变分包含的迭代解及其应用
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摘要
本文主要研究了含有不同算子的变分包含的迭代解及其应用和逐次渐近Φ-强半压缩型有限算子簇的多步迭代程序的收敛性问题,在条件更弱的情况下,我们的结果推广或改进现有的相关结果.具体阐述如下：
在第一章中,主要介绍了问题的研究背景及本课题已有的相关成果.
在第二章中,主要研究了一类含有(A,η)-单调算子的广义非线性隐拟变分包含组的迭代解,在不满足一致光滑或自反等条件的一般巴拿赫空间中,利用邻近映射的技巧,构造了新的迭代算法来逼近改变分包含组的解,证明了解的存在性和算法的强收敛性.其结果是现有一些成果的推广和改进,详见第二章.
在第三章中,讨论了三类问题的公共解问题,主要通过构造新的迭代算法,求出类含有(α,β)-松弛强制映射的变分不等式的迭代解,且使得该解同时满足广义平衡问题又是无限簇非扩张映射的不动点.我们证明了该新算法生成的迭代序列强收敛到含有(α,β)-松弛强制映射的变分不等式、广义平衡问题和非扩张映射无限簇的公共元,且该元为最小化问题的最优条件.本章的结果是近来相关文献中结果的推广和改进,详见第三章.
在第四章中,主要研究了含严格伪压缩有限簇和非扩张映射无限簇的复合迭代算法问题.通过对通常的Mann迭代算法进行修正,构造了一含有限个严格伪压缩和无限个非扩张映射簇的新迭代算法,证明了该迭代算法强收敛于这有限个严格伪压缩和无限个非扩张映射簇的公共不动点,且该不动点为某变分不等式的解.这些结果本质地推广和改进近来许多已有的相应结果,详见第四章.
最后在第五章中,对一致广义Lipschitz连续的逐次渐近Φ-强半压缩型有限算子簇研究了在一致光滑Banach空间中具误差的修正多步Noor迭代序列强收敛于该算子簇的公共不动点问题,作为所得结果的应用,得到了2007年Huang Z.Y.在相同空间框架中所建立的逼近具有有界值域的逐次Φ-强伪压缩算子的不动点具误差的修正Mann迭代和具误差的修正Ishikawa迭代两者的收敛是等价的这一结果,而且所用的方法不同于Huang Z. Y.,同时,它还改进和推广了Rhoades B. E和Soltuz S. M.、Huang Z.Y., BuF.W.和Noor M. A.、Huang Z.Y.和Bu F.W.、Su K.、Yao Y. H., Chen R. D.和Zhou H. Y.、Liu Z.Q., Kim J. K., Kim K. H.、Liu L.S.、Ni R. X.和Xu Y.G.等人的近期相应结果.
In this paper,we studied the iterative solution of variational inclusions with different operators and its application and the c convergence of mulit-step iterations for a finite fam-ily of successively asymptotically stronglyΦ-hemicontractive type operations.In the case of weaker conditions, our results extendor improve the existing relevant results.Specifically stated as follows.
In the chapter l,we introduces the research background and the relevant results.
In the chapter 2,we give the notion of proximal mapping associated with the (A,η)-monotone operator and study a new system of generalized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operator in Banach spaces without usual uniform smoothness or reflexiveness. By using the proximal mapping technique,we con-structed some iterative algorithms to approximate the solutions of a new system of gen-eralized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operators in Banach spaces without usual uniform smoothness or reflexiveness.Then proved the exis-tence of solutions and the convergence of the sequences generated by the algorithms.The results in this chapter extend and improve some well-known results in the literature.More details please read the chapter 2.
In the chapter 3,we introduce an iterative scheme for finding a common element of the set of a generalized equilibrium problems,the set of common fixed points of a fam-ily of infinitely non expansive mappings and the set of the variational inequality for a relaxed (α,β)-coercive mapping in Hilbert space.We prove strong convergence of the iter-ative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality,which is the optimality condition for the minimization problem.The results in this paper extend and improve some well-known results in the literature.More details please read the chapter 3.
