One class of singular linear functional differential equations (Q1759244)

Consider the functional differential equation \[ x'(t)+ a(t)x(t) +(Tx)(t) = f(t), 0\leq t\leq b, \] where \(a(t)\) has the form \(a(t) = \frac{k}{t} + \tilde{a}(t)\) with certain conditions, and the linear operator \(T\) from the space of absolutely continuous functions on \([0, b]\) to the Banach space \(L^p[0, b]\) is completely continuous. Conditions are obtained for the Fredholm property and the solvability of this equation.