算子不动点逼近理论及其应用
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摘要
本文主要在不同的空间框架下讨论几类算子不动点的迭代逼近问题,研究其算法设计、算法的收敛性以及它们在变分包含问题中的应用。同时,我们利用逼近理论和方法研究了一类多个多值随机算子的随机对合点和随机不动点的存在性问题。本文共分为六章。
在第一章引言中,我们介绍了研究背景和主要研究结果。
第二章是预备知识,主要包括几类算子的定义以及后面证明所需要的若干引理。
第三章是在一般的实赋范线性空间中讨论了多值Φ-半压缩映射不动点的逼近问题。我们引入并研究了新的变系数的带误差项的Ishikawa类迭代算法,在映射值域和定义域均没有有界性要求的情况下,给出了一致连续的多值Φ-半压缩映射不动点的迭代逼近定理。
第四章是在一致光滑的实Banach空间框架下讨论几类算子不动点的逼近问题。
在§4.1中,引入并研究了新的变系数的带误差项的最速下降算法,在映射无连续性假设下,获得了Φ-半增生类映射零点的迭代序列的收敛性。然后通过增生类映射零点和压缩类映射不动点的转化关系,得到了Φ-半压缩类映射的一个新的不动点定理。
在§4.2中,在p-一致光滑空间框架下,我们讨论了多值广义Φ-半压缩映射不动点的逼近问题,引入并研究了新的变系数的带误差项的Ishikawa类迭代算法,在映射值域和定义域均没有有界性要求的条件下、对映射也没有连续性假设的情况下,采用了新的证明方法,获得了多值广义Φ-半压缩映射不动点的逼近定理。然后利用该方法,在映射值域有界的条件下,把结果推广到一致光滑空间中。
在§4.3中,在Hilbert空间框架下,我们讨论了渐近κ-严格伪压缩映射不动点的逼近问题,引入并研究了新的变系数的CQ类算法,采用了新的证明方法,获得了渐近κ-严格伪压缩映射不动点的逼近定理,取消了近期文献对映射的某种有界性要求。
第五章是在可分的完备度量空间中,用逼近理论和方法讨论了一类多个多值随机算子的随机对合点和随机不动点的存在性问题。首先我们证明了一个选子定理,然后给出一些新的随机对合点和随机不动点定理。由于去除了算子的紧性条件,所获得的结果即使是在非随机情形下,也是对已有结果的改进和推广。
在最后一章中,我们主要利用不动点逼近理论和方法来讨论如下两个变分包含问题,获得了两个变分包含问题解的逼近定理。
1、设T,A:X_→X,N(·,·):X×X→X,g:X→X~*,η:X~*×X~*→X~*是五个映射,φ:X~*→R∪{+∞}为具有η-次微分α_ηφ的真凸泛函。对给定的f∈X,求u∈X,使得
2、设T, A:X→2~X,g:X→X~*,η:X~*×X~*→X~*是四个映射,φ:X~*→R∪{+∞}具有η-次微分α_ηφ的的真凸泛函。对给定的f∈X求x~*∈X,u∈Tx~*,v∈Ax~*,记作(x~*,u,v),使得
In this thesis, we discuss mainly the iterative approximation problems of fixed points of several classes of operators, including their algorithm designs, the con-vergence of their algorithms and their applications in variational inclusions in the framework of several different spaces. Also, using the approximation theory and methods, we study the existence of random coincidence points and random fixed points of a class of random multi-valued operators. This thesis is divided into six chapters.
In Chapter 1, we introduce the research backgrounds and the main results of this thesis.
Chapter 2 is preliminaries, mainly including some definitions and lemmas which will be used in the following.
In Chapter 3, we discuss the iterative approximation of fixed points of multi-valuedΦ-hemicontractive mappings in real normed linear spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. General results on the iterative approximation of fixed points of uni-formly continuousΦ-hemicontractive mappings without boundedness conditions on the ranges and the domains are given in real normed linear spaces.
In Chapter 4, we discuss the iterative approximation of fixed points of several classes of operators in the framework of uniformly smooth real Banach spaces.
In§4.1, we introduce and study a new steepest descent approximation algo-rithm with error items and variable coefficients. A result on the convergence of the iterative sequencess of zero points ofΦ-hemiaccretive mappings without continuity assumption is given. Since the zeros of Φ-hemiaccretive mappings and the fixed points of correspondingΦ-hemicontractive mappings may transform mutually, we obtain a new fixed point theorem forΦ-hemicontractive mappings.
