严格伪压缩映像不动点迭代的逼近问题
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摘要
非线性算子不动点理论和均衡理论是非线性泛函分析的重要组成部分,尤其是非线性算子方程解的迭代逼近问题已成为非线性泛函分析领域近年来研究的活跃课题.该文对严格伪压缩映象的不动点问题和均衡问题进行深入研究,建立了更有效的迭代过程来逼近一族严格伪压缩映象的公共不动点集的公共元和一族严格伪压缩映象的公共不动点集与均衡问题解集的公共元,所得结果改进和推广了许多作者的最新结果.全文共分四章.第一章介绍了非线性算子理论及迭代算法的背景及本文的主要工作.第二章在q一致光滑Banach空间框架下讨论了无限个λi-严格伪压缩自映象的带误差的修改的Ishikawa迭代过程的强收敛问题.第三章在q一致光滑Banach空间框架下讨论了有限个λi-严格伪压缩非自映象的一般迭代过程的收敛性问题.第四章在Hilbert空间框架下讨论了找有限个依中间意义的渐进ki-严格伪压缩映象的公共不动点集和均衡问题解集的公共元问题.
The fixed point theory of nonlinear operations and equilibrium problems are important parts of nonlinear functional analysis. Especially, the problem of approximating to solutions of nonlinear operator equations becomes the topic that people study in the recently years. In this paper, we construct effective schemes to approximate the common fixed point of a family ofλi—strictly pseudocontractive mappings and find a common element of the set of common fixed points of a finite family of asymptotically ki—strictly pseudo-contractions in the intermediate sense and the set of solutions of an equilibrium problem. Results presented in this paper improve and extend many authors'recent results. This paper includes fives chapters. Now we will describe them briefly one by one.
Chapter 1 recalls the history and present situation of nonlinear operator equations in Banach spaces and we also give a summary of this work.
Chapter 2 mainly concerns about the convergence problems of modified Ishikawa iter-ative process with errors for an infinite family of strict pseudocontractions in q—uniformly smooth Banach spaces.
Chapter 3 mainly discusses about the convergence problems of a general iterative pro-cess for a finite family ofλi—strict pseudocontractions in q—uniformly smooth Banach spaces.
Chapter 4 mainly considers convergence analysis for equilibrium problems and fixed point problems of a finite family of asymptotically ki—strictly pseudocontractive mappings in the intermediate sense in Hilbert spaces.
The fixed point theory of nonlinear operations and equilibrium problems are important parts of nonlinear functional analysis. Especially, the problem of approximating to solutions of nonlinear operator equations becomes the topic that people study in the recently years. In this paper, we construct effective schemes to approximate the common fixed point of a family ofλi—strictly pseudocontractive mappings and find a common element of the set of common fixed points of a finite family of asymptotically ki—strictly pseudo-contractions in the intermediate sense and the set of solutions of an equilibrium problem. Results presented in this paper improve and extend many authors'recent results. This paper includes fives chapters. Now we will describe them briefly one by one.
Chapter 1 recalls the history and present situation of nonlinear operator equations in Banach spaces and we also give a summary of this work.
Chapter 2 mainly concerns about the convergence problems of modified Ishikawa iter-ative process with errors for an infinite family of strict pseudocontractions in q—uniformly smooth Banach spaces.
Chapter 3 mainly discusses about the convergence problems of a general iterative pro-cess for a finite family ofλi—strict pseudocontractions in q—uniformly smooth Banach spaces.
Chapter 4 mainly considers convergence analysis for equilibrium problems and fixed point problems of a finite family of asymptotically ki—strictly pseudocontractive mappings in the intermediate sense in Hilbert spaces.
引文
[1]Mann W.R.Mean value methods in iteration[J].Proceedings of the American Mathe-matical Society,1953,4:506-510.
[2]Browder F.E,Petryshyn W.V. Construction of fixed points of nonlinear mappings in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,1967,20:197-228.
[3]Reich S. Asymptotic behavior of contractions in Banach spaces[J]. Journal of Mathe-matical Analysis and Applications,1973,44:57-70.
[4]Ishikawa S.Fixed points by a new iteration [J]. Proceedings of the American Mathe-matical Society,1974,44:147-150.
[5]Reich S.Weak convergence theorems for nonexpansive mappings in Banach spaces [J]. Journal of Mathematical Analysis and Applications,1979,75:274-276.
