Necessary and sufficient condition for mann iteration converges to a fixed point of Lipschitzian mappings

Suppose that e is a real normed linear space,c is a nonempty convex subset of e,t:c→c is a lipschitzian mapping,and x ∈c is a fixed point of t. for given x0∈c,suppose that the sequence {xn}⊂c is the mann iterative sequence defined by xn+1=(1-αn)x n+αntxn,n≥0,where {αn} is a sequence in [0,1],∑n=0 ∞αn 2<∞,∑n=0 ∞αn=∞. we prove that the sequence {xn} strongly converges to x if and only if there exists a strictly increasing function φ:[0,∞)→[0,∞) with φ(0)=0 such that lim supn→∞infj(xn-x) ∈j(xn-x){〈txn-x,j(xn-x )〉-∥xn-x ∥2+φ(∥xn-x ∥)}≤0. © 2012 chang-he xiang et al.