Fixed Point and Convergence Theorems for Firmly Nonexpansive－－Type Mappings in Banach Spaces

絕對非擴張型映射在巴拿赫空間上的定點及收斂性定理

本論文在巴拿赫空間上用加托可微函數定義絕對非擴張型映射，並同時討論非線性映射的定點定理及相關的收斂性理論。定點定理正好將香坂及高橋學者著名的絕對非擴張型映射定點定理在希式空間推展到更廣的巴拿赫空間上。此外，還運用本論文中的定點理論並參考索洛多夫和斯費特爾學者2000年提出的強收斂理論去開發搜尋單調映射零根的演算法。這個新開發的演算法使用較廣義的布雷格曼距離來取代原有索洛多夫和斯費特爾理論中的範數。論文最後舉例說明好的布雷格曼距離會較範數顯著的提升整個演算過程中收斂到零根的速度。

In this paper we define the class of firmly nonexpansive-type mappings relative to a G?teaux differentiable function ? in a Banach space, and study the fixed point and convergence problem for this type of nonlinear mapping. We properly extend the Kohsaka-Takahashi fixed point theorem for firmly nonexpansive-type mappings to this much wider class of nonlinear mappings. As an application of our new fixed point theorem, the relation between the zero points of a monotone mapping and the fixed points of its associated resolvent relative to a Bregman function is studied. Using the Bregman distance associated with a well-chosen Bregman function instead of the norm in the scheme of a proximal-type algorithm, we extend the Solodov-Svaiter strong convergence theorem in Hilbert spaces to Banach spaces.