摘要
本文讨论了一些非线性算子迭代方法的等价性问题以及有限个严格伪压缩映射的收敛定理,全文分为四章,摘要如下：
第一章为绪论,介绍了非线性算子迭代方法以及不动点定理的实际意义、研究现状及本文所用到的相关知识和本文的创新之处.
第二章主要研究了新的Ishikswa-Halpern迭代与粘性迭代方法之间的等价性问题,这里最主要把Meir-Keeler压缩映射(简称为MKC)这种算子融入到粘性迭代算法中
第三章主要研究了在Bananch空间中一族有限个λi—严格伪压缩映射的收敛的定理。
最后,在第四章对本文进行了总结和展望,并指出作为进一步研究的某些课题.
The paper mainly studies the equivalence of some nonlinear iterations and the conver-gence theorems of finite nonexpansive mappings.The paper includes four chapters.
The first chapter as an introduction, modeling process of nonlinear iterations and theconvergence theorems of fixed points, its real meaning, the current research status and themain work of this paper are addressed.
In the second chapter,it is to establish some groups of equivalent theorems of conver-gence between modified Ishikswa-Halpern iteration and viscosity approximation method.Inthis chapter,it considers the viscosity approximation method with Meir-Keeler contrac-tion(MKC, for short)
In the third chapter,we consider the convergence theorems of finite λi strictly pseudo-contractive mappings in Banach space.
Finally, the conclusion is showed in the fourth chapter, and some problems we needstudy later are given.
引文
[1] Shuang Wang, Changsong Hu,Equivalent theorems of the convergence betweenIshikawa-Halpern iteration and viscosity.23(2010)693-699.
[2] S.Reich, Weak convergence theorems for nonexpansive mappings in Banachspaces,J.Math.Anal.Appl.67(1979)274-276.
[3] A.Genel, J.Lindenstrass, An example concerning fixed points,Israel J. Math.22(1975)81-86.
[4] B.Halpern, Fixed points of nonexpansive maps,Bull.Amer.Math.Soc.73(1967)957-961.
[5] K. Nakajo, W. Takahashi, Strong convergence theorms for nonexpansive semi-groups,J. Math. Anal. Appl.279(2003)372-379.
[6] T. H. Kim, H.K.Xu,Strong convergence of modified Mann iterations, Nonlinear Anal.61(2005)51-60.
[7] Y.H.Yao,R.D.Chen,J.C.Yao,Strong convergence and certain control condition formodified Mann iteration, Nonlinear Anal.68(2008)1687-1693.
[8] Y.J.Cho,S.M.Kang,X.Qin,Approximation of common fixed points of an infinite familyof nonexpansive mappings in Banach spaces,Comp.Math.Appl.56(2008)-2064.
[9] x.Qin,Y.J.Cho,J.I.Kang,S.M.Kang,Strong convergence theorems for an infinite familyof nonexpansive mappings in Banach,J.comput.Appl.Math.(2009)doi:10.1016
[10] Y.H.Yao, Y.C.Liou,S.M.Kang,Strong convergence of an iterative algorithm on an infi-nite countable family of nonexpansive mappings,Appl.Math.Comput.208(2009)211-218.
[11] T.Suzki,Moudafi‘s viscosity approximations with Meir-Keeler contractions,J. Math.Anal. Appl.325(2007)342-352.
[12] B.Halpern, Fixed point of nnexpansive maps, Bull..Amer.Math.Soc.73(1967),957-961.
[13] N.Shioji, W.Takahashi, Strong convergence of approximated sequences for non-expansive mappings in Banach spaces, Proc.Am.Math. Soc.125(1997)3641-3645.
[14] T.H.Kim and H.K.Xu, Robustness of Mann‘s algorithm for nonexpansive mappings,J.Math.Anal.Appl.327(2007)
[15] P.L.Lions, Approximation de points fixs de contractions, C.R.Acad.Sci. Pairs Ser. A-B284(1997). A1357-A1359.
[16] A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math.Anal. Appl.241(2000),46-55.
[17] W.Takahashi, Y.Ueda, On Reichs strong convergence theorems for resolvents of ac-cretive operators, J.Math. Anal. Appl.104(1984)546-553.
[18] T.Suzuki, Strong convergence of approximated sequences for nonexpansive mappingsin Banach spaces, Proc.Amer. Math. Soc.135(2007),99-106.
[19] Q.L.Dong, S.N.He and F.Su, Strong convergence of an intertive algorithms for aninfinite family of strict pseudo-constructions in Banach space, Applied Mathematicesand Computation216(2010),959-969.
[20] Yisheng Song, Rudong Chen, Strong convergence of an interative method for nonex-pansive mappings,Math Nachr,accepted for publition.
[21] W.Takahashi, Non-linear Function Analysis-Fixed Point Theory and its Applications,Yokohama Pnblishers Inc, Yokohama,2000(in Japanese).
[22] J.S.Jung, Viscosity approximation methods for nonexpansive mappings in Banachspaces,Nonline.
[23] H.K.Xu, Interative algorithms for non-linear operators, J. Lond. Math. Soc.66(2002)240-256.
[24] Yisheng Song, Rudong Chen, Interative approximation to common fixed points ofnon-expansive mapping squences in reflexive Banach spaces, Non-linear Anal.66(2007)591-603.
