Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the $p$-Laplace operator, the $\varphi$-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator ${\left(S(t,u^{\prime })\right)}^{\prime }$, where ${}^{\prime }=\frac{d}{dt}$ and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator ${\left(S(t,u^{\prime })\right)}^{\prime }$ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in $C^1$ spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.