เอกสารประชุมวิชาการระดับขาติมหาวิทยาลัยทักษิณ ครั้งที่ 28 2561

1091 งานวิจัยและนวัตกรรมเพื่อสังคมที่มั่นคง มั่งคั่ง และยั่งยืน (Research and Innovation for Social Stability, Prosperity and Sustainability) Introduction Let 9 be a Banach space with the norm ˜ . Let ^ `  0 1 # x 9 : x be the unit sphere of 9 . A Banach space 9 is said to be smooth if the limit  0 U x UZ x MJN U (1) exists for any  0 x Z # . A Banach space 9 is said to be uniformly smooth if the limit (1) is attained uniformly for x Z in u 0 0 # # . A Banach space E is said to be strictly convex if 1 2 x Z for all  x Z 9 with 1 x Z and z x Z (see [1] for more details). For a sequence ^ ` O x of a Banach space 9 and a point  x 9 , the weak convergence of ^ ` O x to x and the strong convergence of ^ ` O x to x are denoted by O x x and  O x x respectively. Lemma 1. [2] Let 9 be a Banach space and ^ `  d ! 0 1 S # x 9 : x S S be a closed ball of 9 . Then 9 is uniformly convex if and only if there exists a continuous, strictly increasing and convex function > > f  f 0 0 H with 0 0 H such that O O O O O O d 2 2 2 1 1 1 x Z x Z H x Z , for all  0 S x Z # and > @ O  0 1 . Lemma 2. [3] Let 9 be a uniformly convex Banach space and D O be a sequence in > @ D D 1 for some D  0 1 . Suppose that sequences ^ ` O x and ^ ` O Z in 9 are such that f f d d O O O O MJNTVp x D MJNTVp Z D and D D f 1 O O O O O MJNTVp x Z D for some t 0 D . Then f 0 O O O MJN x Z . Let C be a closed convex subset of a Banach space 9 , Recall that a point  x C is a fixed point of a mapping  T : C C if Tx x . Denoted by F T the set of fixed point of T . A point p in C is said to be a strongly asymptotic fixed point of T [4] if C contains a sequence ^ ` O x which converges strong to p such that f 0 O O O MJN x Tx . The set of strong asymptotic fixed point of T denoted by F T .