| Abstract: |
| We consider nonlinear systems of boundary-value problems in one dimension of the form
$$
(\mathcal S(t,u'))'=f(t,u,u',\lambda),
$$
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 $x\in\mathbb R^N$, the function $f$ is Caratheodory. 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 global continuum of nontrivial solutions depending on the parameter $\lambda$. Our approach builds on a degree-theoretic alternative of Fitzpatrick, Massabo, 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 $t$. |
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