Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function . It is shown that if maps a modular space into subsets of a modular space , then is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that we have .
We present some results concerning the problem , in , , where , , and is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.
This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?