Abstract-valued Orlicz spaces of range-varying type

Abstract

Abstract 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 L ϱ θ(⋅) (I, X θ(⋅)) and L + ϱ θ ( ⋅ ) $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $ (I, X θ(⋅)), and discuss their geometrical properties as well as the representation of L + ϱ θ ( ⋅ ) $\begin{array}{} L_{+}^{\varrho_{\theta(\cdot)}} \end{array} $ (I, X θ(⋅))*. We also investigate some functionals and operators on L ϱ θ(⋅) (I, X θ(⋅)), giving expression for the subdifferential of the convex functional generated by another continuous modular net. After making some investigations on the Bochner-Sobolev spaces W 1, ϱ θ(⋅) (I, X θ(⋅)) and W per 1 , ϱ θ ( ⋅ ) $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $ (I, X θ(⋅)), and the intersection space W per 1 , ϱ θ ( ⋅ ) $\begin{array}{} W_{\textrm{per}}^{1,\varrho_{\theta(\cdot)}} \end{array} $ (I, X θ(⋅)) ∩ L φ ϑ(⋅) (I, V ϑ(⋅)), a second order differential inclusion together with an anisotropic nonlinear elliptic equation with nonstandard growth are also taken into account.