Nonlinear problem by using fixed point theory

使用不动点理论的非线性问题

• 批准号: 09640160
• 负责人: TAKAHASHI Wataru
• 金额: $ 1.86万
• 依托单位: TOKYO INSTITUTE OF TECHNOLOGY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640160/
• 关键词: functional analysis; nonlinear operators.; ergodic theorem; evolution equation; convex analysis; fixed point theorem; mini-max theorem; variational inequality; 非線形作用素

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.

利用非线性泛函分析和不动点理论研究了非线性演化方程、数理经济学、数学规划和图像恢复等非线性问题。我们首先证明了 Banach 空间中非扩张映射族的一些不动点定理。接下来，我们证明了非扩张映射的非线性半群的Baillon型非线性遍历定理。特别是，我们通过将 Takahashi 的结果和 Rode 的结果扩展到 Banach 空间来得到一个适合的非扩张映射半群，对 1996 年希腊雅典第二次世界非线性分析大会上提出的开放问题给出了答案。此外，我们还为非扩张映射族建立了曼型弱收敛定理。我们还证明了非扩张映射族的 Halpern 类型的强收敛定理。最后，利用这些结果，我们讨论了非扩张性回缩凸组合的图像恢复问题、寻找非扩张性映射通勤族的公共不动点问题、凸最小化问题等。

项目成果

Wataru TAKAHASHI and Gang-Eun KIM: "Strong Convergence of Approximants to Fixed Points of Nonexpansive Nonself Mappings in Banach Spaces;" Nonlinear Analysis. 32-3. 447-454 (1998)

Wataru TAKAHASHI 和 Gang-Eun KIM：“Banach 空间中非扩张非自映射不动点近似的强收敛性；”

Wataru Takahashi: "Strong Convergence of Approximants to Fixed Points of Nonexpansive Noself Mappings in Banach Spaces" Nonlinear Analysis. 32-3. 447-454 (1998)

Wataru Takahashi：“巴拿赫空间中非扩张性自身映射的近似值的强收敛性”非线性分析。

Wataru TAKAHASHI and Takayuki TAMURA: "Convergence Theorems for a Pair of Nonexpansive Mappings" J.Convex Analysis. 5-1. 45-56 (1998)

Wataru TAKAHASHI 和 Takayuki TAMURA：“一对非扩张映射的收敛定理”J.Convex Analysis。

Wataru TAKAHASHI and Naoki SHIOJI: "Strong Convergence of Approximated Sequences for Nonexpansive Mappings in Banach Spaces" Proc.Amer.Math.Soc.125-12. 3641-3645 (1998)

Wataru TAKAHASHI 和 Naoki Shioji：“Banach 空间中非扩张映射的近似序列的强收敛性”Proc.Amer.Math.Soc.125-12。

Wataru TAKAHASHI & Osamu KADA: "Strong Convergence and Nonlinear Ergodic Theorems for Commutative Semigroups of Nonexpansive Mappings" Nonlinear Analysis,Theory,Methods & Applications. 28-3. 495-511 (1997)

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