Advanced Search

In this paper we consider some spaces of differentiable multifunctions, in particular the generalized Orlicz-Sobolev spaces of multifunctions, we study completeness of them, and give some theorems.

We introduce the Musielak-Orlicz space of multifunctions $X_{m,\phi }$ and the set $S_F^{\phi }$ of φ-integrable selections of F. We show that some decomposable sets in Musielak-Orlicz space belong to $X_{m,\phi }$. We generalize Theorem 3.1 from [6]. Also, we get some theorems on the space $X_{m,\phi }$ and the set $S_F^{\phi }$.

We introduce the spaces $M_{Y,\varphi }^1$, $M_{Y,\varphi }^{o,n}$, ${\tilde{M}}_{Y,\varphi }^o$ and $M_{Y,𝐝,\varphi }^o$ of multifunctions. We prove that the spaces $M_{Y,\varphi }^1$ and $M_{Y,𝐝,\varphi }^o$ are complete. Also, we get some convergence theorems.

We introduced the notion of $(𝐗,dist,𝒱)$-boundedness of a filtered family of operators in the Musielak-Orlicz sequence space $X_{\varphi }$ of multifunctions. This notion is used to get the convergence theorems for the families of $𝐗$-linear operators, $𝐗$-dist-sublinear operators and $𝐗$-dist-convex operators. Also, we prove that $X_{\varphi }$ is complete.