【作者】 刘霞；

【导师】 崔艳兰；

【作者基本信息】 延安大学 ， 基础数学， 2010， 硕士

【摘要】 非扩张映像的不动点问题在近代数学分支中有着广泛的应用,如非线性半群,遍历定理和单调算子理论中.最小范数不动点是满足最小范数性质的不动点,它的应用也非常广泛,如在最小二乘法问题,有界线性算子的伪逆是用最小范数解定义的,同时它也应用于最小化问题中,因此研究非扩张映像的最小范数不动点的重要性是不言而喻的.全文的主要研究内容及取得成果如下：1.介绍非线性泛函分析的发展及不动点的概况,同时给出国内外对不动点研究的动态以及本文的研究方案和内容.2.在Hilbert空间中,利用著名的Browder和Halpern方法,讨论了非扩张自映像的最小范数不动点,并给出了最小范数不动点在极小化问题中的应用,除此之外对上述两方法作了补充.3.在Hilbert空间中,借助弱内项条件的性质,对非扩张非自映像的最小范数不动点问题作了研究.4.在Hilbert空间中引入并研究了一类新型的非扩张映像,即有限族非扩张映像,分别利用循环算法和平行算法讨论了这类非扩张映像的公共最小范数不动点.更多还原

【Abstract】 The fixed point of nonexpansive mappings is widely used in modern mathematics, such as in nonlinear hemigroups, ergodic theorem and the theorem of monotone operator. Mminimum-norm fixed point is a kind of fixed point that satisfies the property of lest-norm, which is used broadly as well as fixed point. For instance, in the least-squares problems, the least-norm solutions are used to define the pseudoinverse of bounded linear operators, further more, it is aslo used in finding the solutions of minimization problem. So it is improtant for us to do the reserch in this field. The main contents of this paper are as follows:1.After the introduction of Nonlinear Function Analysis, the realm of fixed point and the research approach of this paper are presented, respectively.2.In Hilbert space, the famous methods of Browder’s and Halpern’s are used in finding the minimum-norm fixed point of nonexpansive self-mapping,meanwhile, an application in minimization problem is also included, at the same time, the above two methods are expanded.3.In Hilbert space, under the discussion of 1, another question arising about finding the minimum-norm fixed point of a nonexpansive non-self mapping which satisfies the weak inwardness condition.4.In Hilbert space, a new kind of nonexpansive mappings (that is a finite family of nonexpansive mappings) is introduced and studied, both cyclic and parallel iteration methods are introduced to find the common minimal-norm fixed points of a finite family of nonexpansive mappings.更多还原