Extremal solutions and comparison principle for nonlinear integro-differential equations in a Banach space

In this paper, we consider an initial value problem for nonlinear integro-differential equations in a Banach space. First, we give a comparison result between the under and over functions and some comparison principles. Then, using these results and the Kuratowski measure of noncompactness, we establish the existence theorem of extremal solutions between the under and over functions, and prove that there exists a unique solution between the lower and upper solutions under an additional Lipschitz's condition. Project supported by the Natural Science Foundation of Shandong Province.