非扩张映像不动点的迭代逼近及应用
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摘要
在本文中,首先在Hilbert空间H上研究了非扩张非自映象下的显式均值迭代过程,并得到了强收敛结果.这些结果推广和改进了文献Y.S.Song,R.D.Chen[Viscosity approximation methods for nonexpansive nonself-mappings,J.Math.Anal.Appl.321(2006)316-326],S.Matsushita,D.Kuroiwa[Strong convergence ofaveraging iterations of nonexpansive mappings,J.Math.Anal.Appl.294(2004)206-214]相应的结果.
接着,研究了Hilbert空间H的非空闭凸子集C上的渐进非扩张非自映像,利用CQ方法构造迭代序列{x_n),{y_n),并得到了强收敛结果.该结果推广和改进了文献T.H.Kim,H.-K.Xu[Strong convergence of modified Mann iterationsfor asympototically nonexpansive mappings and semigroups,Nonlinear Anal.64(2006)1140-1152]相应的结果.
接着,在q—一致光滑的Banach空间上,利用向阳非扩张映象特点构造迭代算法,来逼近变分不等式组的解的问题.该结果推广和改进了R.U.Verma[Generalconvergnece analysis for two-step projection methods and applications to variationalproblems,Appl.Math.Lett.18(2005)1286-1292]相应的结果.
最后,初步尝试利用计算机对相应算法进行算法实现即比较了Picard迭代序列,Mann迭代序列及Ishikawa迭代序列的收敛速度的问题,具有一定的理论意义和实际意义.
In this paper,firstly in Hilbert spaces H,we consider the explicit averaging iterations of nonexpansive nonself mappings,and get strong convergent results. The results presented extend and improve the corresponding ones of Y.S. Song and R.D.Chen[Viscosity approximation methods for nonexpansive nonselfmappings, J.Math.Anal.Appl.321(2006)316-326]and S.Matsushita,D.Kuroiwa [Strong convergence of averaging iterations of nonexpansive mappings,J.Math. Anal.Appl.294(2004)206-214].
Then,we consider another iteration,that is,the iteration of asymptotically nonexpansive mappings of the nonempty closed convex subset C to Hilbert spaces H.We use CQ methods to construct the iterations {x_n},{y_n},and get strong convergent results.The results presented extend and improve the corresponding ones of T.H.Kim,H.-K.Xu[,Strong convergence of modified Mann iterations for asympototically nonexpansive mappings and semigroups,Nonlinear Anal. 64(2006)1140-1152].
Next,in q- uniformly Banach spaces,we construct iteration algorithm through sunny nonexpansive mappings to approach the solutions of a system of two nonlinear variational inequalities problem.The results presented extend and improve the corresponding ones of R.U.Verma[General convergnece analysis for two-step projection methods and applications to variational problems, Appl.Math.Lett.18(2005)1286-1292].
At last,we preliminarily attempt to use computer and compare the speed of the common iterative sequences such as Picard iterative sequence,Mann iterative sequence and Ishikawa iterative sequence.The result has certain theory significance and the practical significance.
接着,研究了Hilbert空间H的非空闭凸子集C上的渐进非扩张非自映像,利用CQ方法构造迭代序列{x_n),{y_n),并得到了强收敛结果.该结果推广和改进了文献T.H.Kim,H.-K.Xu[Strong convergence of modified Mann iterationsfor asympototically nonexpansive mappings and semigroups,Nonlinear Anal.64(2006)1140-1152]相应的结果.
接着,在q—一致光滑的Banach空间上,利用向阳非扩张映象特点构造迭代算法,来逼近变分不等式组的解的问题.该结果推广和改进了R.U.Verma[Generalconvergnece analysis for two-step projection methods and applications to variationalproblems,Appl.Math.Lett.18(2005)1286-1292]相应的结果.
最后,初步尝试利用计算机对相应算法进行算法实现即比较了Picard迭代序列,Mann迭代序列及Ishikawa迭代序列的收敛速度的问题,具有一定的理论意义和实际意义.
In this paper,firstly in Hilbert spaces H,we consider the explicit averaging iterations of nonexpansive nonself mappings,and get strong convergent results. The results presented extend and improve the corresponding ones of Y.S. Song and R.D.Chen[Viscosity approximation methods for nonexpansive nonselfmappings, J.Math.Anal.Appl.321(2006)316-326]and S.Matsushita,D.Kuroiwa [Strong convergence of averaging iterations of nonexpansive mappings,J.Math. Anal.Appl.294(2004)206-214].
