Nonlinear functional analysis and nonlinear problems by using fixed point theory

利用不动点理论研究了非线性泛函分析和一些非线性问题。首先研究了Banach空间中极大单调算子零点的存在性。利用一个新的边界条件证明了极大单调算子的存在性定理。利用其中之一，我们得到了Kakutani多值映象不动点定理的一个推广。在此基础上，我们讨论了Banach空间中极大单调算子零点的迭代求解问题。在Banach空间中，利用广义投影得到了极大单调算子的两个新的预解式，进而得到了极大单调算子预解式的弱收敛定理和强收敛定理.特别地，我们利用广义投影和度量投影得到了Solodov和Svaiter定理的两个推广。在Banach空间中引入了相对非扩张映射的概念，推广了Hilbert空间中的非扩张映射。然后，得到了Banach空间中相对非扩张映象的两个收敛定理。其中之一是Nakajo和Takahash定理的推广。我们还得到了与一个新的预解式有关的向阳广义非扩张收缩的重要例子。其次，我们介绍了在Hilbert空间中寻找非扩张映射的不动点集和逆强单调算子的解集的公共元素的迭代方法。进一步地，我们将这种迭代方法推广到寻找Hilbert空间中非扩张映射的不动点集和平衡问题的解集的公共元素。然后证明了一些弱收敛定理和强收敛定理。利用这些结果改进和推广了Wittmann强收敛定理、赖希弱收敛定理和Baillon非线性遍历定理。

We studied nonlinear functional analysis and some nonlinear problems by using fixed point theory. We first studied the existence of zero points of maximal monotone operators in Banach spaces. We proved existence theorems for maximal monotone operators by a new boundary condition. Using one of them, we obtained a generalization of Kakutani's fixed point theorem for multivalued mappings. Then we considered iteration schemes of finding zero points of maximal monotone operators in Banach spaces. We found two new resolvents for maximal monotone operators and then obtained weak and strong convergence theorems for resolvents of maximal monotone operators in Banach spaces with generalized projections. In particular, we obtained two generalizations of Solodov and Svaiter's theorem by using generalized projections and metric projections. Further, we introduced the notion of relatively nonexpansive mappings in a Banach space which generalize nonexpansive mappings in a Hilbert space. Then, we obtained two convergence theorems for relatively nonexpansive mappings in Banach spaces. One of them is a generalization of Nakajo and Takahash's theorem. We also obtained important examples of sunny generalized nonexpansive retractions which are related to one of new resolvents. Next, we introduced an iteration scheme of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an inverse-strongly-monotone operator in a Hilbert space. Further, we extended this iteration scheme to that of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space. Then we proved some weak and strong convergence theorems. Using these results, we improved and extended Wittmann' strong convergence theorem, Reich's weak convergence theorem and Baillon's nonlinear ergodic theorem which are well known in this field.