R. Hakl and S. Mukhigulashvili

On a Periodic Boundary Value Problem for Third Order Linear Functional Differential Equations

abstract:
Unimprovable effective sufficient conditions are established for the unique solvability of the periodic problem
\begin{align*}
u'''(t) & =\sum_{i=0}^2\ell_i(u^{(i)})(t)+q(t)  \;\;\mbox{for}\;\; 0\leq t\leq\omega, \\ u^{(j)}(0) & =u^{(j)}(\omega)\;\; (j=0,1,2),
\end{align*}
where $\omega>0$, $\ell_i: C\big([0,\omega]\big)\to L\big([0,\omega]\big)$ are linear bounded operators, and $q\in L\big([0,\omega]\big)$.

Mathematics Subject Classification: 34K06, 34K10

Key words and phrases: Linear functional differential equation, ordinary differential equation, periodic boundary value problem