Then,we consider another iteration,that is,the iteration of asymptotically nonexpansive mappings of the nonempty closed convex subset C to Hilbert spaces H.We use CQ methods to construct the iterations {x_n},{y_n},and get strong convergent results.The results presented extend and improve the corresponding ones of T.H.Kim,H.-K.Xu[,Strong convergence of modified Mann iterations for asympototically nonexpansive mappings and semigroups,Nonlinear Anal. 64(2006)1140-1152].
Next,in q- uniformly Banach spaces,we construct iteration algorithm through sunny nonexpansive mappings to approach the solutions of a system of two nonlinear variational inequalities problem.The results presented extend and improve the corresponding ones of R.U.Verma[General convergnece analysis for two-step projection methods and applications to variational problems, Appl.Math.Lett.18(2005)1286-1292].
At last,we preliminarily attempt to use computer and compare the speed of the common iterative sequences such as Picard iterative sequence,Mann iterative sequence and Ishikawa iterative sequence.The result has certain theory significance and the practical significance.
引文
[1]L.E.J.Brouwer,Uber Abbildung von Manigfaltigkeiten,Math.Ann.1912,71:97-115.
[2]S.Banach,Sur les operations dans les ensembles abstraits et leur application aux equations integrals,Fund.Math.1922,3:133-181.
[3]K.Goebel,W.A.Kirk,A fixed point theorem for asymptotieally nonexpansive mappings,Proc.Amer.Math.Soc.1972,35(1):171-174.
[4]W.A.Kirk,Fixed Point theorems for non-Lipschitziau mappings of asymptotically nonexpansive type,Israel J.Math.1974,11:339-346.
[5]W.R.Mann,Mean value methods in iteration,Proe.Ame.rMath.Soc,1953,4:506-510.
[6]S.Ishikawa,Fixed points by a new iteration method,Proc.Amer.Math.Soc,1974,44:147-150.
[7]Browder.F.E.,Nonlinear mappings of nonexpansive and accretive type in Banach spaces,Bull.Amer.Math.Soc.,1967,73:875-882.
[8]Kato.T,Nonlinear semigroups and evolution equation,J.Math.Soc.Japan,1967,18/19:502-508.
[9]H.Y.Zhou,Iterative solution of nonlinear equations involving strongly accretive operators without the Lipschitz assumption,J.Math.Anal.Appl.,1997,213:296-307.
[10]H.Y.Zhou,Y.J.Cho,S.S.Chang,Approximating the fixed points of φ-hemicontractiongs by the Ishikawa iterative process with mixed errors in normed linear spaces,Nonlinear Anal,TMA,2001,47(7):4819-4826.
[11]C.E.Chidume,Convergence theorems.for strongly pseudo-contractive and strongly accretive maps,J.Math.Anal.Appl.,1998,228:254-264.
[12]S.S.Chang,Convergence of iterative methods for accretive and pseudocontractive type mappings in Banach spaces,Nonlinear Functional Anal.and Appl.,1999,4:1-23.
[13]M.O.Oslike,Stability of the Mann and Ishikawa iteration procedures for φ-strongly pseudo-contractions and nonlinear equations of the φ-strongly accretive type,J.Math.Anal.Appl.,1999,227:319-334.
[14]M.O.Oslike,Stability of the Ishikawa iteration methods for quasi-contractive maps,Indian J.Pure Appl.Math.,1997,28:1251-1265.
[15]X.P.Ding,A new class of generalized strongly nonlinear quasi-variational inequalities and quasi-complementarity problems,Indian J.Pure.Appl.Math.,1994,25(11):1115-1128.
[16]N.J.Huang,Mann and Ishikawa type perturbed iterative algorithms.for generalized nonlinear implicit quasi-variational inclusions,ComPut.Math.Appl.,1998,35:1-7.
[17]K.R.Kazmi,Mann and Ishikawa type perturbed iterative algorithms of generalized quasi-variational inclusions,J.Math.Anal.Appl.,1997,209:572-584.
[18]L.C.Zeng,Iterative algorithm.for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequalities,J.Math.Anal.Appl.,1996,201:180-194.
[19]C.Z.Zhang,Y.J.Cho and S.M.Wei,Variational inequalities and subjectivity for multi-valued monotone mappings,Topological Methods on Nonlinear Anal.,1998,12:169-178.
