伪压缩族公共不动点的复合隐格式迭代逼近问题

设H是一实Hillber空间,K是H之一非空间凸子集,设{Ti}Ni=1是N个Lipschitz伪压缩映象使得F=∩Ni=1F(Ti)≠0,其中F(Ti)={x∈K:Tix=x}并且{αn}n∞=1,{βn}∞n=10,1]是满足如下条件的实序列(i)∑∞n=1(1-αn)2= ∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)< ∞;(iv)(1-αn)L2<1,n1;(v)αn(1-βn)2 αnβn L(1-βn)]2<1,其中L1是{Ti}iN=1的公共Lipschitz常数,对于x0∈K,设{xn}n∞=1是由下列定义的复合隐格式迭代xn=αnxn-1 (1-αn)Tnyn,yn=βnxn (1-βn)Tnxn,其中Tn=TnmodN,则(i)limn→∞‖xn-p‖存在,对于所有的p∈F;(ii)limn→∞d(xn,f)存在,其中d(xn,F)=infp∈F‖xn-p‖;(iii)liminfn→∞‖xn-Tnxn‖=0.本文的结果推广并且改进H-K.Xu和R.G.Ori在2001年的结果和Osilike在2004年的结果,并且在这篇文章中,主要的证明方法也不同与H-K.Xu和Osilike的方法.

Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Pseudocontractive Maps

Let H be a real Hilbert space and let K be a nonempty closed convex subset of H.Let{T,}Ni=l be N Lipschitz pseudocontractive self-maps of K such that F= N∩i=1F(Ti)≠(),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-βn)＜+∞;(iv)(1-αn)L2＜1,()n≥1; (v)αn(1-βn)2+βn+L(l-βn]2＜1, where L≥1 is common Lipschitz constant of {Ti}Ni=1.For x0 ∈ K,let {xn}∞n=1 be the composite implicit process defined by {xn=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(xa,F)=inf p∈Fxn-p; (iii)limn→∞ infxn-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.