[25] Ya.I. Alber, S. Guerre-Delabriere, L. Zelenko, Principle of weakly contractive mapsin metric spaces, Comm. Appl. Nonlinear Anal.5(1)(1998)45-68.
[26] C.E.Chidume,C.O,Intretive appoximation of fixed points of nonexpansive mappings,J. Math. Anal.Appl.318(2006)288-295.
[27] F. E. Browder, W. V. Petryshyn,Construction of fixed points of nonlinear mappings,J.Math. Anal. Appl.20(1967)197-228.
[28] C.Byrne,A unifed treatment of some iterative algorithms in singal processing and im-age reconstruction,Inverse Problems20(2004)103-120.
[29] C.I.Podilchuk,R.J.Mammone, Image recovery by convex projections using a least-squares constraint,J.Opt. Soc.Amer.7(1990)517-521.
[30] H.H.Bauschke.The appoximation of fixed points of compositions of nonexpansivemappings in Hilbert space,J.Math.Anal.Appl.202(1996),150-129.
[31] C.Byrne, A unified treatment of some interative algorithms in signal processing andimage reconstruction, Invers Problems20(2004),103-120.
[32] P.L.Combettes, The convex feasibility problem in image recovery, in: P.Hawkes(Ed),Advances in Imaging and Electron Physices, New York Academic95(1996), pp.155-270.
[33] K.Goebel and S.Reich, Uniform convexity, Hyperbolic Geometry and NonexpansiveMappings, M-arcel Dekker,New York and Basel (1984).
[34] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math.Anal. Appl.241(2000)2683-2693.
[35] H.H.Bauschke,J.M.Borwein, On projection algorithms for solving convex feasibilityproblems, SIAM Rev.38(1996)367-426.
[36] H.H.Bauschke, The approximation of fixed points of compositions of nonexpansivemappings in Hilber space,J.Math.Aal.202(1996)150-159.
[37] D.C.Youla, Mathematics theory of image restoration by the method of convex pro-jections, in:H. Stark(Ed),Image Recovery:Theory and Applications,Academic Press,Florida,1987,pp.29-77.
[38] Y.song,Equivalent theorems of the convergence between Proximal type algorithms,Nonlinear Anal.(2008)doi:10.1016.
[39] P.L. Combettes, A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlin-ear Convex Anal.6(2005)117-136.
[40] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings inHilbert space, J. Math. Anal. Appl.20(1967)197-228.
[41] P.L. Combettes, A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlin-ear Convex Anal.6(2005)117-136.
[42] A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem anda nonexpansive mapping, J. Optim. Theory Appl.133(2007)359-370.
[43] A. Moudafi, M. Thera, Proximal and Dynamical Approaches to Equilibrium Prob-lems, in: Lecture Notes in Economics and Mathematical Systems, vol.477, Springer,1999, pp.187-201.
[44] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium prob-lems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl.331(2007)506-515.
[45] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappingsand monotone mappings, J. Optim. Theory Appl.118(2003)417-428.
[46] S. Takahashia, W. Takahashi, Strong convergence theorem for a generalized equilib-rium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal.69(2008)1025-1033.
[47] S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixedpoint problems of nonlinear mappings and monotone mappings, Appl. Math. Comput.197(2008)548-558.
[48] Yao,J.C. Yao, On modified iterative method for nonexpansive mappings and monotonemappings, Appl. Math. Comput.186(2)(2007)1551-1558.
[49] S.-s. Chang, et al., A new method for solving equilibrium problem fixed point prob-lem and variational inequality problem with application to optimization, NonlinearAnalysis (2008),doi:10.1016/j.na.2008.04.035.
[50] V. Colao et al,An iterative method for finding common solutions of equilibrium andfixed point problems, J.Math.Anal.Appl.344(2008)340-352.
[51] Y. Liu, A general iterative method for equilibriumproblems and strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis (2009), doi:10.1016/j. na.2009.03.060.
[52] T. Suzuki, Strong convergence of Krasnoselskii and Mannn s type sequences foroneparameter nonexpansive semigroups without Bochner integrals, J. Math. Anal.Appl.305(2005)227-229.
[53] Y.Song, Equivalent theorems of the convergence between Proximal type algo-rithms,Nonlinear Anal(2008), doi:10.1016/j.na.2008.10.067.
[54] H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc.66(2002)240-256.
[55] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans.Amer. Math. Soc.149(1970)75-88.
[56] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinitenonexpansive mappings and applications,Taiwanese J. Math.5(2)(2001)387-404.
[57] E. Blum, S. Oettli, From optimization and variational inequalities to equilibrium prob-lems, Math. Student63(1&4)(1994)123-145.
[58] H. Zegeye, N. Shahzad, Strong Convergence Theorems for a common zero of a count-ably infinite family of α-inverse strongly accretive Mappings, Nonlinear Analysis(2008),doi:10.1016/j.na.2008.10.091.
[59] S. Reich, Approximation fixed points of non-expansive mappings, Panam. Math.59(1992)486-491.
[60] R. Wittmann, Approximation fixed points of non-expansive mappings, Arch.Math. J.4(2)(1994)23-28.
[61] Yisheng Song, Rudong Chen,Strong convergence theorems on an iterative method fora family of finite non-expansive mappings,Appl.Math. Comput.180(2006)275-287.
[62] Yisheng Song, Rudong Chen, Viscosity approximation methods for non-expansivenonself-mappings,J.Math. Anal. Appl.321(2006)316-326.