In the chapter 4,Mainly Studied the composite iterative algorithm for finite family of strict pseudo-contractions and infinite family of nonexpansive mappings in Hilbert spaces. by modifying the normal Mann's iteration process,We construct a new iterative algo-rithm for finite family of strict pseudo-contractions and infinite family of non expansive mappings in Hilbert spaces. Under suitable conditions, a strong convergence theorem for approximating a common fixed point of the above two sets are obtained and this fixed point is also a solution of the variational inequality. Our results extend and improve the recent ones announced by many others.More details please read the chapter 4.
In the chapter 5, the problems which modified multi-step Noor iterations with er-rors converges strongly to a common fixed point are investigated for a finite family of uniformly generalized Lipschitz continuous and successively asymptotically stronglyΦ-hemicontractive type operators in uniformly smooth Banach spaces. A s application, it is obtained that the result of Huang Zhenyu at 2007 concerning the equivalence of the convergence criteria between modified Mann iterations with errors and modified multi-step Noor iterations with errors for successivelyΦ-strongly pseudo-contractive operators with bounded range in same spaces. Furthermore, the methods of proofs are quite differ-ent from Huang Zhenyu's. Meanwhile, these results improve and generalize many recent corresponding results obtained by Rhoades B. E., Soltuz S. M.、Huang Z.Y., Bu F.W., Noor M. A.、Huang Z.Y., Bu F.W.、Su K.、Yao Y. H., Chen R. D., Zhou H. Y.、Liu Z.Q., Kim J. K., Kim K.H.、Liu L.S.、Ni R. X.、Xu Y.G. and the other authors.More details please read the chapter 5.
在第一章中,主要介绍了问题的研究背景及本课题已有的相关成果.
在第二章中,主要研究了一类含有(A,η)-单调算子的广义非线性隐拟变分包含组的迭代解,在不满足一致光滑或自反等条件的一般巴拿赫空间中,利用邻近映射的技巧,构造了新的迭代算法来逼近改变分包含组的解,证明了解的存在性和算法的强收敛性.其结果是现有一些成果的推广和改进,详见第二章.
在第三章中,讨论了三类问题的公共解问题,主要通过构造新的迭代算法,求出类含有(α,β)-松弛强制映射的变分不等式的迭代解,且使得该解同时满足广义平衡问题又是无限簇非扩张映射的不动点.我们证明了该新算法生成的迭代序列强收敛到含有(α,β)-松弛强制映射的变分不等式、广义平衡问题和非扩张映射无限簇的公共元,且该元为最小化问题的最优条件.本章的结果是近来相关文献中结果的推广和改进,详见第三章.
在第四章中,主要研究了含严格伪压缩有限簇和非扩张映射无限簇的复合迭代算法问题.通过对通常的Mann迭代算法进行修正,构造了一含有限个严格伪压缩和无限个非扩张映射簇的新迭代算法,证明了该迭代算法强收敛于这有限个严格伪压缩和无限个非扩张映射簇的公共不动点,且该不动点为某变分不等式的解.这些结果本质地推广和改进近来许多已有的相应结果,详见第四章.
最后在第五章中,对一致广义Lipschitz连续的逐次渐近Φ-强半压缩型有限算子簇研究了在一致光滑Banach空间中具误差的修正多步Noor迭代序列强收敛于该算子簇的公共不动点问题,作为所得结果的应用,得到了2007年Huang Z.Y.在相同空间框架中所建立的逼近具有有界值域的逐次Φ-强伪压缩算子的不动点具误差的修正Mann迭代和具误差的修正Ishikawa迭代两者的收敛是等价的这一结果,而且所用的方法不同于Huang Z. Y.,同时,它还改进和推广了Rhoades B. E和Soltuz S. M.、Huang Z.Y., BuF.W.和Noor M. A.、Huang Z.Y.和Bu F.W.、Su K.、Yao Y. H., Chen R. D.和Zhou H. Y.、Liu Z.Q., Kim J. K., Kim K. H.、Liu L.S.、Ni R. X.和Xu Y.G.等人的近期相应结果.
