On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Abstract

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem $$ u^{\prime }_{i} (t) = {\ell }_{i} (u_{{i + 1}} )(t) + q_{i} (t)\quad (i = \overline{{1,n - 1}} ), $$ $$ u^{\prime }_{n} (t) = {\ell }_{n} (u_{1} )(t) + q_{n} (t), $$ $$ u_{j} (0) = u_{j} (\omega )\quad (j = \overline{{1,n}} ), $$ where ω > 0, l i : C([0, ω])→ L([0,ω]) are linear bounded operators, and q i ∈L([0, ω]).