Abstract-valued Orlicz spaces of range-varying type

This paper mainly deals with the abstract-valued Orlicz spaces of range-varying type. Using notions of Banach space net and continuous modular net etc., we give definitions of Lϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)), and discuss their geometrical properties as well as the representation of $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅))*. We also investigate some functionals and operators on Lϱθ(⋅)(I, Xθ(⋅)), giving expression for the subdifferential of the convex functional generated by another continuous modular net. After making some investigations on the Bochner-Sobolev spaces W1, ϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)), and the intersection space $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $(I, Xθ(⋅)) ∩ Lφϑ(⋅)(I, Vϑ(⋅)), a second order differential inclusion together with an anisotropic nonlinear elliptic equation with nonstandard growth are also taken into account.