Course title
6M005300
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. To understand the assertion of fixed point theorems
  2. To understand proofs of fixed point theorems
  3. To apply fixed point theorems to nonlinear problems
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． Brouwer fixed point theorem (1) proof Preparation and review of the contents 190minutes
9． Brouwer fixed point theorem (2) sequel Preparation and review of the contents 190minutes
10． Brouwer fixed point theorem (3) example 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 (1) proof Preparation and review of the contents 190minutes
14． Application to Peano existence theorem (2) sequel Preparation and review of the contents 190minutes
Total. - - 2711minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Report Total.
1. 40% 40%
2. 30% 30%
3. 30% 30%
Total. 100% -
Evaluation method and criteria
Submit papers for examination questions on fixed point theorems.
Textbooks and reference materials
Kyuya Masuda, ``Hisenkei-sugaku'', Asakura shoten (Japanese)
Wataru Takahashi, ``Hisenkei Kansu Kaisekigaku'', Kindaikagakusha (Japanese)
Prerequisites
Differential/Integral calculus, the basics of functional analysis.
Office hours and How to contact professors for questions
  • Lunchtime on every Tuesday.
Relation to the environment
Non-environment-related course
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
Active-learning course
More than one class is interactive
Last modified : Wed Oct 17 07:45:16 JST 2018