Summary: We consider the abstract semilinear nonlocal second-order Cauchy problem \[ \begin{gathered} u''(t)= Au(t)+ f(t,u(t), u'(t)),\quad t\in (0,T],\qquad u(0)= x_0,\\ u'(0)+ \sum^p_{i=1} h_i u(t_i)= x_1,\end{gathered} \] where \(A\) is a linear operator from a real Banach space \(X\) into itself, \(u:[0,T]\to X\), \(f:[0,T]\times X^2\to X\), \(x_0,x_1\in X\), \(h_i\in\mathbb{R}\) \((i= 1,2,\dots, p)\) and \(0< t_1< t_2<\cdots< t_p\leq T\).
We prove two theorems on the existence and uniqueness of mild and classical solutions. For this purpose we apply the theory of strongly continuous cosine families of linear operators in a Banach space. We also apply the Banach contraction theorem and the Bochenek theorem.