馬劍鵬,何中全
(西華師范大學(xué) 數(shù)學(xué)與信息學(xué)院 四川 南充 637000)
廣義漸近φ-半壓縮映射的強(qiáng)收斂定理
馬劍鵬,何中全
(西華師范大學(xué) 數(shù)學(xué)與信息學(xué)院 四川 南充 637000)
在任意實(shí)Banach空間中,研究了依中間意義漸近k-嚴(yán)格偽壓縮的廣義漸近φ-半壓縮映射帶誤差的修正Ishikawa迭代序列的強(qiáng)收斂性。 所得結(jié)果改近和推廣了這類問題的最新研究結(jié)果。
廣義漸近φ-半壓縮映射;不動(dòng)點(diǎn);依中間意義的漸近k-嚴(yán)格偽壓縮;帶誤差的修正Ishikawa迭代序列;實(shí)Banach空間
由于迭代過程往往不是精確的,存在誤差的,因而帶誤差項(xiàng)的迭代算法的收斂性問題的研究是十分有意義的工作。最近Yang[1]在Banach空間中,研究了漸近φ-半壓縮映射的帶誤差的Mann及Ishikawa迭代序列的收斂性。Kim[2]在實(shí)Banach空間中,研究了幾乎一致L-Lipschitzian的廣義漸近φ-半壓縮映射帶誤差的修正Mann迭代序列的收斂性。Zeng[3]和Wang[4]在任意實(shí)的Banach空間中研究了依中間意義漸近非擴(kuò)張的漸近偽壓縮映射的修正Ishikawa迭代序列的收斂性。受上面文獻(xiàn)啟發(fā),將漸近偽壓縮映象射推廣到廣義漸近φ-半壓縮映射,研究了依中間意義漸近k-嚴(yán)格偽壓縮的廣義漸近φ-半壓縮映射帶誤差的修正Ishikawa迭代序列的收斂性。本文證明的方法跟Wang的有根本的區(qū)別。改進(jìn)了Kim[2]、Zeng[3]和Wang[4]等人的研究結(jié)果。
本文設(shè)E是實(shí)的Banach空間,E*是其對(duì)偶空間,D是E的非空凸子集,F(T)={x∈D:T(x)=x}是映射T在D中的所有不動(dòng)點(diǎn)之集。正規(guī)對(duì)偶映射J:E→2E*定義為:
J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2}},?x∈E。
其中〈·,·〉表示E和E*之間的廣義對(duì)偶組。
眾所周知,若E是光滑的Banach空間,則J是單值的。若E是一致光滑的Banach空間,則J在E的任一有界子集上是一致連續(xù)。用j表示單值的正規(guī)對(duì)偶映射。
定義1.1 設(shè)T:D→D是一個(gè)映射。
〈Tnx-Tny,j(x-y)〉≤kn‖x-y‖2-φ(‖x-y‖)。
〈Tnx-q,j(x-y)〉≤kn‖x-q‖2-φ(‖x-q‖)。
(3)T稱為一致L-Lipschitz的,如果存在常數(shù)L≥0,滿足‖Tnx-Tny‖≤L‖x-y‖?x,y
∈D,n∈N。
下面給出關(guān)于映射T的帶誤差項(xiàng)的修正Ishikawa迭代算法:
設(shè)任意x0∈D,{un},{vn}是D上的有界序列{βn},{γn},{bn},{cn}是[0,1]上的實(shí)數(shù)列,則{xn}是由下式定義的序列:
(1)
稱為含誤差項(xiàng)的修正Ishikawa迭代序列。特別,若在(1)中取bn=cn=0,?n≥0,則{xn}是由下式定義的序列:
xn+1=(1-βn-γn)xn+βnTnyn+γnun
(2)
稱為含誤差項(xiàng)的修正Mann序列。
引理1.2[7]設(shè)E是實(shí)的Banach空間,則對(duì)?x,y∈E有下面不等式成立
‖x-y‖2≤‖x‖2+2〈y,j(x-y)〉,?j(x+y)∈J(x+y)。
引理2.1 設(shè){an},{bn},{cn}是三個(gè)非負(fù)實(shí)序列,又存在非負(fù)整數(shù)n0,使得
證明: 由題意知
綜上可知{an}有界。
證明根據(jù)(1)以及定理的條件知
(3)
又由T:D→D是依中間意義漸近k-嚴(yán)格偽壓縮映射,則
‖Tnxn+1-Tnyn‖2
由引理1.4知
‖Tnxn+1-Tnyn‖→0 (n→∞)
(4)
根據(jù)(1)和引理1.2,知
‖xn+1-q‖2=‖(1-βn-γn)(xn-q)+βn(Tnyn-q)+γn(un-q)‖2
≤(1-βn-γn)2‖xn-q‖2+2βn〈Tnyn-q,j(xn+1-q)〉+2γn〈un-q,j(xn+1-q)〉
=(1-βn-γn)2‖xn-q‖2+2βn〈Tnyn-Tnxn+1,j(xn+1-q)〉
+2βn〈Tnxn+1-q,j(xn+1-q)〉+2γn〈un-q,j(un+1-q)〉
+2βnkn‖xn+1-q‖2-2βnφ(‖xn+1-q‖)+2γn‖un-q‖‖xn+1-q‖
則由上式子
(5)
結(jié)合(5),得:
(6)
現(xiàn)在令
則由(6)得:
再根據(jù)定理?xiàng)l件假設(shè)知,
由(5)知
+4(γn‖un-q‖+βn‖Tnyn-Tnxn+1‖)‖xn+1-q‖
+4(γn‖un-q‖+βn‖Tnyn-Tnxn+1‖)‖M1
(7)
現(xiàn)在再令
則由(7)可得
下面證明不動(dòng)點(diǎn)的唯一性。
由于T是廣義漸近φ-半壓縮映射,則T的不動(dòng)點(diǎn)集非空。設(shè)x*∈F(T),如果q∈F(T),那么存在j(q-x*)∈J(q-x*),使得
‖q-x*‖2=
≤‖q-x*‖2-φ(q-x*),
則T有唯一的不動(dòng)點(diǎn)。證明完畢。
則{xn}強(qiáng)收斂于T的不動(dòng)點(diǎn)q。
[1] Yang L P. The equivalence between the convergence of Ishikawa and Mann iterative approximations for an asymptoticallyφ-hemicontractive mapping[J].Far East J Math Sci,2005,18(2):197-212.
