The authors use a fixed point theorem due to R.I. Avery and A. C. Peterson [Comput. Math. Appl. 42, 313–322 (2001; Zbl 1005.47051)] in order to establish the existence of at least three positive solutions for differential equations of the type \[ \bigl(\varphi_{p}(u')\bigr)'+q(t)f(t,u,u')=0, \; t \in (0,1), \] subject to the boundary conditions \[ \begin{gathered} u(0)=\sum_{i=1}^{n-2}\alpha_{i}u(\xi_{i}),\quad u'(1)=\sum_{i=1}^{n-2}\beta_{i}u'(\xi_{i}),\\ u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\quad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}). \end{gathered} \] Here \(f\) and \(q\) are nonnegative continuous functions, \(\varphi_{p}(s)=| s| ^{p-2}\), \(p>1\), \(0 < \xi_{1} < \xi_{2} < \cdots <\xi_{n-2} < 1\) and \(\alpha_{i}\), \(\beta_{i}\) are nonnegative parameters such that \(\sum_{i=1}^{n-2}\alpha_{i}<1\) and \(\sum_{i=1}^{n-2}\beta_{i}<1\).