Some results in the α-norm for some nonlinear second order differential equation with finite delay in Banach space

Abstract

This paper investigates the existence, regularity and compactness property in the α-norm for some second order nonlinear differential equations with finite delay in Banach spaces. The theory of the cosine family, the contraction principle, and Schauder’s fixed point theorem are used to establish global existence, continuous dependence on initial data, blowing up of solutions, local existence, and compactness of the flow. Furthermore, some sufficient conditions are given to ensure the regularity of the solutions. Finally, an example is given to illustrate the theoretical results.