关于伪压缩映象不动点迭代的一点注记

Journal of Mathematical Research&ExpositionAug.,2008,Vol.28,No.3,pp.740–744DOI:10.3770/j.issn:1000-341X.2008.03.040A Note on Approximating Fixed Points ofPseudocontractive MappingsYAO Yong Hong(Department of Mathematics,Tianjin Polytechnic University,Tianjin300160,China)(E-mail:yuyanrong@;yaoyonghong@)Abstract Let K be a nonempty bounded closed convex subset of a real reflexive Banach spaceE with a uniformly Gateaux differentiable norm.Let T:K→K be a uniformly continuouspseudocontractive mapping.Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings.Let{λn}⊂(0,1Received date:2006-08-04;Accepted date:2006-10-12Foundation item:the National Natural Science Foundation of China(No.10771050).A note on approximating fixed points of pseudocontractive mappings 741Theorem IS [7]If K is a compact convex subset of a Hilbert space H ,T :K →K is a Lipschitzian pseudocontractive mapping.For x 0∈K ,define the sequence {x n }byx n +1=(1−αn )x n +αn T y n ;y n =(1−βn )x n +βn T x n ,n ≥0,where {αn },{βn }are sequences of positive numbers satisfying the conditions:(i)0≤αn ≤βn <1;(ii)lim n →∞βn =0;(iii) ∞n =1αn βn =∞.Then the sequence {x n }converges strongly to a fixed point of T .We note that βn =0for all n ≥0in Theorem IS.In connection with the iterative ap-proximation of fixed points of pseudocontractions,the following question is still open:Does the Mann iterative algorithm always converge for continuous pseudocontractions or even Lipschitz pseudocontractions?However,in 2001,Chidume and Mutangadura [8]provided an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iterative algorithm failed to converge.In 2003,Chidume and Zegeye [9]introduced a new iterative scheme for approximating the fixed points of pseudocontractive mappings.Theorem CZ [9]Let E be a real reflexive Banach space with a uniformly Gateaux differentiable norm.Let K be a nonempty closed convex subset of E .Let T :K →K be an L -Lipschitz pseudocontractive mapping such that F (T )=∅.Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings.Let {λn }and {θn }be two sequences in (0,1]satisfying the following conditions:(i)lim n →∞θn =0;(ii)λn (1+θn )≤1, ∞n =0λn θn =∞,lim n →∞λn θn −1)(n +1)a ,θn=1n ,θn =1I K (x )(742YAO Y HFurthermore,if E is a reflexive Banach space with a uniformly Gateaux differentiable norm and every nonempty closed convex bounded subset of K has thefixed point property for nonexpansive self-mappings.Then as t→1,y t converges strongly to afixed point of T.Lemma1.2[11](i)If E is smooth Banach space,then the duality mapping J is single valued and strong-weak∗continuous.(ii)If E is a Banach space with a uniformly Gˆa teaux differentiable norm,then the duality mapping J:E→E∗is single valued and norm to weak star uniformly continuous on bounded sets of E.Lemma 1.3[12]Let{a n}be a sequence of nonnegative real numbers satisfying a n+1≤(1−αn)a n+αnβn,n≥0,where{αn}⊂(0,1)and{βn}are two sequences of real numbers such that ∞αn=∞and lim sup n→∞βn≤0.Then{a n}converges to zero as n→∞.n=02.Main resultsTheorem2.1Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm.Let T:K→K be a uniformly continuous pseudocontractive mapping.Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings.Let{λn}⊂(0,1],we have{x n}⊂K,∀n≥1.Take p∈F(T).Let S=I−T.Then we2haveSx−Sy,j(x−y) ≥0.