Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MO266	Special Topics in Modern Mathematics (1)	

Section	Semester	Hours: C+S+L	Category	Type	
Algebra and Geometry - in English	1	2+1+0		compulsory	
Real and Compex Analysis - in English	1	2+1+0		compulsory	

Teaching Staff in Charge	Prof. COBZAS Stefan, Ph.D., scobzas@math.ubbcluj.ro
Assoc.Prof. ANISIU Valeriu, Ph.D., anisiu@math.ubbcluj.ro	
Aims	To present some fundamental results in fixed point theory in connection with variational principles and with applications to critical point theory, best approximation, ergodic theory,  minimax and game theory.
	
Content	Banach@s contraction principle and  Edelstein@s Fixed Point Theorem (FPT) for contractive mappings.
 Examples: isometries without fixed points, contractive mappings without fixed points, an incomplete metric space having the  fixed point property for contractions.
Partially ordered sets, Zorn&#8217;s Lemma and the Axioma of Choice (AC). Applications of the abstract fixed point theorems.  Fixed points in partially ordered sets: the theorems of  Zermelo, Knaster-Tarski, Tarski-Kantorovich, Birkhoff-Tarski.
  The Brezis-Browder maximality principle, Caristi-Kirk fixed point theorem, 
       Ekeland@s Variational Principle(EkVP) .Applications of EkVP to critical points. Completeness .
The best approximation problem in  normed spaces : existence, strict convexity and uniqueness,
        the distance function and the metric projection.Examples related to metric projections in concrete Banach spaces.

The metric projection in Hilbert space: existence, variational characterization and the   
       nonexpansiveness  of metric projection onto closed convex sets. Von Neumann&#8217;s ergodic theorem.
 Brouwer&#8217;s FPT &#8211; preliminary results: retractions, contractibility, equivalent conditions.
 Milnor&#8217;s proof of Brouwer FPT and of the Hairy Ball Theorem.
Brouwer FPT for finite dimensional closed bounded convex subsets of  Hausdorff topological  
       vector spaces.
 Applications of Brouwer FPT: the convexity of Chebyshev sets in Rn, the  subsets of Rn with the unique farthest  point property are singletons.

Fixed Points in Infinite Dimensional Spaces

Kakutani and Leray examples on the failure of Brouwer fixed point principle in infinite  dimensional normed spaces. The failure of Brouwer FPT in any infinite dimensional normed space. 
Schauder projections and Schauder FPT.
Set-valued mappings &#8211;upper and lower semicontinuity (usc and lsc). The fixed point theorems of  
         Kakutani and Ky Fan.  Applications to minimax and game theory- von Neumann&#8217;s theorem.
 Fixed points for nonexpansive mappings in Hilbert space and Baillon&#8217;s ergodic theorem.
Iterative methods for nonexpansive mappings in Hilbert space: Krasnoselski, Mann, Ishikawa.
 Applications to recurrent sequences.
The Pompeiu-Hausdorff (PH) metric &#8211; definition, properties and completeness with respect to PH metric of various classes of subsets of a metric space (e.g., compact subsets) or of a normed space (convex closed bounded, convex compact or weakly compact).
Nadler&#8217;s FPT for set-valued contractions and some related results.
	
Bibliography	1. Y. Benyamini and  J. Lindenstrauss, Geometric Nonlinear Functional Analysis, AMS 2000
2. V. Berinde, Iterative Approximation of Fixed Points,  LNM 1912,  Springer-Verlag, Berlin - Heidelberg, 2007.                                            
3.  M. Fabian et al., Functional Analysis and Infinite Dimensional Geometry, Springer 2000
4.    K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge   UP 1998                                                  
5.  V. I. Istratescu, Introducere in Teoria Punctelor Fixe, Ed. Academiei, Bucuresti, 1973, [English translation: D. Reidel Publishing   Co., Boston 1981].     
6.  M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Wiley, New York, 2001.                                                                                                      
7. R.R. Phelps, Convex Functions, Monotone Operators and Differentiability. 2nd edition. LNM 1364, Springer-Verlag 1993.
	
Assessment	Seminar activities 25%; the presentation of a topic at seminar 25%; written examination 50%