THE SET OF COMMON FIXED POINTS OF AN n-PARAMETER CONTINUOUS SEMIGROUP OF MAPPINGS

Throughout this paper, we denote by N, Z, Q and R the sets of all positive integers, all integers, all rational numbers and all real numbers, respectively. We put R+ = [0,∞)n and ej = (0, 0, · · · , 0, 0, (j) 1 , 0, 0, · · · , 0) ∈ R for j ∈ N with 1 ≤ j ≤ n. Let C be a subset of a Banach space E, and let T be a nonexpansive mapping on C, i.e., ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We know that T has a fixed point in the case that E is uniformly convex and C is bounded, closed and convex; see [4, 9]. See also [2, 3, 13] and others. We denote by F (T ) the set of fixed points of T . Let τ be a Hausdorff topology on E. A family of mappings {T (p) : p ∈ R+} is called an n-parameter τ -continuous semigroup of mappings on C if the following are satisfied: (sg 1) T (p+ q) = T (p) ◦ T (q) for all p, q ∈ R+; (sg 2) for each x ∈ C, the mapping p 7→ T (p)x from R+ into C is continuous with respect to τ . As a topology τ , we usually consider the strong topology of E. Also, a family of mappings {T (p) : p ∈ R+} is called an n-parameter τ -continuous semigroup of nonexpansive mappings on C (in short, an n-parameter nonexpansive semigroup) if (sg 1), (sg 2) and the following (sg 3) are satisfied: (sg 3) for each p ∈ R+, T (p) is a nonexpansive mapping on C. We know that an n-parameter nonexpansive semigroup {T (p) : p ∈ R+} has a common fixed point in the case that E is uniformly convex and C is bounded, closed and convex; see Browder [4]. Moreover, in 1974, Bruck [7] proved that an n-parameter nonexpansive semigroup {T (p) : p ∈ R+} has a common fixed point in the case that C is weakly compact, convex, and has the fixed point property for nonexpansive mappings.