[20]Takahashi W.,Nonlinear Functional Analysis Fixed point Theory and its Appliciation,Japanese:Yokohama Publishers,2000.
[21]Reich S.,An iterative procedure for constructing zeros of accretive set in Banach spaces,Nonlinear Anal,1978,2:85-92
[22]Xu H.K.,Inequalities in Banach spaces with application,Nonlinear Anal.,1991,16:1127-1138
[23]Kirk W A,Sims B.Handbook of Metric Fixed Point Theory,Netherlands:Kluwer Academic Publishers,2001
[24]谷峰,高伟,田巍.不动点定理及非线性算子的迭代收敛性.哈尔滨:黑龙江科学技术出版社,2002
[25]Diestel J.,Geometry of Banach spaces-selected topics,lecture notes in Mathematics.NewYork:Springer,1975,485
[26]Jong Soo Jung,Iterative approaches to common.fixed points of nonexpansive mappings in Banach spaces,J.Math.Anal.Appl.302(2005):509-520.
[27]H.-K.Xu,Approximating curves of nonexpansive nonself mappings in Banach spaces,in:Mathematical Analysis,C.R.Acad.Sci.Paris 325(1997)151-156.
[28]R.U.Verma,A class of projection-contraction methods applied to monotone variational inequalities,Appl.Math.Lett.13(2000):55-62.
[29]H.Bauschke,The appproximation of fixed points of compositions of nonexpansive mappings in Hilbert space,J.Math.Anal.Appl.202(1996):150-159.
[30]Y.S.Song,R.D.Chen,Viscosity approximation methods.for nonexpansive nonself-mappings,J.Math.Anal.Appl.321(2006)316-326.
[31]S.Matsushita,D.Kuroiwa,Strong convergence of averaging iterations of nonexpansive mappings,J.Math.Anal.Appl.294(2004)206-214.
[32]H.-K.Xu,Viscosity approximation methods.for nonexpansive mappings,J.Math.Anal.Appl.298(2004)279-291.
[33]H.Bauschke,The appproximation of fixed points of compositions of nonexpansive mappings in Hilbert space,J.Math.Anal.Appl.202(1996):150-159.
[34]T.-H.Kim,H.-K.Xu,Strong convergence of modified Mann iterations for asympototically nonexpansive mappings and semigroups,Nonlinear Anal.64(2006)1140-1152.
[35]K.Goebel,W.A.Kirk,A fixed point theorem for asympototically nonexpausive mappings,J.Math.Anal.Appl.35(1972)171-174.
[36]C.E.Chidume,E.U.Ofoedu,H.Zegeye Strong and weak convergence theorems for asymptotically nonexpansive mappings,J.Math.Anal.Appl.280(2003)364-374.
[37]W.Takahashi,G.-E.Kim,Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces,Nonlinear Anal.32(1998)447-454.
[38]S.Y.Matsuhita,D.Kuroiwa,Strong convergence of averaging iteration of nonexpansive nonself mappings,J.Math.Anal.Appl.294(2004)206-214.
[39]W.Kaczor,Weak convergence of almost orbits of asymptoticall nonexpansive semigroup,J.Math.Anal.Appl.272(2002)565-574.
[40]J.S.Jung,S.S.Kim,Strong convergence theorems for nonexpansive nonself mappings in Banach spaces,Proc.Amer.Math.Soc.73(3)(1998)321-329.
[41]C.E.Chidume,N.Shahzad,Strong convergence of an implicit iteration process.for finite family of nonexpansive mappings,Nonlinear Anal.62(6)(2005)1149-1156.
[42]N.Shahzad,Approximating fixed points of nonself nonexpansive mappings in Banach spaces,Nonlinear Anal.61(2005):1031-1039.
[43]S.Matsushita,D.Kuroiwa,Strong convergence of averaging iterations of nonexpansive mappings,J.Math.Anal.Appl.294(2004)206-214.
[44]R.U.Verma,General convergnece analysis for two-step projection methods and applications to variational pwblems,Appl.Math.Lett.18(2005)1286-1292.
[45]R.U.Verma,Projection methods,algorithms and a new system of nonlinear variational inequalities,Comput.Math.Appl.41(2001)1025-1031.
[46]V.Berindee,Iterative Approximation of Fixed Point,Springer-Verlag Berlin Heidelberg,New York.2007.
