On asymptotic centres and fixed point theorems for set-valued nonexpansive mappings (Q1065341)

First a fixed point theorem concerning set-valued nonexpansive and weakly inward mappings on unbounded domain is proved by imposing a mild boundedness condition on the mappings. This theorem extends a result of \textit{W. A. Kirk} and \textit{W. O. Ray} [Stud. Math. 64, 125-127 (1979; Zbl 0412.47033)] for single valued mapping. Next a fixed point theorem of a set-valued mapping on nonconvex domain is proved. \textit{K. Yanagi} [Pac. J. Math. 87, 233-240 (1980; Zbl 0408.47042)] has proved a fixed point theorem in this direction where the domain of the mapping is weakly compact starshaped subset of a uniformly convex Banach space. With Lami Dozo's (1983) new concepts of asymptotic centre and radius, the authors have proved a similar theorem replacing the uniform convexity assumption on the space with some geometric assumption on the domain.