广义变分不等式的若干类算法
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摘要
近年来,变分不等式理论已成为研究大量纯粹数学和应用科学领域中非线性问题的有效工具,如数学规划,最优化,力学,弹性理论,运输,经济平衡,渗流介质以及数学与工程科学许多别的分支。由于自身的发展和应用到别的学科,利用各种新颖的技巧,变分不等式已朝不同方向被推广。本文分别在Hilbert空间、Banach空间框架下,研究了变分不等式(组)(包含(组))的解的存在性和迭代算法的收敛性。具体内容如下:
1.简要叙述了变分不等式理论研究的历史背景。
2.回顾了文中将要用到的一些基本概念和理论。
3.在自反Banach空间中引入和研究了一类新的完全广义拟似变分包含,利用J~η邻近映射给出了此类变分包含近似解的迭代算法,并证明所构造的迭代算法生成的迭代序列的收敛性。
4.用更弱的弱压缩映射来代替压缩映射,我们引入了分别由(4.1.1.1)和(4.2.1.1)定义的两类隐式粘性迭代序列{x_t)和{z_m],并证明了这两个序列都收敛于变分不等式(4.1.1.2)的唯一解。
5.在Hilbert空间中引入和研究了一类新的完全广义强非线性混合似变分不等式组,并证明了其辅助变分不等式问题解的存在唯一性。基于该辅助问题,我们构造了一个迭代算法,分析了由该算法产生的迭代序列的收敛性。
6.我们给出(H,η)-增生算子和广义(A,η)-增生算子定义,并引入和研究了含(H,η)-增生算子的集值变分包含组和含广义(A,η)-增生算子的集值非线性变分包含。利用与(H,η)-增生算子有关的和与广义(A,η)-增生算子有关的预解算子,构造了迭代算法并给出了由这两个算法生成的迭代序列的收敛性。
In recent years,variational inequality theory has been become very effective and powerful tools for studying a wide class of nonlinear problems arising in many diverse fields of pure mathematics and applied sciences,such as mathematical programming,optimization,mechanics, elasticity,transportation,economic equilibrium,fluid flow through porous media and many other branches of mathematical and engineering science.Variational inequalities have been extended and generalized in different directions by using novel and innovative techniques both for own sake and for its application.We arrange this dissertation as follows:
1.The historic background of variational inequality theory is recalled briefly.
2.We recall some basic concepts and theories.
3.A new class of completely generalized quasi-variational-like inclusions in reflexive Banach spaces is introduced and studied.Using the J~η—proximal mapping,two iterative algorithms to compute approximate solutions for this class of completely generalized quasi-variational-like inclusions are suggested and analyzed,and the convergence of the iterative sequences generated by the algorithms is also proved.
4.We study the strong convergence for the viscosity iterative sequences {x_t} and {z_m} defined by(4.1.1.1) and(4.2.1.1),respectively.We prove that {x_t} and {z_m} converges strongly to some p∈F(T),where p is a unique solution to the variational inequality(4.1.1.2).
5.In real Hilbert spaces,a new system of completely generalized strongly nonlinear mixed variational-like inequalities(SCGSNMVLI) is introduced.We establish an existence and uniqueness theorem of solutions to the auxiliary variational inequality problems for the SCGSNMVLI. Based on the auxiliary problems,we construct an iterative algorithm to compute the approximate solutions of the SCGSNMVLI.And also we give the convergence analysis of the iterative sequences generalized by the algorithm.
6.We first introduce and study a new system of multi-valued variational inclusions involving (H,η)-accretive operators in Banach spaces.Using the resolvent operator associated with (H,η)-accretive operators,we construct an algorithm of this system and prove the convergence of the iterative sequences generated by the algorithm.Then,we introduce a new concept of generalized (A,η)-accretive mappings,study some properties of generalized(A,η)-accretive mappings and define resolvent operators associated with generalized(A,η)-accretive mappings.In terms of the new resolvent operator technique,we construct an algorithm for a class of multi-valued nonlinear variational inclusions involving generalized(A,η)-accretive mappings and prove the convergence of the iterative sequences generated by the algorithm.