[6]Youla D.C. Mathematical theory of image restoration by the method of convex projec-tions [M]. in:H.Stark(Ed.),Image Recovery:Theory and Applications. Florida:Aca-demic Press,1987.29-77.
[7]Xu H.K.Inequalities in Banach spaces with applications [J]. Nonlinear Analysis,1991, 16:1127-1138.
[8]Tan K.K, Xu H.K. Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process[J]. Journal of Mathematical Analysis and Applications, 1993,178:301-308.
[9]Bauschke H.H.The approximation of fixed points of compositions of nonex-pansive mappings in Hilbert space [J]. Journal of Mathematical Analysis and Applications,1996,202:150-159.
[10]Bauschke H.H.Borwein J.M. On projection algorithms for solving convex feasibility problems[J], SIAM Review,1996,38:367-426.
[11]Moudafi A. Viscosity approximation mehods for fixed points problems[J]. Journal of Mathematical Analysis and Applications,2000,241:46-55.
[12]Shimoji K,Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications[J]. Taiwanese Journal of Mathematics, 2001,5:387-404.
[13]Xu H.K,Ori M.G. An implicit iterative process for nonexpansive mappings[J]. Nu-merical Functional Analysis and Optimization,2001,22:767-773.
[14]Xu H.K. An iterative approach to quadratic optimization[J]. Journal of Optimization Theory and Applications,2003,116:659-678.
[15]Xu H.K. Viscosity approximation methods for nonexpansive mappings[J]. Journal of Mathematical Analysis and Applications,2004,298:279-291.
[16]Kim T.H,Xu H.K. Strong convergence of modified Mann iterations [J]. Nonlinear Analysis,2005,61:51-60.
[17]Suzuki T. Strong convergence of Krasnoselskii and mann's type sequences for one parameter nonexpansive semigroups without bochner integrals[J]. Journal of Mathe-matical Analysis and Applications,2005,305:227-239.
[18]Shang M,Su Y,Qin X. Strong convergence theorems for a finite family of non-expansive mappings[J]. Fixed Point Theory and Applications,2007(2007)Art.ID 76971,9pages.
[19]Yao Y,Liou Y.C,Yao J.C. Convergence theorem for equilibrum problems and fixed problems of infinite family of nonexpansive mappings [J]. Fixed Point Theory and Applications,2007(2007)Art.ID 64363,12 pages.
[20]Zhou H. Convergence theorems for λ—strict pseudo-contractions in 2-uniformly smooth Banach spaces[J]. Nonlinear Analysis,2008,69:3160-3173.
[21]Yao Y,Chen R,Yao J.C. Strong convergence and certain control conditions for modi-fied Mann iteration[J]. Nonlinear Analysis,2008,68(6):1687-1693.
[22]Cho Y.J,Kang S.M,Qin X. Approximation of common fixed points of an infinite fam-ily of nonexpansive mappings in Banach spaces[J]. Computers and Mathematics with Applications,2008,56(8):2058-2064.
[23]Cho Y,Qin X. Convergence of a general iterative method for nonexpansive map-pings in Hilbert spaces [J]. Journal of Computational and Applied Mathematics, 2009,228(1):458-465.
[24]Zhang H,Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces[J]. Nonlinear Analysis,2009,70(9):3236-3242.
[25]Gossez J.P, Lami Dozo E. Some geomethic properties related to the fixed point theory for nonexpansive mappings[J]. Pacific Journal of Mathematics,1972,40:565-573.
[26]Rhoades B.E. Fixed point iterations using infinite matrices[J].Transactions of the American Mathematical Society,1974,196:162-176.
[27]Wittmann R. Approximation of fixed points of nonexpansive mappings [J]. Archiv der Mathematik,1992,58:486-491.
[28]Nakajo K,Takahashi W. Strong convergence theorems for nonexpansive map-pings and nonexpansive semigroups[J]. Journal of Mathematical Analysis and Applications,2003,279:372-379.
[29]Jung J.S. Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces[J]. Journal of Mathematical Analysis and Applications,2005,302:509-520.
[30]Iiduka H,Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings[J]. Nonlinear Analysis,2005,61:341-350.
[31]Marino G,Xu H.K. A general iterative method for nonexpansive mappings in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,2006,318:43-52.
