By using the theory of strongly continuous cosine families and the properties of completely continuous maps, we study the existence of mild, strong, classical and asymptotically almost periodic solutions for a functional second order abstract Cauchy problem with nonlocal conditions.

It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma$-th order Cesàro mean $C_t^{\gamma }:=\gamma t^{-\gamma }{\int }_0^t{(t-s)}^{\gamma -1}T\left(s\right)ds$ and Abel mean $A_{\lambda }:=\lambda {\int }_0^{\infty }{e}^{-\lambda s}T\left(s\right)ds$ of the uniformly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in ℝ$, satisfy that (a) $\parallel C_t^{\gamma }\parallel \sim t^{k-\gamma }(t\to \infty )$ for all $0<\gamma \le k+1$; (b) $\parallel C_t^{\gamma }\parallel \sim t^{-1}(t\to \infty )$ for all $\gamma \ge k+1$; (c) $\parallel A_{\lambda }\parallel \sim \lambda (\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function ${\left(C\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by ${(iaI+N)}^2$.