Generalizations to some Integro-differential equations embodying powers of a differential operator

The abstract equations containing the operators of the second, third and fourth degree are investigated in this work.
The necessary conditions for the solvability of the abstract equations, containing the operators of the second and fourth degree, are proved without using linear independence of the vectors included in these equations. Previous authors have essentially used the linear independence of the vectors to prove the necessary solvability condition.
The present paper also gives the correctness criterion for the abstract equation, containing the operators of the third degree with arbitrary vectors, and its exact solution in terms of these vectors in a Banach space.
The theory presented here, can be useful for investigation of Fredholm integro-differential equations embodying powers of an ordinary differential operator or a partial differential operator.