Let \(E\) be a Banach space and let \(I= [0,a]\). Denote \(B= \{x\in E:\| x\|\leq b\}\). For \(t\in(0, a)\) and \(r>0\) put \(I_{tr}= (t- r,t+r)\cap I\). Let \(f: I\times B\to E\) be a bounded continuous function. Consider the Cauchy problem of the form: (1) \(x^{(m)}= f(t,x)\), (2) \(x(0)= 0\), \(x'(0)= x_1,\dots, x^{(m- 1)}(0)= x_{m- 1}\), where \(x_1,\dots, x_{m-1}\in E\). Assume that \(u\) is a continuous function on \((0,a]\) such that \(u(t)>0\) for \(t>0\), \(u(0)=\cdots= u^{(m- 1)}(0)= 0\), \(u^m(t)\) is positive and Lebesgue integrable and \[ \lim_{t\to 0+} \alpha(f(I_{tr}\times X))\leq (u^{(m)}(t)/u(t)) \alpha(X) \] for \(X\subset B\), where \(\alpha\) denotes the Kuratowski measure of noncompactness. Under some additional technical assumptions it is shown that the problem (1)–(2) has a local solution.