Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings (Q2045206)

Let \(X\) be an arbitrary linear space. A functional \(\rho: X\rightarrow [0,\infty]\) is called a convex modular if for any \(x,y\in X\), \begin{itemize} \item \(\rho (x)=0\) if and only if \(x=0\); \item \(\rho(\alpha x)=\rho(x)\) for every scalar \(\alpha\) with \(| \alpha |=1\); \item \(\rho (\alpha x+\beta y )\leq \alpha \rho(x)+\beta \rho(y)\) if \(\alpha, \beta \geq 0\) and \(\alpha +\beta =1\). \end{itemize} Then, the corresponding modular space is defined as \[ X_\rho:=\{ x\in X: \rho(\frac{1}{\lambda} x) < \infty, \mbox{for some}\,\, \lambda > 0 \}.\] In particular, the formula \(\|x\|:= \inf \{\lambda >0: \rho(\frac{1}{\lambda} x)<1\}\) defines a norm in \(X_\rho\) which is called the Luxemburg norm. Modular spaces were introduced by \textit{H. Nakano} [Modulared semi-ordered linear spaces. Tokyo: Maruzen (1950; Zbl 0041.23401); Topology of linear topological spaces. Tokyo: Maruzen (1951)] and [Nakano, H.: Topology of Linear Topological Spaces. Maruzen Co. Ltd, Tokyo (1951)] and developed by \textit{J. Musielak} and \textit{W. Orlicz} [Stud. Math. 18, 49--65 (1959; Zbl 0086.08901)]. This paper is concerned with some fixed point results for multivalued mappings in modular spaces. For this purpose, the authors study the uniform noncompact convexity, a geometric property for modular spaces, which is similar to nearly uniform convexity in the Banach spaces setting. Moreover, in order to investigate the fixed point property in modular spaces, they introduce the property (R) (Definition 2.12) which can be considered as a modular counterpart of the reflexivity in the sense of Banach spaces. In Section 4, ``uniform noncompact convexity'' is introduced in modular spaces as well as Example 4.7 shows that not every uniformly \(\rho\)-noncompact convex space is \(\rho\)-uniformly convex. The main fixed point theorem of the paper, Theorem 6.7, extends Theorem 6.2 which is in fact itself the modular version of the Kirk-Massa theorem.