The first part of this paper deals with the autonomous problem \[ (\varphi(u'))'+g(u)=0,\quad u'(0)=0,\quad u'(1)=0. \tag{1} \] Here, \(\varphi\) is an odd homeomorphism asymptotic to a q-Laplacian at the origin and to a p-Laplacian at infinity. The function \(g\) is locally Lipschitzian and such that \(g(u)u >0\) for all \(u\neq 0\). The main assumptions concern the behaviour of \(g\) for \(u\) near \(-\infty\), \(0\) and \(+\infty\), i.e. \[ \lim_{u\to 0}\frac{g(u)}{\varphi(u)}=h \geq 0, \quad \lim_{u\to -\infty}\frac{g(u)}{\varphi(u)}=\beta > 0, \quad \lim_{u\to +\infty}\frac{g(u)}{\varphi(u)}=+\infty. \] Using time-map techniques and a generalized Fučík spectrum, the author obtains the existence of at least \(2k\) nontrivial solutions to (1), where \(k\) is obtained from \(h\) and \(\beta\), i.e. from the behaviour of the nonlinearity for \(u\to 0\) and \(u\to -\infty\). Examples illustrate the results. A second result concerns the problem \[ (|u'_i|^{p_i-2}u'_i)'+g(u_i)=h_i(t,u,u'), \quad u_i'(0)=0,\;u_i'(1)=0, \quad i=1,\ldots,N,\tag{2} \] which is a weakly-coupled system of problems such as (1). Within the same framework, conditions are given on the coupling so that problem (2) has two solutions with exactly \(k\) zeros. This follows from degree theory, a continuation theorem and estimates on the number of zeros of solutions to (1). The particular case \(p=2\) and \(N=1\) extends known results to the Neumann problem and nonlinearities with friction terms.