[2] Kim J K,Sahu D R, Nam Y M. Convergence theorem for fixed points of nearly uniformlyL-Lipschitz asymptotically generalized φ-hemicontractive mappings[J].Nonlinear Anal,2009,71(12):2833-2838.
[3] Zeng L C. On the strong convergence of an iterative method for non-Lipschitzian asymptotically pseudocontractive mappings[J].Acta Math APPL Sin,2004,27(3):230-239.
[4] Wang S R,Yang Z H,Xing M.Iterative Approximations of Fixed Points for Non-Lipschitzian Asymptotically Pseudocontractive Mappings[J].Mathematica Applicata,2012,25(1):214-219.
[5] OSILIKE M O. Iterative approximations of fixed points asymptotically demicontractive mappings[J].Indian J Pure Appl Math,1998,29(12):1291-1300.
[6] Sahu D R,Xu H K,Yao J C.Asymptotically strict pseudocontractive mappings in the intermediate sense[J].Nonlinear Analysis,2009,70(10):3502-3511.
[7] Chang S S. Some problems and results in the study of nonlinear analysis[J].Nonlinear Anal TMA,1997,30(7):4197-4208.
[8] Moore C, Nnoli B V. Iterative solution of nonlinear equations involving set-valued uniformly accretive operators[J].Comput Math Appl,2001,42(1):131-140.
[9] Yang L P. Strong convergence of iterative sequence for non-lipschitz generalized asymptotically φ-hemicontractive mappings[J].J Sys Sci & Mathe Scis,2010,30(12):1661-1668.
[10] Xue Z Q, Lv G W, Rhoades B E. The convergence theorems of ishikawa iterative process with errors for φ-hemicontractive mappings in uniformly smooth banach spaces[J].Fixed Point Theory Appl,2012,206:1687-1812.
[11] Huang Z Y. Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively φ-pseudocontractive mappings without Lipschitzian assumptions[J].J Math Anal Appl,2007,329(2):935-947.
[12] Deimling K. Nonlinear Functional Analysis[M].Berlin:Springer,1985:205-207.
[13] Chidume C E, Chidume C O.Convergence theorems for fixed points of uniformly continuous generalized φ-hemicontractive mappings[J].J Math Anal Appl,2005,303(2):545-554.
[14] Ozdemir M,Akbulut S.On the equivalence of some fixed point iterations[J].Kyungpook Math J,2006,46(2):211-217.
[15] Su K.Three-step iterations with errors for nonlinear strongly accretive operator equations[J].Acta Math Applicatae Sinica(English Series),2005,21(4):565-570.
(責(zé)任編輯:張凱兵)
StrongConvergenceTheoremsofGeneralizedAsymptoticallyφ-hemicontractiveMappings
Ma Jianpeng,He Zhongquan
(SchoolofMathematicsandInformation,ChinaWestNormalUniversity,Nanchong,Sichuan637000,China)
Strong convergence of modified Ishikawa iteration algorithm with errors for asymptotically k-strict pseudocontractive mapping in the intermediate sense and generalized asymptoticallyφ-hemicontractive mappings are obtained in arbitrary real Banach space. The result of this paper improves and extends the results of the latest research in this field.
generalized asymptotically;φ-hemicontractive mappings; fixed point; asymptotically k-strict pseudocontractive papping in the intermediate sense; modified Ishikawa iterative sequence with errors;real Banach space
O177.91
A
2095-4824(2013)06-0079-04
2013-05-11
四川省省級(jí)精品課程泛函分析項(xiàng)目資助((2008)359-20)
馬劍鵬(1986- ),男,四川樂至人,西華師范大學(xué)數(shù)學(xué)與信息學(xué)院碩士研究生。
何中全(1955- ),男,四川南充人,西華師范大學(xué)數(shù)學(xué)與信息學(xué)院教授。