1.简要叙述了变分不等式理论研究的历史背景。
2.回顾了文中将要用到的一些基本概念和理论。
3.在自反Banach空间中引入和研究了一类新的完全广义拟似变分包含,利用J~η邻近映射给出了此类变分包含近似解的迭代算法,并证明所构造的迭代算法生成的迭代序列的收敛性。
4.用更弱的弱压缩映射来代替压缩映射,我们引入了分别由(4.1.1.1)和(4.2.1.1)定义的两类隐式粘性迭代序列{x_t)和{z_m],并证明了这两个序列都收敛于变分不等式(4.1.1.2)的唯一解。
5.在Hilbert空间中引入和研究了一类新的完全广义强非线性混合似变分不等式组,并证明了其辅助变分不等式问题解的存在唯一性。基于该辅助问题,我们构造了一个迭代算法,分析了由该算法产生的迭代序列的收敛性。
6.我们给出(H,η)-增生算子和广义(A,η)-增生算子定义,并引入和研究了含(H,η)-增生算子的集值变分包含组和含广义(A,η)-增生算子的集值非线性变分包含。利用与(H,η)-增生算子有关的和与广义(A,η)-增生算子有关的预解算子,构造了迭代算法并给出了由这两个算法生成的迭代序列的收敛性。
In recent years,variational inequality theory has been become very effective and powerful tools for studying a wide class of nonlinear problems arising in many diverse fields of pure mathematics and applied sciences,such as mathematical programming,optimization,mechanics, elasticity,transportation,economic equilibrium,fluid flow through porous media and many other branches of mathematical and engineering science.Variational inequalities have been extended and generalized in different directions by using novel and innovative techniques both for own sake and for its application.We arrange this dissertation as follows:
1.The historic background of variational inequality theory is recalled briefly.
2.We recall some basic concepts and theories.
3.A new class of completely generalized quasi-variational-like inclusions in reflexive Banach spaces is introduced and studied.Using the J~η—proximal mapping,two iterative algorithms to compute approximate solutions for this class of completely generalized quasi-variational-like inclusions are suggested and analyzed,and the convergence of the iterative sequences generated by the algorithms is also proved.
4.We study the strong convergence for the viscosity iterative sequences {x_t} and {z_m} defined by(4.1.1.1) and(4.2.1.1),respectively.We prove that {x_t} and {z_m} converges strongly to some p∈F(T),where p is a unique solution to the variational inequality(4.1.1.2).
5.In real Hilbert spaces,a new system of completely generalized strongly nonlinear mixed variational-like inequalities(SCGSNMVLI) is introduced.We establish an existence and uniqueness theorem of solutions to the auxiliary variational inequality problems for the SCGSNMVLI. Based on the auxiliary problems,we construct an iterative algorithm to compute the approximate solutions of the SCGSNMVLI.And also we give the convergence analysis of the iterative sequences generalized by the algorithm.
6.We first introduce and study a new system of multi-valued variational inclusions involving (H,η)-accretive operators in Banach spaces.Using the resolvent operator associated with (H,η)-accretive operators,we construct an algorithm of this system and prove the convergence of the iterative sequences generated by the algorithm.Then,we introduce a new concept of generalized (A,η)-accretive mappings,study some properties of generalized(A,η)-accretive mappings and define resolvent operators associated with generalized(A,η)-accretive mappings.In terms of the new resolvent operator technique,we construct an algorithm for a class of multi-valued nonlinear variational inclusions involving generalized(A,η)-accretive mappings and prove the convergence of the iterative sequences generated by the algorithm.
引文
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[9]L.Ameria,Continuous solution of problem of a string vibrating against an obstacle,Rend.Sem.Mat.Univ.,Padova,59:67-96,1974.
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[16]B.He,A new method for a class of linear variational inequalities,Math.Programming,66:137-144,1994.
[17]B.He,Solving a class of linear projection equations,Number Math.,68:71-80,1994.
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[35]M.A.Noor,Iterative methods for general mixed quasivariational inequalities,J.Optim.Theoty Appl.,119:123-136,2003.
[36]A.Moudafi,Viscosity approximation methods for fixed-points problems,J.Math.Anal.Appl.,241:46-55,2000.
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