In this course we shall introduce the subject of Nonlinear Functional Analysis. First we
shall present the degree theory in finite dimension space, especially the Brauer degree
and its applications for instance Brauer Fixed Point Theorem, Fundamental Theorem of
Algebra. Then we study the degree theory of infinite dimensional space, especially the
Leray-Schauder degree of I-K, where K is a compact operator. We shall study the
existence of solutions of nonlinear elliptic PDE. Crandall-Rabinowitz bifurcation from
simple eigenvalues and Rabinowitz global bifurcation Theorem will be proved. The
applications will contain the existence of solutions of nonlinear local and nonlocal
elliptic PDE.

(1) Sze-Bi Hsu , Ordinary Differential Equations with Applications, World Scientific Press
,2013, 2nd edition ,Chapter 8 , Index Theory and Brouwer degree
(2) Melvyn Berger and Marion Berger, Perspective in Nonlinearity, An introduction to
Nonlinear Analysis
(3) Hwai-Chiuan Wang, Nonlinear Analysis, 2003, NTHU Press
(4) Topics in Nonlinear Functional Analysis, L. Nirenberg, Courant Institute of
Mathematics Science

Students will be asked to present some important papers in the class.