赋范线性空间中渐近伪压缩映象不动点迭代逼近的充要条件

Banach空间多值Φ-强伪压缩映像不动点的迭代逼近
刘丽莉;师涌江;刘桂霞
【期刊名称】《河北师范大学学报：自然科学版》
【年(卷),期】2006(30)1
【摘要】在一般Banach空间中,利用多值映像一致连续的性质,研究了多值Φ强伪压缩映像不动点的具误差的Ishikawa及Mann迭代逼近问题,得出了Ishikawa,Mann迭代序列强收敛的一个充分条件.由于单值映像是多值映像的特殊情况,故该结果改进和推广了近期相关结果.
【总页数】4页(P10-12)
【关键词】多值Ф-强伪压缩映像;具谩差的Ishikawa迭代序列;具谩差的Mann迭代序列
【作者】刘丽莉;师涌江;刘桂霞
【作者单位】河北建筑工程学院数学物理系
【正文语种】中文
【中图分类】O177.91
【相关文献】
1.多值Ф-强伪压缩映像公共不动点的Ishikawa迭代逼近 [J], 冉凯;惠存阳;赵凤群
2.一致光滑Banach空间中多值Ф-伪压缩映象不动点的带随机混合型误差的迭代逼近 [J], 张树义
3.多值φ-强伪压缩映像不动点的迭代逼近 [J], 冉凯
4.多值Ф-强伪压缩映像不动点的迭代逼近 [J], 李德瑾;赵凤群
5.多值Φ-强伪压缩映像不动点的集合序列的Ishikawa迭代逼近 [J], 冉凯;赵凤群因版权原因，仅展示原文概要，查看原文内容请购买。

Lipschitz局部严格伪压缩映象的迭代逼近
邓磊;丁协平
【期刊名称】《应用数学和力学》
【年(卷),期】1994(15)2
【摘要】设Ｋ是一致光滑Ｂａｎａｃｈ空间Ｋ的非空子集，Ｔ：Ｋ→Ｘ是Ｌｉｐｓｃｈｉｔｚ局部严格伪压缩映象。

本文给出一个迭代序列强收敛到Ｔ的唯一不动点，并给出一个涉及Ｌｉｐｓｃｈｉｔｚ局部强增殖映象Ｔ的非线性方程Ｔｘ＝ｆ的解的迭代逼近。

【总页数】5页(P115-119)
【关键词】压缩现象;不动点;迭代逼近
【作者】邓磊;丁协平
【作者单位】重庆师范专科学校,四川师范大学
【正文语种】中文
【中图分类】O189.2
【相关文献】
1.Banach空间中Lipschitz局部严格伪压缩映象带误差的Ishikawa迭代过程 [J], 杨永琴
2.Lipschitz严格伪压缩映象的具误差的迭代逼近 [J], 金茂明
3.Banach空间中Lipschitz严格伪压缩映象的带误差的Ishikawa型迭代逼近 [J], 王黎明;崔艳兰
4.非Lipschitz的渐近弱伪压缩映象不动点的迭代逼近 [J], 沈霞;孟京华;刘文军
5.Banach空间中Lipschitz严格伪压缩映象的迭代逼近 [J], 曾六川; 杨亚立因版权原因，仅展示原文概要，查看原文内容请购买。

