On Firmly Nonexpansive Mappings

It is shown that any A-firmly, 0 < A < 1 , nonexpansive mapping T: C —> C has a fixed point in C whenever C is a finite union of nonempty, bounded, closed convex subsets of a uniformly convex Banach space. Let C be a nonempty subset of a Banach space X, and let X £ (0, 1). Then a mapping T: C —> X is said to be X-firmly nonexpansive if (1) \\Tx Ty\\ < ||(1 X)(x y)+X(Tx Ty)\\ for all x, y £ C. In particular, if (1) holds for every X £ (0, 1), then T is said to be firmly nonexpansive. It is clear that every A-firmly nonexpansive mapping is nonexpansive; i.e., for each x and y in C, we have \\Tx Ty\\ < II* y ||. Conversely, to each nonexpansive T: C —> C one can associate a firmly nonexpansive mapping with the same fixed-point set whenever C is closed and convex [3]. Moreover, from the point of view of fixed-point theory for the class of all closed convex subsets C, firmly nonexpansive mappings T: C -> C do not exhibit better behavior than nonexpansive mappings in general [2]. However, this behavior is completely different in the class of nonconnected subsets C. Indeed, in this note we prove the following result: Theorem. Let X be a uniformly convex Banach space, let C = \j"k=x Ck be a union of nonempty, bounded, closed convex subsets Ck of X, and suppose T: C —> C is X-firmly nonexpansive for some X £ (0, 1). Then T has a fixed point in C. This theorem is no longer true if T is merely nonexpansive, even in onedimensional space X = R. For example, if C = [-2, -1] U [1, 2], then the mapping T: C —> C defined by Tx = —x is nonexpansive and fixed-point free. On the other hand, if Cx = ■■■ = Cn = C, then the theorem is true for each nonexpansive mapping T : C —► C, which is the celebrated result of Browder [1], Göhde [4], and Kirk [5]. More generally, the same result is also valid when rt=1 q * 0 [6]Received by the editors April 23, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H09, 47H10.