On a boundary value problem for scalar linear functional differential equations

Abstract

Theorems on the Fredholm alternative and well-posedness of the linear boundary value problem $u^{\prime }\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right)$, $h\left(u\right)=c$, where $\ell :C\left(\left[a,b\right];ℝ\right)\to L\left(\left[a,b\right];ℝ\right)$ and $h:C\left(\left[a,b\right];ℝ\right)\to ℝ$ are linear bounded operators, $q\in L\left(\left[a,b\right];ℝ\right)$, and $c\in ℝ$, are established even in the case when $\ell$ is not a strongly bounded operator. The question on the dimension of the solution space of the homogeneous equation $u^{\prime }\left(t\right)=\ell \left(u\right)\left(t\right)$ is discussed as well.