本试卷需：答题纸 5 页、草稿纸 2 页 试卷审核时间： 年 月 日考试时间：120分钟 考试方式：闭卷（提示：答案必须依试题顺序做在答题册上，并标明大、小题号，否则不予计分）一、单项选择题（每小题3分，共15分）1. 设X 是赋范线性空间，X y x ∈,，T 是X 到X 中的压缩映射，则下列哪个式子成立（ ）.A ．10<<-≤-αα，　y x Ty Tx B. 1≥-≤-αα，　y x Ty Tx C. 10<<-≥-αα，　y x Ty Tx D. 1≥-≥-αα，　y x Ty Tx2. 设X 是线性空间，X y x ∈,，实数x 称为x 的范数,下列哪个条件不是应满足的条件：（ ）.A. 0等价于0且,0==≥x x xB. ()数复为任意实,αααx x =C. y x y x +≤+D. y x xy +≤3．下列关于度量空间中的点列的说法哪个是错误的（ ）.A ．收敛点列的极限是唯一的 B. 基本点列是收敛点列C ．基本点列是有界点列 D. 收敛点列是有界点列4. 设X 是复内积空间则下列哪个式子不成立（ ）.A ．X y x x y y x ∈∀=,,,,B. ααα,,,,,X y x y x y x ∈∀=为复数C. X x x x x ∈∀=,,2D. βαβααβα,,,,,,,,X z y x z x y x z y x ∈∀+=+为复数5. 设f 是度量空间X 到度量空间Y 上的连续影射，则下列说法哪个不成立（ ）.A ． Y 中()f x 的邻域U 的原像()1f U -是X 中x 的邻域续影的连续影YY y X x y A y Ax ∈∀∈∀=*,,,, B. *=A A C. D. ()A A =**二、判断题（正确的打“√”，错误的打“×”，每小题3分，共15分）1．度量空间中的收敛点列是有界点集 （ ）课程考核试卷装订线2．度量空间中的柯西点列是收敛点列. （ ）3. 任何空间中压缩映射都有唯一的不动点. （ ）4. 内积空间是一种特殊的赋范线性空间. （ ）5. 强收敛一定弱收敛，但弱收敛不一定强收敛. （ ）三、填空题（每小题4分，共20分）1． 称为Banach 空间.2．设X 是内积空间，X y x ∈,，如果 ，则称x 与y 相互正交或垂直.3. 用极限来描述连续映射：设T 度量空间X 到Y 中的映射，那么T 在X x ∈0连续的充要条件为.4.设X 是赋范线性空间，f 是X 上的线性泛函，()f N 表示f 的零空间，那么f 是X 上的连续线性泛函的充要条件是()f N 是X 中的 .5.设X 是内积空间，X y x ∈,，请写出Schwarz 不等式.四、证明题（每小题10分，共50分）1．证明：设X 是实内积空间，X y x ∈∀,，有下式成立： ⎪⎭⎫ ⎝⎛--+=2241,y x y x y x . 2．设{}n x是内积空间X 中点列，若()∞→→n x x n ，且对一切X y ∈有 ()∞→→n y x y x n ,,，证明：()∞→→n x x n .3．证明：设T 是赋范线性空间X 到赋范线性空间Y 中的线性算子，则T 为有界线性算子的充要条件为T 是X 上连续线性算子.4. 设n j i a ij ,2,1,,=为一组实数，适合条件()121,<-∑=n j i ijij a δ，其中ij δ当j i =时为1，否则为0.证明：代数方程组有⎪⎪⎩⎪⎪⎨⎧=+++=+++=+++nn nn n n n n n n b x a x a x a b x a x a x a b x a x a x a22112222212111212111 对任何一组固定的,,,21n b b b 必有唯一解.,,21n x x x5.设n X X X ,,21 是一列Banach 空间，{} n x x x x ,,21=是一列元素，其中n n X x ∈, 并且∞<∑∞=1n p nx ，这种元素列全体记成X ，类似通常数列的加法和数乘，在X 中引入线性运算.若令p n p n x x 11⎪⎪⎭⎫ ⎝⎛=∑∞= 证明：当1≥p 时，X 是Banch 空间.。

严格伪压缩映象不动点的近似逼近
赵宇;王素霞
【期刊名称】《石家庄铁路职业技术学院学报》
【年(卷),期】2004(003)001
【摘要】证明当T是Q一致光滑Banach空间X的有界闭凸子集到自身的严格伪压缩映象时,Ishikawa迭代法强收敛到T的唯一不动点;又当T∶XX是强增生算子时,Ishikawa迭代法强收敛到方程Tx=f的唯一解.
【总页数】3页(P54-56)
【作者】赵宇;王素霞
【作者单位】西藏自治区山南地区教体局,山南,856000;石家庄市无线电三厂,石家庄,050002
【正文语种】中文
【中图分类】O177.91
【相关文献】
1.凸度量空间中两无限族严格拟渐近伪压缩映象公共不动点的逼近 [J], 刘敏
2.关于广义变分不等式的解和严格伪压缩映象不动点的迭代逼近 [J], 李付成;谷峰
3.迭代算法逼近严格伪压缩映象最小范数不动点与变分不等式解 [J], 徐卫;李冰冰;屠国燕;董力强
4.φ强伪压缩映象不动点的近似逼近 [J], 野金花;崔云安
5.均衡问题与无限族k-严格伪压缩映象的公共不动点的迭代逼近 [J], 周光亚
因版权原因，仅展示原文概要，查看原文内容请购买。

