The purpose of this work is to investigate the existence and regularity of solutions in the α-norm for some second order partial neutral functional differential equations in Banach
spaces with finite delay using fractional -power and the theory of the cosine family. As result, we obtain a generalization of work of Herman R. Henriquez et al. (Journal of Mathematics, Vol. 41, No. 6 (2011)) without alpha norm and regularity. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.

Cosine family, Mild and strict solutions, Neutral equations, α--norm, second order functional differential equations.