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

(3)Reference:

Melvin Berger & Marion Berger :Perspectives in Nonlinearity

(4) Teaching Method

Lectures

(5)Syllabus:

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%