有限个广义φ-伪压缩映像不动点的迭代逼近冯媛媛;崔艳兰;许霞【摘要】在Banach空间中,给出了有限个广义φ-伪压缩映像的迭代逼近,并证明了一个强收敛定理,此结果推广了原有的相关结果.【期刊名称】《延安大学学报（自然科学版）》【年(卷),期】2010(029)002【总页数】3页(P22-24)【关键词】Banach空间;广义φ-伪压缩映像;不动点;迭代序列【作者】冯媛媛;崔艳兰;许霞【作者单位】延安大学,数学与计算机科学学院,陕西,延安,716000;延安大学,数学与计算机科学学院,陕西,延安,716000;延安大学,数学与计算机科学学院,陕西,延安,716000【正文语种】中文【中图分类】O177.91在Banach空间中，关于φ－伪压缩映像不动点的迭代逼近问题已经被许多作者深入研究，并获得一系列很好的结果。

2008年，胡洪萍［1］等引入了广义φ－伪压缩映像，并证明了迭代序列的收敛定理。

本文在此基础上，定义了有限个迭代程序，并证明了其迭代的收敛定理。

设E是一个实的Banach空间，〈·，·〉和‖·‖为其广义对偶对和范数，E*是 E的对偶空间，映像 Jp：E→2E*（其中1＜p＜∞）是由下式定义的对偶映像：注1 当p＝2时，对应的J2就是正规对偶映像，即很显然，Jp（x）＝‖x‖p－2J2（x），∀x∈E且x≠0。

定义1 设 D是 E的一非空子集，T：D→E是一映像。

φ：［0，∞］→［0，∞）是一严格递增的函数，满足φ（0）＝0。

（1）T称为φ－伪压缩的［2］，如果存在一点x*∈ D，使得∀x∈D，存在j（x －x*）∈J（x－x*），满足〈Tx－x*，j（x－x*）〉≤‖x－x*‖2－φ（‖x－x*‖）；（2）T称为广义φ－伪压缩的［1］，如果存在一点x*∈D，使得∀x∈D，存在∀jp（x－x*）∈Jp（x－x*）满足〈Tx－x*，jp（x－x*）〉≤‖x－x*‖p－φ（‖xx*‖）；注2 φ－伪压缩映像是广义φ－伪压缩映像当 p＝2时的特例。

Lipschitzф—半压缩映象的不动点迭代逼近
周海云
【期刊名称】《《数学年刊：A辑》》
【年(卷),期】1999(20A)003
【摘要】设X为一致光滑的Banach空间，K为X的非空凸子集，T：
K→KLipschitzφ半压缩映象，设和为[0，1]中的实数列且满足一定条件，则Ishikawa迭代序列强收敛于T的唯一不动点。

【总页数】4页(P399-402)
【作者】周海云
【作者单位】石家庄军事工程学院基础部
【正文语种】中文
【中图分类】O189.2
【相关文献】
1.一致L-Lipschitz的渐近伪压缩映象不动点的迭代逼近? [J], 王绍荣;何彩香;杨泽恒;熊明
2.非Lipschitz渐近伪压缩映象不动点的迭代逼近 [J], 张树义;宋晓光;万美玲;李丹
3.Lipschitz ψ-半压缩映象不动点的迭代逼近 [J], 张树义
4.非Lipschitz的渐近弱伪压缩映象不动点的迭代逼近 [J], 沈霞;孟京华;刘文军
5.一致L-Lipschitz的渐近伪压缩非自映象不动点的迭代逼近 [J], 张芳;向长合因版权原因，仅展示原文概要，查看原文内容请购买。

