Existence results for some nonlocal partial integrodifferential equations without compactness or equicontinuity

In this work, we provide a new approach to address the solvability of a class of partial integrodifferential equations with nonlocal conditions and two linear parts, the first one being the generator of a strongly continuous semigroup, while the second one is of integral type. We introduce new ideas to handle the measure of noncompactness of a class of integral operators. The methods developed in this paper are of independent interest and enable us to evolve a general existence theory for a large class integrodifferential equations and to give positive answers to some questions raised in earlier works. In our considerations the semigroup generated by the first linear part is not necessarily compact or equicontinuous, which is an extra interesting feature. To demonstrate the usefulness of our results we illustrate three examples.