In this course we shall introduce the subject of Nonlinear Functional Analysis. First we 
<br>shall present the degree theory in finite dimension space, especially the Brauer degree 
<br>and its applications for instance Brauer Fixed Point Theorem, Fundamental Theorem of 
<br>Algebra. Then we study the degree theory of infinite dimensional space, especially the 
<br>Leray-Schauder degree of I-K, where K is a compact operator. We shall study the 
<br>existence of solutions of nonlinear elliptic PDE.  Crandall-Rabinowitz bifurcation from 
<br>simple eigenvalues and Rabinowitz global bifurcation Theorem will be proved. The 
<br>applications  will contain the existence of solutions of  nonlinear local and nonlocal 
<br>elliptic PDE.
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<br>References:
<br>(1) Sze-Bi Hsu , Ordinary Differential Equations with Applications, World Scientific Press 
<br>,2013, 2nd edition ,Chapter 8 , Index Theory and Brouwer degree
<br>(2) Melvyn Berger and Marion Berger, Perspective in Nonlinearity, An introduction to 
<br>Nonlinear Analysis 
<br>(3) Hwai-Chiuan Wang, Nonlinear Analysis, 2003, NTHU Press
<br>(4) Topics in Nonlinear Functional Analysis, L. Nirenberg, Courant Institute of 
<br>Mathematics Science
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<br>Students will be asked to present some important papers in the class.
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