[47]T.Zamfirescu,Fix point theorems in metric spaces,Archiv der Mathematik 23(1992),292-298.
[48]V.Berinde,Picard iteration converges faster than Mann iteration for a class of quasi-contravtive operators,Fixed Point Theroy and Applications(2004),no,2,97-105.
[49]G.BaBu and K.Vara Prasad,Mann Iteration Converges Faster than Ishikawa Iteration for the Class of Zamfirescu Operators,Fixed Point Theory of Applications,2006,1-6;
[50]L.Deng.,Iteration process.for nonlinear Lipschtizian stongly accretive mappings in L_p spaces,J.Math.Anal.Appl.188(1994)128-140.
[2]S.Banach,Sur les operations dans les ensembles abstraits et leur application aux equations integrals,Fund.Math.1922,3:133-181.
[3]K.Goebel,W.A.Kirk,A fixed point theorem for asymptotieally nonexpansive mappings,Proc.Amer.Math.Soc.1972,35(1):171-174.
[4]W.A.Kirk,Fixed Point theorems for non-Lipschitziau mappings of asymptotically nonexpansive type,Israel J.Math.1974,11:339-346.
[5]W.R.Mann,Mean value methods in iteration,Proe.Ame.rMath.Soc,1953,4:506-510.
[6]S.Ishikawa,Fixed points by a new iteration method,Proc.Amer.Math.Soc,1974,44:147-150.
[7]Browder.F.E.,Nonlinear mappings of nonexpansive and accretive type in Banach spaces,Bull.Amer.Math.Soc.,1967,73:875-882.
[8]Kato.T,Nonlinear semigroups and evolution equation,J.Math.Soc.Japan,1967,18/19:502-508.
[9]H.Y.Zhou,Iterative solution of nonlinear equations involving strongly accretive operators without the Lipschitz assumption,J.Math.Anal.Appl.,1997,213:296-307.
[10]H.Y.Zhou,Y.J.Cho,S.S.Chang,Approximating the fixed points of φ-hemicontractiongs by the Ishikawa iterative process with mixed errors in normed linear spaces,Nonlinear Anal,TMA,2001,47(7):4819-4826.
[11]C.E.Chidume,Convergence theorems.for strongly pseudo-contractive and strongly accretive maps,J.Math.Anal.Appl.,1998,228:254-264.
[12]S.S.Chang,Convergence of iterative methods for accretive and pseudocontractive type mappings in Banach spaces,Nonlinear Functional Anal.and Appl.,1999,4:1-23.
[13]M.O.Oslike,Stability of the Mann and Ishikawa iteration procedures for φ-strongly pseudo-contractions and nonlinear equations of the φ-strongly accretive type,J.Math.Anal.Appl.,1999,227:319-334.
[14]M.O.Oslike,Stability of the Ishikawa iteration methods for quasi-contractive maps,Indian J.Pure Appl.Math.,1997,28:1251-1265.
[15]X.P.Ding,A new class of generalized strongly nonlinear quasi-variational inequalities and quasi-complementarity problems,Indian J.Pure.Appl.Math.,1994,25(11):1115-1128.
[16]N.J.Huang,Mann and Ishikawa type perturbed iterative algorithms.for generalized nonlinear implicit quasi-variational inclusions,ComPut.Math.Appl.,1998,35:1-7.
[17]K.R.Kazmi,Mann and Ishikawa type perturbed iterative algorithms of generalized quasi-variational inclusions,J.Math.Anal.Appl.,1997,209:572-584.
[18]L.C.Zeng,Iterative algorithm.for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequalities,J.Math.Anal.Appl.,1996,201:180-194.
[19]C.Z.Zhang,Y.J.Cho and S.M.Wei,Variational inequalities and subjectivity for multi-valued monotone mappings,Topological Methods on Nonlinear Anal.,1998,12:169-178.
[20]Takahashi W.,Nonlinear Functional Analysis Fixed point Theory and its Appliciation,Japanese:Yokohama Publishers,2000.
[21]Reich S.,An iterative procedure for constructing zeros of accretive set in Banach spaces,Nonlinear Anal,1978,2:85-92
[22]Xu H.K.,Inequalities in Banach spaces with application,Nonlinear Anal.,1991,16:1127-1138
[23]Kirk W A,Sims B.Handbook of Metric Fixed Point Theory,Netherlands:Kluwer Academic Publishers,2001
[24]谷峰,高伟,田巍.不动点定理及非线性算子的迭代收敛性.哈尔滨:黑龙江科学技术出版社,2002
[25]Diestel J.,Geometry of Banach spaces-selected topics,lecture notes in Mathematics.NewYork:Springer,1975,485
[26]Jong Soo Jung,Iterative approaches to common.fixed points of nonexpansive mappings in Banach spaces,J.Math.Anal.Appl.302(2005):509-520.