(3) From(2),we obtainx n=x n+1+λn x n−λn T x n+λn x n−λn x1=x n+1+λn x n+λn Sx n−λn x1=x n+1+λn[x n+1+λn x n+λn Sx n−λn x1]+λn Sx n−λn x1=(1+λn)x n+1+λ2n(x n+Sx n)−λ2n x1+λn Sx n−λn x1=(1+λn)x n+1+λn Sx n+1+λ2n(x n+Sx n)−λ2n x1+λn(Sx n−Sx n+1)−λn x1.(4)A note on approximatingfixed points of pseudocontractive mappings743From(4),we havex n−p=(1+λn)(x n+1−p)+λn(Sx n+1−Sp)+λ2n(x n+Sx n)−λ2n x1+λn(Sx n−Sx n+1)+λn(p−x1).(5) Combining(3)and(5),we havex n−p−λ2n(x n+Sx n)+λ2n x1−λn(Sx n−Sx n+1)+λn(x1−p),j(x n+1−p)=(1+λn) x n+1−p 2+λn Sx n+1−Sp,j(x n+1−p)≥(1+λn) x n+1−p 2.(6)Next we prove that lim sup n→∞ x1−p,j(x n−p) ≤0.Indeed,taking z=x1in Lemma1.1,we have z t−x n=(1−t)(T z t−x n)+t(x1−x n).Hencez t−x n 2=(1−t) T z t−x n,j(z t−x n) +t x1−x n,j(z t−x n)=(1−t) T z t−T x n,j(z t−x n) +(1−t) T x n−x n,j(z t−x n) +t x1−z t,j(z t−x n) +t z t−x n 2≤ z t−x n 2+(1−t) T x n−x n z t−x n +t x1−z t,j(z t−x n) . Therefore,z t−x1,j(z t−x n) ≤1−ttT x n−x n ,where M>0is some constant such that z t−x n ≤M for t∈(0,1],n≥1.Letting n→∞, we havelim supn→∞z t−x1,j(z t−x n) ≤0.From Lemma1.1,we know z t→p(as t→0).Since the duality mapping J:E→E∗is norm to weak star uniformly continuous,we have from Lemma1.2lim supn→∞x1−p,j(x n−p) ≤0.(7) From(6),we get(1+λn) x n+1−p 2≤ x n−p−λ2n(x n+Sx n)+λ2n x1−λn(Sx n−Sx n+1)+λn(x1−p),j(x n+1−p)≤ x n−p x n+1−p +Mλ2n+Mλn Sx n+1−Sx n +λn x1−p,j(x n+1−p)≤13λn] x n−p 2+2Mλ2n+2Mλn Sx n+1−Sx n +2λn x1−p,j(x n+1−p) .(8)744YAO Y HNote thatx n+1−x n ≤λn x n +λn T x n +λn x n−x1 →0(n→∞).By the uniform continuity of T,we haveSx n+1−Sx n →0(n→∞).Finally applying Lemma1.3to(8),we can conclude that x n→p.This completes the proof.2 Theorem2.2Let E be a uniformly smooth Banach space and K a nonempty bounded closedconvex subset of E.Let T:K→K be a uniformly continuous pseudocontractive mapping.Let {λn}⊂(0,1for all n≥1.It is worthy of mentioningnthat our methods are different from those in the literature.References[1]CHIDUME C E,MOORE C.The solution by iteration of nonlinear equations in uniformly smooth Banachspaces[J].J.Math.Anal.Appl.,1997,215(1):132–146.[2]MANN W R.Mean value methods in iteration[J].Proc.Amer.Math.Soc.,1953,4:506–510.[3]OSILIKE M O.Iterative solution of nonlinear equations of theφ-strongly accretive type[J].J.Math.Anal.Appl.,1996,200(2):259–271.[4]LIU Qihou.The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings[J].J.Math.Anal.Appl.,1990,148(1):55–62.[5]REICH S.Iterative Methods for Accretive Sets[M].Academic Press,New York,1978:317-326.[6]ZHANG Shisheng.On the convergence problems of Ishikawa and Mann iterative processes with error forΦ-pseudo contractive type mappings[J].Appl.Math.Mech.,2000,21(1):1–12.[7]ISHIKAW A S.Fixed points and iteration of a nonexpansive mapping in a Banach space[J].Proc.Amer.Math.Soc.,1976,59(1):65–71.[8]CHIDUME C E,MUTANGADURA S A.An example of the Mann iteration method for Lipschitz pseudo-contractions[J].Proc.Amer.Math.Soc.,2001,129(8):2359–2363[9]CHIDUME C E,ZEGEYE H.Approximatefixed point sequences and convergence theorems for Lipschitzpseudocontractive maps[J].Proc.Amer.Math.Soc.,2004,132(3):831–840.[10]MORALES C,JUNG J S.Convergence of paths for pseudocontractive mappings in Banach spaces[J].Proc.Amer.Math.Soc.,2000,128(11):3411–3419.[11]CHANG Shihsen,CHO Y J,ZHOU Haiyun.Iterative Methods for Nonlinear Operator Equations in BanachSpaces[M].Nova Science Publishers,Inc.,Huntington,NY,2002.[12]KIM T H,XU Hongkun.Strong convergence of modified Mann iterations[J].Nonlinear Anal.,2005,61(1-2):51–60.。