In this paper,we studied the iterative solution of variational inclusions with different operators and its application and the c convergence of mulit-step iterations for a finite fam-ily of successively asymptotically stronglyΦ-hemicontractive type operations.In the case of weaker conditions, our results extendor improve the existing relevant results.Specifically stated as follows.
In the chapter l,we introduces the research background and the relevant results.
In the chapter 2,we give the notion of proximal mapping associated with the (A,η)-monotone operator and study a new system of generalized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operator in Banach spaces without usual uniform smoothness or reflexiveness. By using the proximal mapping technique,we con-structed some iterative algorithms to approximate the solutions of a new system of gen-eralized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operators in Banach spaces without usual uniform smoothness or reflexiveness.Then proved the exis-tence of solutions and the convergence of the sequences generated by the algorithms.The results in this chapter extend and improve some well-known results in the literature.More details please read the chapter 2.
In the chapter 3,we introduce an iterative scheme for finding a common element of the set of a generalized equilibrium problems,the set of common fixed points of a fam-ily of infinitely non expansive mappings and the set of the variational inequality for a relaxed (α,β)-coercive mapping in Hilbert space.We prove strong convergence of the iter-ative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality,which is the optimality condition for the minimization problem.The results in this paper extend and improve some well-known results in the literature.More details please read the chapter 3.
In the chapter 4,Mainly Studied the composite iterative algorithm for finite family of strict pseudo-contractions and infinite family of nonexpansive mappings in Hilbert spaces. by modifying the normal Mann's iteration process,We construct a new iterative algo-rithm for finite family of strict pseudo-contractions and infinite family of non expansive mappings in Hilbert spaces. Under suitable conditions, a strong convergence theorem for approximating a common fixed point of the above two sets are obtained and this fixed point is also a solution of the variational inequality. Our results extend and improve the recent ones announced by many others.More details please read the chapter 4.
In the chapter 5, the problems which modified multi-step Noor iterations with er-rors converges strongly to a common fixed point are investigated for a finite family of uniformly generalized Lipschitz continuous and successively asymptotically stronglyΦ-hemicontractive type operators in uniformly smooth Banach spaces. A s application, it is obtained that the result of Huang Zhenyu at 2007 concerning the equivalence of the convergence criteria between modified Mann iterations with errors and modified multi-step Noor iterations with errors for successivelyΦ-strongly pseudo-contractive operators with bounded range in same spaces. Furthermore, the methods of proofs are quite differ-ent from Huang Zhenyu's. Meanwhile, these results improve and generalize many recent corresponding results obtained by Rhoades B. E., Soltuz S. M.、Huang Z.Y., Bu F.W., Noor M. A.、Huang Z.Y., Bu F.W.、Su K.、Yao Y. H., Chen R. D., Zhou H. Y.、Liu Z.Q., Kim J. K., Kim K.H.、Liu L.S.、Ni R. X.、Xu Y.G. and the other authors.More details please read the chapter 5.
引文
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[5]X.P. Ding, H.R. Feng, Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclutions in Banach spaces[J], Applied Mathematics and Computation,2009,208:547-555.
[6]S.Plutbieng, R. Punpaeng, Ageneral iterative method for equlibrium problems and fixed point problems in Hilbert spaces[J], J.Math. Anal. Appl.2007,336:445-469.
[7]V.Colao, G.Marino, H.K.Xu, An iterative method for finding common solutions of equilibrium and fixed point problems[J], J.Math. Anal. Appl,2008,344:340-352.
[8]X.Qin, Y.Cho, Convergence of a general iterative method for nonexpansive mapping in Hilbert spaces[J], Journal of Computational and Applied Mathematics,2009,228: 458-465.
[9]Y. Yao, R. Chen, J.C. Yao, Strong convergence and certain control conditions for modified Mann iteration[J], Nonlinear Anal.2008,68:1687-1693.
