Nonlinear problem by using fixed point theory

利用非线性泛函分析和不动点理论研究了非线性发展方程、数学经济学、数学规划和图像恢复等方面的一些非线性问题。我们首先证明了Banach空间中非扩张映射族的几个不动点定理。其次，我们证明了非扩张映射的非线性半群的Baillon型遍历定理。特别地，我们通过将Takahashi的结果推广到Banach空间来回答1996年在希腊雅典举行的第二届世界非线性分析家大会上提出的公开问题，并将其结果推广到Banach空间上的服从非扩张映射的半群。进一步，我们建立了非扩张映射族的Mann型弱收敛定理。我们还证明了非扩张映射族的Halpern型强收敛定理。最后，利用这些结果，我们讨论了非扩张收缩的凸组合的图像恢复问题，寻找交换的非扩张映射族的公共不动点问题，凸极小化问题等。

We studied some nonlinear problems concerning nonlinear evolution equation, mathematical economics, mathematical programming and image recovery by using nonlinear functional analysis and fixed point theory. We first proved some fixed point theorems for families of nonexpansive mappings in a Banach space. Next, we proved nonlinear ergodic theorems of Baillon's type for nonlinear semigroups of nonexpansive mappings. In particular, we gave an answer to the open problem posed during the Second World Congress on Nonlinear Analysts, Athens, Greece, 1996, by extending Takahashi's result and Rode's result to a Banach space for an amenable semigroup of nonexpansive mappings. Further, we established weak convergence theorems of Mann's type for families of nonexpansive mappings. We also proved strong convergence theorems of Halpern's type for families of nonexpansive mappings. Finally, using these results, we discussed the problem of image recovery by convex combinations of nonexpansive retractions, the problem of finding a common fixed point of a commuting family of nonexpansive mappings, the convex minimization problem and so on.