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   Necessary and sufficient condition for mann iteration converges to a fixed point of Lipschitzian mappings  
   
نویسنده xiang c.-h. ,zhang j.-h. ,chen z.
منبع journal of applied mathematics - 2012 - دوره : 2012 - شماره : 0
چکیده    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.
آدرس college of mathematics,chongqing normal university, China, school of management,shandong university,shandong, China, college of mathematics,chongqing normal university, China
 
     
   
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