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.

References:

(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.