Compactness and compact operators on Musielak-Orlicz spaces of multifunctions (Q1359062)

We introduce the notions of compact operator on Musielak-Orlicz spaces of multifunctions \(X_{d,S(X),\varphi}\). The aim of this note is to obtain some theorems on compact operators on \(X_{d,S(X),\varphi}\). We apply well-known notation on total boundedness of a set, compactness of a set and compactness of an operator to \(X_{d,S(X),\varphi}\). Also, we study some compactness criteria for the subsets of these spaces and we compare the space \(X_{d,S(X),\varphi}\) with the space \(B_L(L^\varphi(\Omega,\Sigma,\mu))\) of all nonempty, bounded and closed subsets of Musielak-Orlicz space \(L^\varphi(\Omega,\Sigma,\mu)\). Let \(X=\{F:\Omega\to 2^{\overline R}:F(t)\) is nonempty for every \(t\in\Omega\), compact for \(\mu\)-a.e. \(t\in\Omega\}\), \(\underline f(F)(t)= \inf_{x\in F(t)}x\), \(\overline f(t)=\sup_{x\in F(t)}x\) for every \(t\in\Omega\), \[ {\mathbf d}(F,G)(t)= \begin{cases} 0,\text{ if }F(t)= G(t)\\ \text{dist}(F(t),G(t)),\text{ if }F(t),\;G(t)\text{ are compact }\\ \infty,\text{ if }F(t)\neq G(t)\text{ and }F(t)\text{ or }G(t)\text{ is noncompact}\end{cases} \] for all \(F,G\in X\) and \(t\in\Omega\). -- The set of all simple multifunctions is denoted by \(S(X)\). \(X_{d,S(X)}=\{F\in X:\lim_{n\to\infty}{\mathbf d}(F,F_n)(t)= 0\) for \(\mu\)-a.e. \(t\in\Omega\), for some \(F_n\in S(X)\}\), \(X_\varphi=\{F\in X:\underline f(F)\), \(\overline f(F)\in L^\varphi(\Omega,\Sigma,\mu)\}\), \(X_{d,S(X),\varphi}=X_\varphi\cap X_{d,S(X)}\). At the end of this note, by Schauder fixed point principle, we get three fixed point theorems for operators from \(X_{d,S(X),\varphi}\) to \(X_{d,S(X),\varphi}\).