[32]Zhou H. Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert space[J]. Nolinear Anal,2008,69:456-462.
[33]Qin X,Su Y. Approximation of a zero point of accrtive operator in Banach spaces[J]. Journal of Mathematical Analysis and Applications,2007,329:415-424.
[34]Yao Y. A general iterative method for a finite family of nonexpansive mappings[J]. Nolinear Anal,2007,66:2627-2687.
[35]Marino G,Xu H.K. Weak and strong convergence theorems for κ-strict pseudo-cnotractions in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications, 2007,329:336-349.
[36]Qin X,Shang M,Kang S.M. Strong convergence theorems of modified Mann it-erative process for strict K-pseudo-contractions in Hilbert spaces [J]. Nolinear Anal,2009,70(3):1257-1264.
[37]Zhou H. Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces[J]. Nolinear Anal,2008,68:2977-2983.
[38]Matsushita S.Takahashi W. Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions [J]. Nolinear Anal,2008,68:412-419.
[39]Zhang H,Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces[J]. Nolinear Analysis,2009,70(9):3236-3242.
[40]Zhou H. Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,2008,343:546-556.
[41]Chidume C.E,Mutangadura S.A. An example on the Mann iteration method for Lip-schitz pseudo-contractions [J]. Proceeding of the American Mathematical Society, 2001,129:2359-2363.
[42]Bruck R.E,Kuczumow T,Reich S. Convergence of iterates of asymptotically nonex-pansive mappings in Banach spaces with the uniform Opial property [J]. Colloquium Mathematicum,1993,65:169-179.
[43]Kim T.H,Xu H.K. Convergence of the modified Mann's iteration method for asymp-totically strict pseudocontractions[J]. Nolinear Analysis,2008,68:2828-2836.
[44]Sahu D.R, Xu H.K,Yao J.C. Asymptotically strict pseudocontractive mappings in the intermediate sense[J]. Nolinear Analysis,2009,70:3502-3511.
[45]Blum E,Oettli W. From optimization and variational inequalities to equilibrium prob-lems[J]. Mathematics Student-India,1994,63:123-145.
[46]Flam S. D,Antipin A. S. Equilibrium progamming using proximal-link algolithms [J]. Mathematical Programming,1997,78:29-41.
[47]Moudafi A,Thera M. Proximal and dynamical approaches to equilibrium prob-lems[M].in:Lecture note in Economics and Mathematical Systems. New York: Springer-Verlag,1999,477:187-201.
[48]Combettes P. L,Hirstoaga S.A. Equilibrium programming in Hilbert spaces[J]. Journal of Nonlinear and Convex Analysis,2005,6:117-136.
[49]Xu H.K. Iterative algorithms for nonlinear operators[J]. Journal of the London Math-ematical Society,2002,66:240-256.
[50]Qin X,Cho Y.J,Kang S.M etal.A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces[J]. Nolinear Anal,2009,70; 1902-1911.
[51]Tada A,Takahashi W. weak and strong convergence theorems for a nonexpansive map-pings and an equilibrium problem[J].Journal of Optimization Theory and Applica-tions,2007,133:359-370.
[52]Opial Z. Weak convergence of the sequence of successive approximation for nonex-pansive mappings[J]. Bulletin of the American Mathematical Society,1967,73: 591-597.
[53]Goebel K,Kirk W.A. Topics in Metric fixed point Theory[M].Cambridge:Cambridge University Press.1990.
[54]Takahashi W. Nonlinear Functional Analysis[M]. Yokohama:Yokohama Publish-ers.2000.
[55]Matinez-Yanes C,Xu H.K. strong convergence of the CQ method for fixed point pro-cesses[J]. Nolinear Anal,2006,64:2400-2411.
[56]Iiduka H,Takahashi W. Weak convergence theorem by Ces'aro means for nonexpan-sive mappings and inverse-strongly monotone mappings [J]. Journal of Nonlinear and Convex Analysis,2006,7:105-113.
[2]Browder F.E,Petryshyn W.V. Construction of fixed points of nonlinear mappings in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,1967,20:197-228.
[3]Reich S. Asymptotic behavior of contractions in Banach spaces[J]. Journal of Mathe-matical Analysis and Applications,1973,44:57-70.
