Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions. (Q1426068)

The authors prove a fixed point theorem for a nonexpansive (compact convex-valued) multimap satisfying the inwardness condition on a bounded closed convex subset of a Banach space whose characteristic of noncompact convexity is less than~1. Reviewer's remarks: Some conditions of the theorem seem to be superfluous: (1) the bounded range of a multimap is the consequence of the boundedness of the domain and nonexpansivity of a multimap; (2) \(1-\chi\)-contractivity of a multimap also follows from its nonexpansivity. (The authors treat this as an open problem but it is the corollary of Proposition 2.2.2 of the book of \textit{M. Kamenskii}, the reviewer and \textit{P. Zecca} [``Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces'' (de Gruyter Series in Nonlinear Analysis and Applications 7, Berlin) (2001; Zbl 0988.34001)].