In§4.2, we discuss the iterative approximation of fixed points of multi-valued generalizedΦ-hemicontractive mappings in the framework of p-uniformly smooth real Banach spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. Some new methods are applied. A new fixed point theorem for generalizedΦ-hemicontractive mappings without continuity assumption and without boundedness conditions on the ranges and the domains of the mappings is obtained in p uniformly smooth real Banach spaces. In case of generalizedΦ-hemicontractive mappings with bounded ranges, the result is extended to uniformly smooth real Banach spaces.
In§4.3, we discuss the iterative approximation of fixed points of asymptoticallyκ-strictly pseudo-contractive mappings in the framework of real Hilbert spaces. We introduce and study some new CQ-type algorithms with variable coefficients. New methods are applied in the proof of the iterative approximation theorem of fixed point of asymptoticallyκ-strictly pseudo-contractive mappings. Some boundedness conditions for the mappings in recent literatures are dropped in this thesis.
In Chapter 5, by using fixed point approximation theory and methods, we discuss the existence of random coincidence points and random fixed points of a class of random multi-valued operators in separable complete metric spaces. First, we prove a selector theorem, then present some new random coincidence point and random fixed point theorems for multifunctions without Compactness conditions. Even in the non-random case, our results also improve and extend some known results.
In the last chapter, fixed point approximation theory and methods are applied to discuss two set-valued variational inclusion problems as follows. Two iterative approximation theorems of solutions to set-valued variational inclusion problems are obtained.
1、Let T,A : X→X, N(·,·) : X×X→X, g : X→X~*η: X~*×X~*→X~* be five mappings andφ: X~*→R∪{+∞} be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding u∈X such thatwhereα_ηφdenotesη-subgradients ofφ.
2、Let T, A : X→2~X, g : X→X~*,η: X~*×X~*→X~* be four mappings andφ:X~*→R∪{+∞}be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding x~*∈X, u∈Tx~*,v∈Ax~*, denoting (x~*,u,v), such thatwhereα_ηφdenotesη-subgradients ofφ.
在第一章引言中,我们介绍了研究背景和主要研究结果。
第二章是预备知识,主要包括几类算子的定义以及后面证明所需要的若干引理。
第三章是在一般的实赋范线性空间中讨论了多值Φ-半压缩映射不动点的逼近问题。我们引入并研究了新的变系数的带误差项的Ishikawa类迭代算法,在映射值域和定义域均没有有界性要求的情况下,给出了一致连续的多值Φ-半压缩映射不动点的迭代逼近定理。
第四章是在一致光滑的实Banach空间框架下讨论几类算子不动点的逼近问题。
在§4.1中,引入并研究了新的变系数的带误差项的最速下降算法,在映射无连续性假设下,获得了Φ-半增生类映射零点的迭代序列的收敛性。然后通过增生类映射零点和压缩类映射不动点的转化关系,得到了Φ-半压缩类映射的一个新的不动点定理。
在§4.2中,在p-一致光滑空间框架下,我们讨论了多值广义Φ-半压缩映射不动点的逼近问题,引入并研究了新的变系数的带误差项的Ishikawa类迭代算法,在映射值域和定义域均没有有界性要求的条件下、对映射也没有连续性假设的情况下,采用了新的证明方法,获得了多值广义Φ-半压缩映射不动点的逼近定理。然后利用该方法,在映射值域有界的条件下,把结果推广到一致光滑空间中。
在§4.3中,在Hilbert空间框架下,我们讨论了渐近κ-严格伪压缩映射不动点的逼近问题,引入并研究了新的变系数的CQ类算法,采用了新的证明方法,获得了渐近κ-严格伪压缩映射不动点的逼近定理,取消了近期文献对映射的某种有界性要求。
第五章是在可分的完备度量空间中,用逼近理论和方法讨论了一类多个多值随机算子的随机对合点和随机不动点的存在性问题。首先我们证明了一个选子定理,然后给出一些新的随机对合点和随机不动点定理。由于去除了算子的紧性条件,所获得的结果即使是在非随机情形下,也是对已有结果的改进和推广。
在最后一章中,我们主要利用不动点逼近理论和方法来讨论如下两个变分包含问题,获得了两个变分包含问题解的逼近定理。
1、设T,A:X_→X,N(·,·):X×X→X,g:X→X~*,η:X~*×X~*→X~*是五个映射,φ:X~*→R∪{+∞}为具有η-次微分α_ηφ的真凸泛函。对给定的f∈X,求u∈X,使得
2、设T, A:X→2~X,g:X→X~*,η:X~*×X~*→X~*是四个映射,φ:X~*→R∪{+∞}具有η-次微分α_ηφ的的真凸泛函。对给定的f∈X求x~*∈X,u∈Tx~*,v∈Ax~*,记作(x~*,u,v),使得
In this thesis, we discuss mainly the iterative approximation problems of fixed points of several classes of operators, including their algorithm designs, the con-vergence of their algorithms and their applications in variational inclusions in the framework of several different spaces. Also, using the approximation theory and methods, we study the existence of random coincidence points and random fixed points of a class of random multi-valued operators. This thesis is divided into six chapters.