[4]Ishikawa S.Fixed points by a new iteration [J]. Proceedings of the American Mathe-matical Society,1974,44:147-150.
[5]Reich S.Weak convergence theorems for nonexpansive mappings in Banach spaces [J]. Journal of Mathematical Analysis and Applications,1979,75:274-276.
[6]Youla D.C. Mathematical theory of image restoration by the method of convex projec-tions [M]. in:H.Stark(Ed.),Image Recovery:Theory and Applications. Florida:Aca-demic Press,1987.29-77.
[7]Xu H.K.Inequalities in Banach spaces with applications [J]. Nonlinear Analysis,1991, 16:1127-1138.
[8]Tan K.K, Xu H.K. Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process[J]. Journal of Mathematical Analysis and Applications, 1993,178:301-308.
[9]Bauschke H.H.The approximation of fixed points of compositions of nonex-pansive mappings in Hilbert space [J]. Journal of Mathematical Analysis and Applications,1996,202:150-159.
[10]Bauschke H.H.Borwein J.M. On projection algorithms for solving convex feasibility problems[J], SIAM Review,1996,38:367-426.
[11]Moudafi A. Viscosity approximation mehods for fixed points problems[J]. Journal of Mathematical Analysis and Applications,2000,241:46-55.
[12]Shimoji K,Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications[J]. Taiwanese Journal of Mathematics, 2001,5:387-404.
[13]Xu H.K,Ori M.G. An implicit iterative process for nonexpansive mappings[J]. Nu-merical Functional Analysis and Optimization,2001,22:767-773.
[14]Xu H.K. An iterative approach to quadratic optimization[J]. Journal of Optimization Theory and Applications,2003,116:659-678.
[15]Xu H.K. Viscosity approximation methods for nonexpansive mappings[J]. Journal of Mathematical Analysis and Applications,2004,298:279-291.
[16]Kim T.H,Xu H.K. Strong convergence of modified Mann iterations [J]. Nonlinear Analysis,2005,61:51-60.
[17]Suzuki T. Strong convergence of Krasnoselskii and mann's type sequences for one parameter nonexpansive semigroups without bochner integrals[J]. Journal of Mathe-matical Analysis and Applications,2005,305:227-239.
[18]Shang M,Su Y,Qin X. Strong convergence theorems for a finite family of non-expansive mappings[J]. Fixed Point Theory and Applications,2007(2007)Art.ID 76971,9pages.
[19]Yao Y,Liou Y.C,Yao J.C. Convergence theorem for equilibrum problems and fixed problems of infinite family of nonexpansive mappings [J]. Fixed Point Theory and Applications,2007(2007)Art.ID 64363,12 pages.
[20]Zhou H. Convergence theorems for λ—strict pseudo-contractions in 2-uniformly smooth Banach spaces[J]. Nonlinear Analysis,2008,69:3160-3173.
[21]Yao Y,Chen R,Yao J.C. Strong convergence and certain control conditions for modi-fied Mann iteration[J]. Nonlinear Analysis,2008,68(6):1687-1693.
[22]Cho Y.J,Kang S.M,Qin X. Approximation of common fixed points of an infinite fam-ily of nonexpansive mappings in Banach spaces[J]. Computers and Mathematics with Applications,2008,56(8):2058-2064.
[23]Cho Y,Qin X. Convergence of a general iterative method for nonexpansive map-pings in Hilbert spaces [J]. Journal of Computational and Applied Mathematics, 2009,228(1):458-465.
[24]Zhang H,Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces[J]. Nonlinear Analysis,2009,70(9):3236-3242.
[25]Gossez J.P, Lami Dozo E. Some geomethic properties related to the fixed point theory for nonexpansive mappings[J]. Pacific Journal of Mathematics,1972,40:565-573.
[26]Rhoades B.E. Fixed point iterations using infinite matrices[J].Transactions of the American Mathematical Society,1974,196:162-176.
[27]Wittmann R. Approximation of fixed points of nonexpansive mappings [J]. Archiv der Mathematik,1992,58:486-491.
[28]Nakajo K,Takahashi W. Strong convergence theorems for nonexpansive map-pings and nonexpansive semigroups[J]. Journal of Mathematical Analysis and Applications,2003,279:372-379.
[29]Jung J.S. Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces[J]. Journal of Mathematical Analysis and Applications,2005,302:509-520.
