On the Xθ(⋠)-valued function space: Definition, property and applications

This paper deals with the Xθ(⋅)-valued function space. After introducing the notions of bounded topological lattice and Banach space net, we give the definitions of two different Xθ(⋅)-valued spaces: Lp(⋅)(I,Xθ(⋅)) and C−(I,Xθ(⋅)), and make some discussions on their properties, including completeness and imbedding relations. At the end of the paper, a class of abstract semilinear evolution equations with domain-varying nonlinearities are taken into account.