(1)Course descriptions
We shall continue the materials in the first semester : Septral theory of compact operators, Sobolev Spaces and variational formulation of elliptic PDE.
Then we shall study the nonlinear functional analysis : Calculus in Banach Space, Implicit function theorem in Banach Space ; Degree Theory in finite and infinite dimesional spaces & Its applications, for examples, Brauer fixed point theorem, Schauder fixed point Theorem ; Bifurcation Thoery : Crandall-Rabinowitz bifurcation from simple eigenvalues.
(2) Text book:
Haim Brezis: Functional Analysis, Sobolev Spaces, and PDE
Louis Nirenberg : Topics in Nonlinear Functional Analysis
Melvin Berger & Marion Berger :Perspectives in Nonlinearity
(4) Teaching Method

For the book of Brezis:
Chapter 6 :Compact operators. Spetral decomposition of self-adjoint compact operators.
Chapter 8: Sobolev Spaces and Variational formulation of Boundary Value Problems in one dimension.

For the book of Nirenberg
Chapter I : Topological Approach . Finite dimension
Chapter II : Topological Degree in Banach Spaces
Chapter III : Bifurcation Theory

(6) Evaluations:
Homework 40%
Midterm Exam. 30%
Final Exam. 30%