Global continuum of solutions for systems of ODEs with periodic boundary conditions and generalized variable-exponent operators

We study nonlinear systems of boundary-value problems in one dimension of the form$ (\mathcal S(t, u(t)'))' = f(t, u(t), u'(t), \lambda), $subject to periodic boundary value conditions. The function $ \mathcal S:[0, T]\times\mathbb{R}^N\to\mathbb{R}^N $ is continuous and satisfies $ \mathcal S(0, x) = \mathcal S(T, x) $ for all $ x\in\mathbb{R}^N $, and the function $ f $ is Carathéodory. Furthermore, we assume $ \mathcal{S} $ satisfies some monotonicity and coercivity conditions, thereby defining a class of generalized variable-exponent operators that strictly contains the Musielak–Orlicz case. Using an abstract continuation framework and the Leray–Schauder degree, we establish the existence of a global continuum of nontrivial solutions depending on the parameter $ \lambda $. Our approach builds on a degree-theoretic alternative of Fitzpatrick, Massabò, and Pejsachowicz, reducing the evaluation of the Leray–Schauder degree to a Brouwer degree. This work shows how monotonicity and coercivity conditions in one dimension allow for continuum of solutions in a setting strictly more general than the Musielak–Orlicz framework. Furthermore, our results are new even if $ \mathcal S $ is independent of $ t $.