Banach 空间中非扩张映象的不动点性质及其迭代算法研究

本项目主要研究 Banach 空间中非扩张映象的不动点存在性及其迭代算法。首先我们在自反 Banach 空间中研究非扩张映象不动点的存在性；其次我们研究两个算子和的不动点性质，也给出在非线性积分方程中的应用；最后我们将 Hilbert 空间的变分不等式问题推广到一般的 Banach 空间中，建立变分不等式问题与不动点问题的等价性命题。我们构成一些迭代算法，证明由这些算法产生的序列强收敛到变分不等式问题解集与非扩张映象不动点集的公共元。

In project,we mainly study the fixed point property of nonexpansive mappings in Banach spaces and iterative algorithms. Firstly, we investigate the fixed point property of nonexpansive mappings in reflexive Banach spaces; Secondly, we study the fixed point property of sums of two operators, we also give some applications to solve some nonlinear integral equations; Finally, we will extend the variational inequality problems from Hilbert spaces to more general Banach spaces, equivalence proposition between variational inequality problems and fixed point problems is given. We introduce some iterative algorithms and prove that the sequences generated by these iterative algorithms converge strongly to a common element of the set of solutions of variational inequality problems and the fixed point sets of nonexpansive mappings.