Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Pseudocontractive Maps
Meijuan Shang · Acta Analysis Functionalis Applicata · 2007
Let H be a real Hilbert space and let K be a nonempty closed convex subset of H.Let {Ti}Ni=1 be N Lipschitz pseudocontractive self-maps of K such that F=∩Ni=1F(Ti)≠ф,where F(Ti)={x∈K:Tix=x} and let {αn}∞n=1 and {βn}∞n=1[0,1] be two real sequences satisfying the conditions(i)∑∞n=1(1-αn)2=+∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)+∞;(iv)(1-αn)L21,n1;(v)αn(1-βn)2+αn[βn+L(1-βn)]21,where L1 is common Lipschitz constant of {Ti}Ni=1.For x0∈K,let {xn}∞n=1 be the composite implicit process defined byxn=αnxn-1+(1-αn)Tnyn,yn=βnxn+(1-βn)Tnxn,where Tn=Tn mod N,then(i)limn→∞‖xn-p‖ exists,for all p∈F;(ii)limn→∞d(xn,F)exists,where d(xn,F)=infp∈F‖xn-p‖;(iii)lim infn→∞‖xn-Tnxn‖=0.The result of this paper improve and extend the results of Osilike in 2004 and that of H.K.Xu and R.G.Ori in 2001.In this paper,the methods of proof of main results are also different from that of Osilike and Xu.