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.
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.
- You can describe the claim of the contraction mapping principle and applications of the theorem.
- You can describe the claim of Brouwer's fixed point theorem and applications of the theorem.
- You can describe the claim of Schauder's fixed point theorem and applications of the theorem.
| 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 |
| Assignment | Total. | |
|---|---|---|
| 1. | 40% | 40% |
| 2. | 30% | 30% |
| 3. | 30% | 30% |
| Total. | 100% | - |
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.
Kyuya Masuda, ``Hisenkei-sugaku'', Asakura shoten (Japanese)
Wataru Takahashi, ``Hisenkei Kansu Kaisekigaku'', Kindaikagakusha (Japanese)
Wataru Takahashi, ``Hisenkei Kansu Kaisekigaku'', Kindaikagakusha (Japanese)
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.
- Course that cultivates an ability for utilizing knowledge
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | N/A |
Last modified : Fri Mar 18 23:23:50 JST 2022