Kumam, Poom Fixed point theorems for nonexpansive mappings in modular spaces. (English) Zbl 1117.47045 Arch. Math., Brno 40, No. 4, 345-353 (2004). The present paper deals with the problem of the fixed point property of nonexpansive mappings in modular spaces. First, the basic geometric properties of modular spaces are recalled. In particular, the natural concepts of the geometry of Banach spaces like uniform convexity, compactness, Fatou property, etc., are extended to modular spaces. Then, using these concepts, some results formulated previously for normed spaces are extended to modular spaces. One of these extensions reads as follows.Theorem. Let \(X_\rho\) be a \(\rho\)-complete modular space whose modular \(\rho\) is convex and satisfies the Fatou property. If \(X_\rho\) is \(\rho _r\)-uniformly convex for all \(r\), then \(X_\rho\) has the fixed point property, i.e., every \(\rho\)-nonexpansive mapping of any \(\rho\)-bounded and \(\rho \)-closed convex subset \(C\subseteq X\) into itself has a fixed point in \(C\). Reviewer: Ondřej Došlý (Brno) Cited in 33 Documents MSC: 47H10 Fixed-point theorems 46A80 Modular spaces 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 46B20 Geometry and structure of normed linear spaces 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:\(\rho\)-nonexpansive mapping; \(\rho\)-normal structure; \(\rho \)-uniform normal structure; \(\rho_r\)-uniformly convex × Cite Format Result Cite Review PDF Full Text: EuDML EMIS