An abstract version of the Cauchy-Kovalevsky theorem for the Cauchy problem \(u'= A(t,u)\), \(u(0) =u_0\) is proved. The nonlinear operator \(A\) is acting in a scale of Banach spaces \(\{X_s\), \(\| \cdot \|_s\}_{0< s<1}\) with respect to \(u\); this is a standard assumption. In opposite to a lot of other papers the author does not assume a Lipschitz property of \(A\) in \(u\). Consequently, only existence results can be proved if one assumes e.g. compactness of \(A\) or a scale of compact embedded Banach spaces. The author uses the interesting idea to describe the behaviour of \(A\) in the scale of Banach spaces by the aid of measures of noncompactness. The obtained existence results are applied to Kirchhoff-type equations of higher order with \(B\)-analytic data. Reviewer's remark: It would be interesting to apply the methods of this paper to Cauchy problems \(u'= A(t,u)\), \(u(0) =u_0\), where \(A\) is not mapping \(X_s\) into \(X_{s'}\) but only a ball around the data of \(X_s\) into \(X_{s'}\), \(0