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[11]X. Qin, Y. Cho, J.I. Kang, S.M. Kang, Strong convergence thorems for an infinite family of nonexpansive mappings in Banach spaces[J], Journal of Computational and Applied Mathematics,2009,230:121-127.
[12]H.R. Feng, X.P. Ding, A new system of generaqlized nonlinear quasi-varitional-like inclutions with A-monotone operators in Banach spaces[J], Journal of Computational and Applied Mathematics,2009,225:365-373.
[13]X.P. Ding, H.R. Feng, Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclutions in Banach spaces [J], Applied Mathematics and Computation,2009,208:547-555.
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[2]F.Q. Xia, N.J. Huang, Variational inclusions with a general H-monotone operator in Banach spaces[J], Comput. Math. Appl,2007,54:24-30
[3]R.U. Verma,A-monotonicity and applications nonlinear inclusion problems[J], J. Appl. Math. Stochastic Anal,2004,17(2):193-195.
[4]H.Y. Lan, A class of nonlinear (A,η)-Monotone operator inclution problems with relaxed cocoercive mappings [J], Adv. Nonlinear Var. Inequal,2006,9(2):1-11.
[5]X.P. Ding, H.R. Feng, Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclutions in Banach spaces[J], Applied Mathematics and Computation,2009,208:547-555.
[6]S.Plutbieng, R. Punpaeng, Ageneral iterative method for equlibrium problems and fixed point problems in Hilbert spaces[J], J.Math. Anal. Appl.2007,336:445-469.
[7]V.Colao, G.Marino, H.K.Xu, An iterative method for finding common solutions of equilibrium and fixed point problems[J], J.Math. Anal. Appl,2008,344:340-352.
[8]X.Qin, Y.Cho, Convergence of a general iterative method for nonexpansive mapping in Hilbert spaces[J], Journal of Computational and Applied Mathematics,2009,228: 458-465.
[9]Y. Yao, R. Chen, J.C. Yao, Strong convergence and certain control conditions for modified Mann iteration[J], Nonlinear Anal.2008,68:1687-1693.
[10]G. Marino, V. Colao, X. Qin, S.M. Kang, Strong convergence of the modified Mann iterative method for strict pseudo-contractions[J], Computers and Mathematics with Applications,2009,57:455-465.
[11]X. Qin, Y. Cho, J.I. Kang, S.M. Kang, Strong convergence thorems for an infinite family of nonexpansive mappings in Banach spaces[J], Journal of Computational and Applied Mathematics,2009,230:121-127.
[12]H.R. Feng, X.P. Ding, A new system of generaqlized nonlinear quasi-varitional-like inclutions with A-monotone operators in Banach spaces[J], Journal of Computational and Applied Mathematics,2009,225:365-373.
[13]X.P. Ding, H.R. Feng, Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclutions in Banach spaces [J], Applied Mathematics and Computation,2009,208:547-555.
[14]H.Iduka, W.Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings[J], Nonlinear Anal,2005,61:341-350.
[15]W.V. Petryshyn, A characterization of strictly of Banch spaces and other uses of duality mappings[J], J.Funct. Anal,1970,6:282-291.
[16]S.B.Nadler, Multivalued contraction mapping[J], Pacific J. Math,1969,30(3):457-488.
[17]Y.P. Fang and N.J. Huang, Iterative algorithm for system of variational inclutions involving H-accretive operators in Banach spaces[J], Acta, Math. Hungar,2005,108: 183-195.
[18]F.Flores-Bazan, Existence theorems for generalized noncoercive equilibrium prob-lems:the quasi-convex case[J], Siam J. Optim,2000,11:675-690.
[19]A.N.Iusem, W.F.Sosa, Iterative algorithms for equilibrium problems[J], Optimiza-tion,2003,52:301-316.
[20]S.D.Flam, A.S.Antipin, Equilibrium programming using proximal-like algorithms [J], Math. Program,1997,78:29-41.
[21]E.Blum, W.Oettli, From optimization and variational inequalities to equilibrium problems[J], Math. Stud,1994,63:123-145.
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