渐近非扩张非自映射的收敛定理(英文)

设E是实的一致凸Banach空间,K是E的一个非空闭凸集,P是E到K上的非扩张的保核收缩映射.设T1,T2,T3:K→E分别是具有数列{hn},{ln},{kn}1,∞)的渐近非扩张非自映射,使得sum (hn-1) from n=1 to ∞<∞,sum ((ln-1)) from n=1 to ∞<∞及sum (n=1(kn-1) from n=1 to ∞<∞,且F=F(T1)∩F(T2)∩F(T3)={x∈K:T1x=T2x=T3x}≠Ф.定义迭代序列{xn}:x1∈K,xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1zn),zn=P((1-γn)xn+γnT3(PT3)n-1xn),其中{αn},{βn},{γn}ε,1-ε],ε是大于零的实数.(i)如果T1,T2,T3中有一个是全连续的或者半紧的,则{xn}强收敛于某一点q∈F;(ii)如果E具有Frechet可微范数或者满足Opial’s条件或者E的对偶空间E~*具有Kadec-Klee性质,则{xn}弱收敛于某一点q∈F.

Convergence Theorems for Nonself Asymptotically Nonexpansive Mappings

Suppose that K is a nonempty closed convex subset of a real uniformly convex Banach space E with P as a nonexpansive retraction.Let T1,T2,T3∶K→E be three nonself asymptotically nonexpansive mappings with sequences {hn},{ln},{kn}1,∞) such that sum (hn-1) from n=1 to ∞<∞,sum ((ln-1)) from n=1 to ∞<∞ and sum (n=1(kn-1) from n=1 to ∞<∞,respectively and F=F(T1)∩F(T2)∩F(T3)={x∈K∶T1x=T2x=T3x}≠Ф.Define the sequence {xn} by x1∈K,xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1zn),zn=P((1-γn)xn+γnT3(PT3)n-1xn),n≥1,where {αn},{βn} and {γn} are three real sequences in ε,1-ε] for some ε>0.(i) If one of T1,T2 and T3 is completely continuous or demicompact,then strong convergence of {xn} to some q∈F are obtained.(ii) If E has a Fréchet differentiable norm or satisfying Opial’s condition or its dual E~* has the Kadec-Klee property,then weak convergence of {xn} to some q∈F are obtained.