Hilbert空间中(Φ)-强伪压缩映像的一个注记张树义;宋晓光【摘要】Abstract; Iteractive approximation problem of fixed points for a class of φstrongly pseudocontractive maps without continuity or even without boundedness in Hilbert space were studied under condtion cn ‖xn - Txn ‖ 2→0(n→∞ ) , the strongly convergence theorem was established.%在cn||xn - Txn‖ 2→0(n→∞)的条件下,在Hilbert空间中研究了一类未必连续,甚至未必有界的(Φ)-强伪压缩映像的不动点的迭代序列逼近问题,获得了一个强收敛定理.【期刊名称】《浙江师范大学学报（自然科学版）》【年(卷),期】2013(036)001【总页数】3页(P28-30)【关键词】Hilbert空间;(Φ)-强伪压缩映像;Mann迭代序列;不动点【作者】张树义;宋晓光【作者单位】渤海大学数理学院,辽宁锦州 121013;渤海大学数理学院,辽宁锦州121013【正文语种】中文【中图分类】O177.91关于φ-强伪压缩映像不动点的迭代逼近问题已在Hilbert空间或Banach空间并在映像具有Lipschitz或有界条件下进行了广泛研究，获得了一系列重要的成果[1-5].文献[2]对既非Lipschitz也非有界的φ-强伪压缩映像不动点的迭代逼近问题进行了研究，其结果为以下定理:定理1 设H为实Hilbert空间,K为H中非空凸子集,T:K→K为φ-强伪压缩映像(未必连续,未必有界).假设T在K中有一个不动点x*,设{cn}为(0,1)中的数列,满足对任意初值x1∈K,迭代地定义序列{xn}n≥1为若‖xn-Txn‖2<∞，则{xn}强收敛于T的唯一不动点x*.文献[2]指出：条件‖xn-Txn‖2<∞是可控的，特别当K为有界子集时，足以保证‖xn-Txn‖2<∞成立.一个问题是:当不收敛，但有cn→0(n→∞)时,就未必能保证‖xn-Txn‖2<∞，此时定理1就不易使用.本文的目的是:在cn‖xn-Txn‖2→0(n→∞)的条件下证明上述结果成立.此时,当K为有界子集时，cn→0(n→∞)足以保证cn‖xn-Txn‖2→0(n→∞).本文的结果是定理1的一种推广. 设H为实Hilbert空间,K为H中非空凸子集,称映像T:K→K为φ-强伪压缩的,如果存在一个严格增加的函数φ:R+=[0,+∞)→R+,满足φ(0)=0,使得对∀x,y∈K,恒有如果φ(t)=kt,k∈(0,1),则称相应的映像T为强伪压缩的;如果φ(t)=0,则称相应的映像T为伪压缩的.伪压缩映像的重要性在于它与增生算子之间的密切联系.记A=I-T,其中I:H→H为恒等映像,则T:K→K为伪压缩的(强伪压缩的,φ-强伪压缩的)当且仅当A:K→H是增生的(强增生的,φ-强增生的).引理1[2] 设H为实Hilbert空间,对于∀λ∈[0,1],∀x,y∈H,有‖x+y‖2=‖x‖2+2〈y,x〉+‖y‖2.