In Chapter 1, we introduce the research backgrounds and the main results of this thesis.
Chapter 2 is preliminaries, mainly including some definitions and lemmas which will be used in the following.
In Chapter 3, we discuss the iterative approximation of fixed points of multi-valuedΦ-hemicontractive mappings in real normed linear spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. General results on the iterative approximation of fixed points of uni-formly continuousΦ-hemicontractive mappings without boundedness conditions on the ranges and the domains are given in real normed linear spaces.
In Chapter 4, we discuss the iterative approximation of fixed points of several classes of operators in the framework of uniformly smooth real Banach spaces.
In§4.1, we introduce and study a new steepest descent approximation algo-rithm with error items and variable coefficients. A result on the convergence of the iterative sequencess of zero points ofΦ-hemiaccretive mappings without continuity assumption is given. Since the zeros of Φ-hemiaccretive mappings and the fixed points of correspondingΦ-hemicontractive mappings may transform mutually, we obtain a new fixed point theorem forΦ-hemicontractive mappings.
In§4.2, we discuss the iterative approximation of fixed points of multi-valued generalizedΦ-hemicontractive mappings in the framework of p-uniformly smooth real Banach spaces. We introduce and study some new Ishikawa-type iterative algorithms with error items and variable coefficients. Some new methods are applied. A new fixed point theorem for generalizedΦ-hemicontractive mappings without continuity assumption and without boundedness conditions on the ranges and the domains of the mappings is obtained in p uniformly smooth real Banach spaces. In case of generalizedΦ-hemicontractive mappings with bounded ranges, the result is extended to uniformly smooth real Banach spaces.
In§4.3, we discuss the iterative approximation of fixed points of asymptoticallyκ-strictly pseudo-contractive mappings in the framework of real Hilbert spaces. We introduce and study some new CQ-type algorithms with variable coefficients. New methods are applied in the proof of the iterative approximation theorem of fixed point of asymptoticallyκ-strictly pseudo-contractive mappings. Some boundedness conditions for the mappings in recent literatures are dropped in this thesis.
In Chapter 5, by using fixed point approximation theory and methods, we discuss the existence of random coincidence points and random fixed points of a class of random multi-valued operators in separable complete metric spaces. First, we prove a selector theorem, then present some new random coincidence point and random fixed point theorems for multifunctions without Compactness conditions. Even in the non-random case, our results also improve and extend some known results.
In the last chapter, fixed point approximation theory and methods are applied to discuss two set-valued variational inclusion problems as follows. Two iterative approximation theorems of solutions to set-valued variational inclusion problems are obtained.
1、Let T,A : X→X, N(·,·) : X×X→X, g : X→X~*η: X~*×X~*→X~* be five mappings andφ: X~*→R∪{+∞} be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding u∈X such thatwhereα_ηφdenotesη-subgradients ofφ.
2、Let T, A : X→2~X, g : X→X~*,η: X~*×X~*→X~* be four mappings andφ:X~*→R∪{+∞}be a proper convex functional such thatφisη-subdifferentiable. For any given f∈X, finding x~*∈X, u∈Tx~*,v∈Ax~*, denoting (x~*,u,v), such thatwhereα_ηφdenotesη-subgradients ofφ.
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