Abstract
Abstract. In this paper, we establish the strong convergence for theMann iterative scheme associated with uniformly continuous pseudocon-tractive mappings in real Banach spaces. Moreover, our technique ofproofs is of independent interest. 1. Introduction and preliminariesLet Ebe a real Banach space and Kbe a nonempty convex subset of E.Let Jdenote the normalized duality mapping from Eto 2 E de ned byJ(x) = ff 2E : hx;fi= jjxjj 2 and jjfjj= jjxjjg;where E denotes the dual space of Eand h;idenotes the generalized dualitypairing. We shall denote the single-valued duality map by j.Let T: D(T) ˆE!Ebe a mapping with domain D(T) in E:De nition 1.1. Tis said to be Lipschitzian if there exists L>1 such that forall x;y2D(T)kTx TykLkx yk:De nition 1.2. T is said to be nonexpansive if for all x, y 2D(T), thefollowing inequality holds:kTx Tykkx ykfor all x;y2D(T):De nition 1.3. T is said to be pseudocontractive if there exists j(x y) 2J(x y) such thathTx Ty;j(x y)ikx yk 2 for all x;y2D(T); n1:
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