Course title
6M0053001
Nonlinear Analysis

 TAKEUCHI Shingo Click to show questionnaire result at 2018
Course content
Fixed point theorems are powerful tools of nonlinear analysis. In particular, it is widely used for showing the existence of solutions of nonlinear equations. This lecture introduces various fixed point theorems with rigorous proofs and their applications to nonlinear problems.
Purpose of class
When you try to solve an equation by using a computer, it will be a waste of time if you let the computer find a solution without restricting an searching area. Also, the solution obtained is an approximate one, not an exact one, but in the first place we should ask whether there really exists the exact solution or not. Fixed point theorems allow us to conclude that there exists the exact solution of the equation in the restricted area. Thanks to the theorem, it suffices to find the solution in the area by the computer.
Goals and objectives
  1. You can describe the claim of the contraction mapping principle and applications of the theorem.
  2. You can describe the claim of Brouwer's fixed point theorem and applications of the theorem.
  3. You can describe the claim of Schauder's fixed point theorem and applications of the theorem.
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Final assignment Total.
1. 40% 40%
2. 30% 30%
3. 30% 30%
Total. 100% -
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1． Introduction Review of the contents 190minutes
2． Metric spaces Preparation and review of the contents 190minutes
3． Contraction mapping theorem (1) proof Preparation and review of the contents 190minutes
4． Contraction mapping theorem (2) sequel Preparation and review of the contents 190minutes
5． Application to the unique existence theorem of ODE Preparation and review of the contents 190minutes
6． Fixed point theorem of non-expansive mapping (1) proof Preparation and review of the contents 190minutes
7． Fixed point theorem of non-expansive mapping (2) sequel Preparation and review of the contents 190minutes
8． Fixed point theorem of non-expansive mapping (3) Browder's strong convergence theorem Preparation and review of the contents 190minutes
9． Brouwer fixed point theorem (1) proof Preparation and review of the contents 190minutes
10． Brouwer fixed point theorem (2) sequel Preparation and review of the contents 190minutes
11． Schauder fixed point theorem (1) proof Preparation and review of the contents 190minutes
12． Schauder fixed point theorem (2) sequel Preparation and review of the contents 241minutes
13． Application to Peano existence theorem Preparation and review of the contents 190minutes
14． Borsk-Ulam's theorem Preparation and review of the contents 190minutes
Total. - - 2711minutes
Evaluation method and criteria
One assignment. As a criterion, if you determine that you understand 60% of the content covered in the class, your final score will be 60 points.
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
The Others 授業内、あるいはScombZ・メール等で行う。
Textbooks and reference materials
Takeuchi Shingo, ``Kansu-kaiseki", Shokabo (Japanese)
Masuda Kyuya, ``Hisenkei-sugaku'', Asakura shoten (Japanese)
Takahashi Wataru, ``Hisenkei Kansu Kaisekigaku'', Kindaikagakusha (Japanese)
Prerequisites
This course requires a good knowledge of differential and integral calculus and a basic knowledge of functional analysis. In particular, students from departments other than the Department of Mathematical Sciences will need to be prepared to actively research basic definitions and theorems on their own as necessary.
Office hours and How to contact professors for questions
  • During class periods, every Wednesday after 4th period (it is advisable to call in advance when visiting).
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Course that cultivates an ability for utilizing knowledge
Active-learning course
More than one class is interactive
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicable
N/A N/A
Education related SDGs:the Sustainable Development Goals
  • 4.QUALITY EDUCATION
  • 9.INDUSTRY, INNOVATION AND INFRASTRUCTURE
Last modified : Wed Nov 06 04:03:52 JST 2024