Integro-differential equations embodying powers of a differential operator

We establish solvability and correctness criteria for two Fredholm type linear integro-differential operators $B_2, B_4$ encompassing up to second and fourth powers, respectively, of a differential operator $\widehat{A}$ with a known inverse $I=\widehat{A}^{-1}$. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the differential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-differential operators are solved.