[30]Iiduka H,Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings[J]. Nonlinear Analysis,2005,61:341-350.
[31]Marino G,Xu H.K. A general iterative method for nonexpansive mappings in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,2006,318:43-52.
[32]Zhou H. Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert space[J]. Nolinear Anal,2008,69:456-462.
[33]Qin X,Su Y. Approximation of a zero point of accrtive operator in Banach spaces[J]. Journal of Mathematical Analysis and Applications,2007,329:415-424.
[34]Yao Y. A general iterative method for a finite family of nonexpansive mappings[J]. Nolinear Anal,2007,66:2627-2687.
[35]Marino G,Xu H.K. Weak and strong convergence theorems for κ-strict pseudo-cnotractions in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications, 2007,329:336-349.
[36]Qin X,Shang M,Kang S.M. Strong convergence theorems of modified Mann it-erative process for strict K-pseudo-contractions in Hilbert spaces [J]. Nolinear Anal,2009,70(3):1257-1264.
[37]Zhou H. Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces[J]. Nolinear Anal,2008,68:2977-2983.
[38]Matsushita S.Takahashi W. Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions [J]. Nolinear Anal,2008,68:412-419.
[39]Zhang H,Su Y. Strong convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces[J]. Nolinear Analysis,2009,70(9):3236-3242.
[40]Zhou H. Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces[J]. Journal of Mathematical Analysis and Applications,2008,343:546-556.
[41]Chidume C.E,Mutangadura S.A. An example on the Mann iteration method for Lip-schitz pseudo-contractions [J]. Proceeding of the American Mathematical Society, 2001,129:2359-2363.
[42]Bruck R.E,Kuczumow T,Reich S. Convergence of iterates of asymptotically nonex-pansive mappings in Banach spaces with the uniform Opial property [J]. Colloquium Mathematicum,1993,65:169-179.
[43]Kim T.H,Xu H.K. Convergence of the modified Mann's iteration method for asymp-totically strict pseudocontractions[J]. Nolinear Analysis,2008,68:2828-2836.
[44]Sahu D.R, Xu H.K,Yao J.C. Asymptotically strict pseudocontractive mappings in the intermediate sense[J]. Nolinear Analysis,2009,70:3502-3511.
[45]Blum E,Oettli W. From optimization and variational inequalities to equilibrium prob-lems[J]. Mathematics Student-India,1994,63:123-145.
[46]Flam S. D,Antipin A. S. Equilibrium progamming using proximal-link algolithms [J]. Mathematical Programming,1997,78:29-41.
[47]Moudafi A,Thera M. Proximal and dynamical approaches to equilibrium prob-lems[M].in:Lecture note in Economics and Mathematical Systems. New York: Springer-Verlag,1999,477:187-201.
[48]Combettes P. L,Hirstoaga S.A. Equilibrium programming in Hilbert spaces[J]. Journal of Nonlinear and Convex Analysis,2005,6:117-136.
[49]Xu H.K. Iterative algorithms for nonlinear operators[J]. Journal of the London Math-ematical Society,2002,66:240-256.
[50]Qin X,Cho Y.J,Kang S.M etal.A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces[J]. Nolinear Anal,2009,70; 1902-1911.
[51]Tada A,Takahashi W. weak and strong convergence theorems for a nonexpansive map-pings and an equilibrium problem[J].Journal of Optimization Theory and Applica-tions,2007,133:359-370.
[52]Opial Z. Weak convergence of the sequence of successive approximation for nonex-pansive mappings[J]. Bulletin of the American Mathematical Society,1967,73: 591-597.
[53]Goebel K,Kirk W.A. Topics in Metric fixed point Theory[M].Cambridge:Cambridge University Press.1990.
[54]Takahashi W. Nonlinear Functional Analysis[M]. Yokohama:Yokohama Publish-ers.2000.
[55]Matinez-Yanes C,Xu H.K. strong convergence of the CQ method for fixed point pro-cesses[J]. Nolinear Anal,2006,64:2400-2411.
[56]Iiduka H,Takahashi W. Weak convergence theorem by Ces'aro means for nonexpan-sive mappings and inverse-strongly monotone mappings [J]. Journal of Nonlinear and Convex Analysis,2006,7:105-113.