[27]H.-K.Xu,Approximating curves of nonexpansive nonself mappings in Banach spaces,in:Mathematical Analysis,C.R.Acad.Sci.Paris 325(1997)151-156.
[28]R.U.Verma,A class of projection-contraction methods applied to monotone variational inequalities,Appl.Math.Lett.13(2000):55-62.
[29]H.Bauschke,The appproximation of fixed points of compositions of nonexpansive mappings in Hilbert space,J.Math.Anal.Appl.202(1996):150-159.
[30]Y.S.Song,R.D.Chen,Viscosity approximation methods.for nonexpansive nonself-mappings,J.Math.Anal.Appl.321(2006)316-326.
[31]S.Matsushita,D.Kuroiwa,Strong convergence of averaging iterations of nonexpansive mappings,J.Math.Anal.Appl.294(2004)206-214.
[32]H.-K.Xu,Viscosity approximation methods.for nonexpansive mappings,J.Math.Anal.Appl.298(2004)279-291.
[33]H.Bauschke,The appproximation of fixed points of compositions of nonexpansive mappings in Hilbert space,J.Math.Anal.Appl.202(1996):150-159.
[34]T.-H.Kim,H.-K.Xu,Strong convergence of modified Mann iterations for asympototically nonexpansive mappings and semigroups,Nonlinear Anal.64(2006)1140-1152.
[35]K.Goebel,W.A.Kirk,A fixed point theorem for asympototically nonexpausive mappings,J.Math.Anal.Appl.35(1972)171-174.
[36]C.E.Chidume,E.U.Ofoedu,H.Zegeye Strong and weak convergence theorems for asymptotically nonexpansive mappings,J.Math.Anal.Appl.280(2003)364-374.
[37]W.Takahashi,G.-E.Kim,Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces,Nonlinear Anal.32(1998)447-454.
[38]S.Y.Matsuhita,D.Kuroiwa,Strong convergence of averaging iteration of nonexpansive nonself mappings,J.Math.Anal.Appl.294(2004)206-214.
[39]W.Kaczor,Weak convergence of almost orbits of asymptoticall nonexpansive semigroup,J.Math.Anal.Appl.272(2002)565-574.
[40]J.S.Jung,S.S.Kim,Strong convergence theorems for nonexpansive nonself mappings in Banach spaces,Proc.Amer.Math.Soc.73(3)(1998)321-329.
[41]C.E.Chidume,N.Shahzad,Strong convergence of an implicit iteration process.for finite family of nonexpansive mappings,Nonlinear Anal.62(6)(2005)1149-1156.
[42]N.Shahzad,Approximating fixed points of nonself nonexpansive mappings in Banach spaces,Nonlinear Anal.61(2005):1031-1039.
[43]S.Matsushita,D.Kuroiwa,Strong convergence of averaging iterations of nonexpansive mappings,J.Math.Anal.Appl.294(2004)206-214.
[44]R.U.Verma,General convergnece analysis for two-step projection methods and applications to variational pwblems,Appl.Math.Lett.18(2005)1286-1292.
[45]R.U.Verma,Projection methods,algorithms and a new system of nonlinear variational inequalities,Comput.Math.Appl.41(2001)1025-1031.
[46]V.Berindee,Iterative Approximation of Fixed Point,Springer-Verlag Berlin Heidelberg,New York.2007.
[47]T.Zamfirescu,Fix point theorems in metric spaces,Archiv der Mathematik 23(1992),292-298.
[48]V.Berinde,Picard iteration converges faster than Mann iteration for a class of quasi-contravtive operators,Fixed Point Theroy and Applications(2004),no,2,97-105.
[49]G.BaBu and K.Vara Prasad,Mann Iteration Converges Faster than Ishikawa Iteration for the Class of Zamfirescu Operators,Fixed Point Theory of Applications,2006,1-6;
[50]L.Deng.,Iteration process.for nonlinear Lipschtizian stongly accretive mappings in L_p spaces,J.Math.Anal.Appl.188(1994)128-140.