定理2 设H为实Hilbert空间,K为H中非空凸子集,T:K→K为φ-强伪压缩映像(未必连续,未必有界).假设T在K中有一个不动点x*,设{cn}为(0,1)中的数列,满足对任意初值x1∈K,Mann迭代定义序列{xn}xn≥1为若cn‖xn-Txn‖2→0(n→∞)，则{xn}强收敛于T的唯一不动点x*.证明由式(1)易知T有唯一不动点x*.由式(2)与引理1得‖xn+1-x*‖2=(1-cn)2‖xn-x*‖2+2cn(1-cn)〈Txn-Tx*,xn-x*〉+c2n‖Txn-Tx*‖2(4)将式(3)代入式(4)得‖xn+1-x*‖2≤(1-cn)‖xn-x*‖2-2cn(1-cn)φ(‖xn-x*‖)‖xn-x*‖+cn‖xn+1-x*‖2+c2n(1-cn)‖xn-Txn‖2.将式(5)中的cn‖xn+1-x*‖2移项后并约去公因子1-cn得令τ.下面证明τ=0.若τ>0,则对∀n≥1,有得φ(‖xn-x*‖)≥φ(τ)，∀n≥1.由cn‖xn-Txn‖2→0(n→∞)知，∃n1>0，对∀n≥n1，有cn‖xn-Txn‖2<τφ(τ), 从而由式(6)知，对∀n≥n1,有由此得进一步‖xn1-x*‖2<∞.与矛盾.故τ=0.从而必存在子列{xnj}⊂{xn}，使得断定{‖xnj-x*‖}有界.若相反,即{‖xnj-x*‖}无界,则必存在子列{‖xnjk-x*‖}⊂{‖xnj-x*‖},使‖xnjk-x*‖→0(k→∞).因此,→1(k→∞).与式(7)矛盾.故{‖xnj-x*‖}有界,从而由于因此,对∀ε>0,∃n2>n1,使得对∀n>n2,有cn‖xn-Txn‖‖xn+1-xn‖进而‖xn+1-xn‖<.又因为‖xnj-x*‖→0(j→∞),所以存在N>n2,使得‖xN-x*‖<ε.下面证明:对∀k≥N,有‖xk-x*‖≤ε.当k=N时,结论显然成立.假设当k≥N时,结论成立.下面证明对k+1时结论也成立.假设结论不成立,则有‖xk+1-x*‖>ε,从而由式(6)有得到矛盾.因此,对∀k≥N,有‖xk-x*‖≤ε.于是‖xk-x*‖≤ε.由ε的任意性知xn→x*(n→∞).定理2证毕.【相关文献】[1]Chang S S,Cho Y J,Zhou H Y.Iterative methods for nonlinear operator equations on Banach spaces[M].New York:Nova Sci Publishers,2002.[2]张明虎,周和月.Hilbert空间中φ-强伪压缩映像不动点的迭代逼近[J].河北师范大学学报:自然科学版,2006,30(5):516-521.[3]谷峰.关于φ-伪压缩型映像的具误差的Ishikawa和Mann迭代过程的收敛性问题[J].数学年刊,2002,23A(1):49-54.[4]张树义.Lipschitz φ-半压缩映像不动点的迭代逼近[J].鲁东大学学报:自然科学版,2007,23(1):14-18.[5]宣渭峰,王元恒.双复合修正的Ishikawa迭代逼近非扩张映像不动点[J].浙江师范大学学报:自然科学版,2009，32(4):401-405.。

