } t\in I:=[a, b] $$ui(mi)(t)=ℓi(ui+1)(t)+qi(t)(i=1,n‾) for t∈I:=[a,b] and $$ u_{i}^{(m_{i})} (t)=F_{i}(u) (t)+q_{0i}(t) \quad (i = \overline{1, n}) \text{ for } t\in I $$ui(mi)(t)=Fi(u)(t)+q0i(t)(i=1,n‾) for t∈I under the periodic boundary conditions $$ u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij} \quad (i=\overline{1, n},j= \overline{0, m_{i}-1}), $$ui(j)(b)−ui(j)(a)=cij(i=1,n‾,j=0,mi−1‾), where $u_{n+1}=u_{1} $un+1=u1, $m_{i}\geq 1$mi≥1, $n\geq 2 $n≥2, $c_{ij}\in R$cij∈R, $q_{i},q_{0i}\in L(I; R)$qi,q0i∈L(I;R), $\ell _{i}:C^{0}_{1}(I; R)\to L(I; R)$ℓi:C10(I;R)→L(I;R) are monotone operators and $F_{i}$Fi 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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
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