A continuation theorem is given for the periodic boundary problem \[ (\phi(u'))'= f(t,u,u'),\qquad u(0)-u(T)=u'(0)-u'(T)=0,\tag{1} \] where \(f:[0,T]\times \mathbb{R}^{2N}\to \mathbb{R}^N\) is a Carathéodory function and \(\phi\) is a homeomorphism between \(\mathbb{R}^N\) and the open unit ball of \(\mathbb{R}^N\) satisfying \[ \phi(x)= w(\| x\| )x\text{ for each }x\in \mathbb{R}^N, \] where \(w:\mathbb{R}^+\to \mathbb{R}^+\) is continuous. More precisely, if for some \(\Omega \subset C_T^1\), the equation \((\phi(u'))'= \lambda f(t,u,u')\) has no \(T\)-periodic solutions on \(\partial\Omega\) for \(\lambda \in (0,1)\), and \(F(a):=\int_0^T f(t,a,0)dt \neq 0\) on some appropriate \(\Omega_2\), with Brouwer degree \(deg_B(F,\Omega_2,0)\neq 0\), problem (1) has a solution in \(\Omega\). An example is given for which the continuation theorem applies, where \(\phi(u')'\) is the one-dimensional mean curvature operator \[ u\mapsto \left(\frac{u'}{\sqrt {1+u'}^{2}}\right)' . \]