Hilbert空间中渐近半压缩型映象的不动点的迭代逼近曾六川;闻涛【摘要】Investigates the convergence criteria of the modified Ishikawa iteration process with errors for the iterative approximation of fixed points of asymptotically hemicontractive type mappings in a real Hilbert space.%研究实Hilbert空间中用于迭代逼近渐近半压缩型映象不动点的带误差的修正的Ishikawa迭代程序的收敛判据.【期刊名称】《上海师范大学学报（自然科学版）》【年(卷),期】2005(034)001【总页数】6页(P6-11)【关键词】Hilbert空间;渐近半压缩型映象;带误差的修正的Ishikawa迭代程序;不动点【作者】曾六川;闻涛【作者单位】上海师范大学,数理信息学院,上海,200234;上海师范大学,数理信息学院,上海,200234【正文语种】中文【中图分类】O177.91Throughout this paper, let H be a real Hilbert space with norm and K be a nonempty convex subset of H. Let T:K → K be a mapping and F( T )denote the set of all fixed points of T in K .Definition 1 A mapping T:K → K is said to be(i) nonexpansive if(1)(ii) an asymptotically hemicontractive type mapping if F( T ) ≠ ∅ and(2)(iii) uniformly L -Lipschitzian if there exists a positive number L>0 such that(iv) compact if T is continuous and maps each bounded subset into a relatively compact subset.It is easy to see that the class of nonexpansive mappings with fixed points is a subclass of the class of asymptotically hemicontractive type mappings. Definition 2 (i) The Modified Ishikawa Iteration Process with Errors. Let K b e a nonempty convex subset of H, T:K → K be a mapping, x0 ∈ K be a given point, and {an},{bn},{cn}, and be four sequences in [0,1 ] . Then the sequence {xn} defined by(3)is called the modified Ishikawa iterative sequence with errors of T, where {un} and {vn} are two bounded sequences in K.(ii) The Modified Mann Iteration Process with Errors. In (3), if = 1 and =0, then yn = xn . The sequence {xn} defined byxn + 1 = an xn + bn Tnxn + cn un , n ≥ 1,(4)is called the modified Mann iterative sequence with errors of T. Recently, Hu [ 1 ] introduced the concept of p - asymptotically hemicontractive mappings in p - uniformly convex Banach spaces, and studied the problem of approximating fixed points of this class of mappings by the modified Mann and Ishikawa iteration processes. His results include Liu′s corresponding results [ 8 ] as special cases. But, unfortunately, his convergence criteria for the modified Mann and Ishikawa iteration processes are hard to be checked since these criteria involve the constant appearing in the norm inequalities in p - uniformly convex Banach spaces.The purpose of this paper is to investigate the convergence criteria for the modified Ishikawa iteration process with errors for the iterative approximation of fixed points of asymptotically hemicontractive type mappings in a real Hilbert space. In the setting of Hilbert spaces, our results generalize, improve and develop Hu′s results[ 1 ] in the following aspects: (i) Our convergence criteria do not involve the constant appearing in the norm inequalities; (ii) Our results extend Hu′s result[ 1 ] from the modified Ishikawa iteration process to the modified Ishikawa iteration process with errors; (iii) Our results remove Hu′s stronger res triction [ 1 ] , 0<ε≤ αn ≤ βn ≤ b<1,imposed on the iterative parameters {αn},{ βn} in the modified Ishikawaiteration process. Moreover, our results also develop and generalize Chidume and Moore′s results [ 4 ] since our results extend their method for studying hemicontractive mappings to that for studying asymptotically hemicontractive type mappings. In addition, in the setting of Hilbert spaces, our results are the extension and improvements of the corresponding results of Tan and Xu [ 2 ] and the author [5 ～7] .We shall make use of the following results as we go on.Lemma 1[ 2 ] Suppose that {ρn},{σn} are two sequences of nonnegative numbers such that for some real number N0 ≥ 1,ρn + 1 ≤ ρn + σn, n ≥ N0 .(i) If then ρn exists.(ii) I f and { ρn} has a subsequence converging to zero, then ρn = 0.We shall also use the following well-known identity for Hilbert spaces H:(5)Lemma 2[ 3 ] Let E be a normed space and K be a nonempty convex subset of E. Let T:K → K be a uni formly L -Lipschitzian mapping and be real sequences in [0,1 ] with an + bn + cn = 1. Let {un},{vn} be bounded sequences in K. For an arbitrary x1 ∈ K, generate a sequence {xn} by the modified Ishikawa iteration process with errors (3). Then for all n ≥ 1,Now, we prove the following main result of this paper.Theorem 1 Let K be a nonempty bounded closed convex subset of a real Hilbert space H, and T:K → K be a uniformly L -Lipschitzian and asymptotically hemicontractive type mapping. Let Tm be compact for some m ≥ 1. Let and be real sequences in [0,1 ] satisfying the following conditions:(i) an + bn + cn = 1 , n ≥ 1;(iv) 0 ≤ αn ≤ βn<1, n ≥ 1, where αn : = bn + cn , βn : .For an arbitrary x1 ∈ K, de fine a sequence n = 1 iteratively by the modified Ishikawa iteration process with errors (3), where {un},{vn} are arbitrary sequences in K. Then, n = 1 converges strongly to a fixed point of T.Proof Since T is an asymptotically hemicontractive type mapping, it is known that F( T ) ≠ ∅. Let x* ∈ K be an arbitrary fixed point of T. Using the identity (5), we obtain the following estimates: For some constants M1 ≥ 0,M2 ≥ 0,Now, utilizing the estimates above and the fact that T is an asymptotically hemicontractive type mapping, we havewhere M3 = M1 + M2 . Therefore, for some constant M4 ≥ 0,(8)where M5 = max (M3 ,M4), since αn ≤ 1. Let M6 = diam( K ). Then from the uniformly L -Lipschitzian continuity of T, we obtainNote that = ∞ and Sincewe deduce that βn = ∞ andSince αn ≤ βn, n ≥ 1, it follows from (8) and (9) that(*)which hence implies that(10)Let = δ≥ 0. We claim that δ = 0. Suppose that this is false, that is, δ>0. Then, there exists an integer N1 ≥ 1 such that ≥ n ≥ N1. Hence, from (10) it follows thatLetting n → ∞ , we get βn <∞. This contradicts the fact that βn = ∞ . Hence, the claim is valid. In view of Lemma 2, we getNext, observe thatHence, = 0. Since {xn} is bounded and Tm is compact, { Tmxn} has a convergent subsequence {Tmxnj } . Suppose Tmxnj = p. Thenwe obtain Tp = p, i e, p ∈ F( T ) .On the other hand, it follows from ( * ) that for any eleme nt x* ∈ F( T ),By Lemma 1, we conclude that exists. In particular, exists. Since = 0, we know that, {xn} converges strongly to the fixed point p ∈ F( T ) . The proof is completed.Remark 1 The conclusion of Theorem 1 remains true if condition (ii) in Theorem 1 is replaced by the condition: βn = ∞ and n <∞. A prototype of our parameters is as follows:for all integers n ≥ 1. Observe thatCorollary 1 Let K be a nonempty compact convex subset of a real Hilbert space H, and T:K → K be a uniformly L -Lipschitzian and asymptotically hemicontractive type mapping. Let and be real sequences in [0,1 ]satisfying the following conditions:(i) an + bn + cn = 1, n ≥ 1;(iv) 0 ≤ αn ≤ βn <∞, ∀n ≥ 1, where αn : = bn +cn ,βn : .For an arbitrary x1 ∈ K, define a sequence n = 1 iteratively by the modified Ishikawa iteration process with errors (3), where {un},{vn} are arbitrary sequences in K. Then, n = 1 converges strongly to a fixed point of T.Proof Th e existence of a fixed point of T follows from Schauder′s fixed point theorem. So, F( T ) ≠ ∅. It is easy to see that T:K → K is compact. Utilizing Remark 1, we can deduce that the conclusion of Corollary 1 is true. Corollary 2 Let ,βn be what we mentioned in Corollary 1. Let T: K → K bea nonexpansive mapping. For an arbitrary x1 ∈ K, define a sequence n =1 iteratively by the modified Ishikawa iteration process with errors (3), where {un},{vn} are arbitrary sequences in K. Then, n = 1 converges strongly to a fixed point of T.Proof The existence of a fixed point of T follows from Schauder′s fixed point theorem. So, F( T ) ≠ ∅. Since T is nonexpansive, T is a uniformly 1 -Lipschitzian and asymptotically hemicontractive type mapping. Therefore, the conclusion of Corollary 2 follows from Corollary 1.Remark 2 Clearly, from Theorem 1, Corollaries 1 and 2, we can immediately obtain the corresponding results on the modified Mann iteration process with errors, respectively.Reference:[1] HU CHANGSONG. Convergence theorems of the iteration processes for asymptotically hemicontractive mappings in p -uniformly convex Banach spaces [J]. Math Applicata, 1999, 12(3): 77- 80.[2] TAN K K, XU H K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process [J]. J Math Anal Appl, 1993, 178: 301-308.[3] OSILIKE M O, ANIAGBOSOR S C. Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings [J]. Math Comput Modelling, 2000, 32: 1181-1191.[4] CHIDUME C E, MOORE C. Fixed point iteration for pseudocontractive maps [J]. Proc Amer Math Soc, 1999, 127(4): 1163-1170.[5] ZENG L C. A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process [J]. J Math Anal Appl, 1998, 226: 245-250.[6] ZENG L C. Iterative approximation of fixed points of (asymptotically) nonexpansive mappings [J]. Appl Math J Chinese Univ, 2001, 16B (4): 402-408.[7] ZENG L C. Weak convergence theorems for nonexpansive mappings in uniformly convex Banach spaces [J]. Acta Math Scientia, 2002, 22A (3): 336-341.[8] LIU Q H. Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive map-pings [J].Nonlinear Analysis, 1996, 26(11): 1835-1842.。