On the periodic boundary value problem for first-order functional-differential equations (Q1768982)

The authors consider the periodic boundary value problem \[ u'(t)=\ell(u)(t)+F(u)(t),\qquad t\in[a,b], \quad u(a)=u(b), \] where \(\ell:C([a,b];\mathbb{R})\to L([a,b];\mathbb{R})\) is a linear bounded operator and \(F:C([a,b];\mathbb{R})\to L([a,b];\mathbb{R})\) is a continuous operator, not necessarily linear. The conditions guaranteeing the solvability and unique solvability of the problem considered are established even in the case when \(\ell\) is not a monotone operator. All results are concretized for the differential equation with deviating arguments. The examples verifying the optimality of the results obtained are given.