WEAK AND STRONG CONVERGENCE THEOREMS FOR COMMUTATIVE FAMILIES OF POSITIVELY HOMOGENEOUS NONEXPANSIVE MAPPINGS IN BANACH SPACES WATARU TAKAHASHI, NGAI-CHING WONG, AND JEN-CHIH YAO Abstract. In this paper, we first prove a weak convergence theorem by Mann’s iteration for a commutative family of positively homogeneous non- expansive mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong con- vergence theorem for such a family of the mappings. These results are new even if the mappings are linear and contractive. 1. Introduction Let N be the set of positive integers. Let E be a real Banach space with norm ∥·∥ and let C be a closed and convex subset of E. Let T be a mapping of C into itself. We denote by F (T ) the set of fixed points of T . A mapping T : C → C is called nonexpansive if ∥Tx − Ty∥≤∥x − y∥ for all x, y ∈ C. Let C be a closed convex cone of E. A mapping T : C → C is called positively homogeneous if T (αx)= αT (x) for all x ∈ C and α ≥ 0. From Reich [27] we know a weak convergence theorem by Mann’s iteration [20] for nonexpansive mappings in a Banach space: Let E be a uniformly convex Banach space with a Fr´ echet differentiable norm and let T : C → C be a nonexpansive mapping with F (T ) ̸= ∅. Define a sequence {x n } in C by x 1 = x ∈ C and x n+1 = α n x n + (1 − α n )Tx n , ∀n ∈ N, where {α n } is a real sequence in [0, 1] such that ∑ ∞ n=1 α n (1 − α n )= ∞. Then, {x n } converges weakly to z ∈ F (T ). In this theorem, the fixed point z is characteraized under any projections in a Banach space. Recently, Takahashi and Yao [45] proved a theorem for positively homogeneous nonexpansive mappings in a Banach space. In the theorem, the limit of weak convergence is characteraized by using a sunny generalized nonexpansive retraction in the sense of Ibaraki and Takahashi [9]. On the other hand, Nakajo and Takahashi [25] proved a strong convergence theorem for nonexpansive mappings in a Hilbert space by using the hybrid method in mathematical programming: Let C be a closed and convex subset of a Hilbert space H and let T : C → C be a nonexpansive mapping with F (T ) ̸= ∅. Let {α n } be a real sequence in [0, 1] such 2000 Mathematics Subject Classification. 47H05, 47H09, 47H20. Key words and phrases. Banach space, nonexpansive mapping, fixed point, generalized non- expansive mapping, hybrid method, Mann’s iteration. 1