International Journal of Modern Mathematics 3(2) (2008), 135–151 c 2008 Dixie W Publishing Corporation, U. S. A. Weak and Strong Convergence for Fixed Points of Nearly Asymptotically Non-Expansive Mappings D. R. Sahu and Ismat Beg Received June 20, 2006; Revised August 1, 2007 Abstract The concept of S-demiclosedness of nonlinear mappings is introduced and analyzed for non-expansive and asymptotically non-expansive mappings. As applications, we prove weak and strong convergence theorems of Mann iteration to fixed points for non- Lipschitzian nearly asymptotically non-expansive and asymptotically non-expansive mappings. Our results improve various celebrated results of fixed point theory in the context of S-demiclosedness principle. Keywords: Demiclosedness principle, fixed point, non-expansive mapping, Banach space. 2000 Mathematics Subject Classification: 47H09, 46B20, 47H10. 1 Introduction Let C be a nonempty subset of a normed space X and T : C → C a mapping. A sequence {x n } in C is said to be an approximating fixed point sequence if lim n→∞ kx n - Tx n k =0. The mapping T is said to be Lipschitzian if for each n ∈ N, there exists a positive number k n such that kT n x - T n yk≤ k n kx - yk for all x, y ∈ C. A Lipschitzian mapping T is said to be uniformly k-Lipschitzian if k n = k for all n ∈ N and asymptotically non-expansive (cf. [9]) if k n ≥ 1 for all n ∈ N with lim n→∞ k n =1. Clearly every non-expansive mapping T (i.e., kTx - Tyk≤kx - yk for all x, y ∈ C) is asymptotically non-expansive with sequence {1} and every asymptotically non-expansive mapping is uniformly k-Lipschitzian with k = sup n∈N k n . The class of asymptotically non-expansive mappings was introduced by Goebel and Kirk [9] as an important generalization of the class of non-expansive mappings. The exis- tence of fixed points of asymptotically non-expansive mappings was proved by Goebel and Kirk [9] as below: