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

Code	Subject	
MMA1005	Applied Functional Analysis	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	3	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. COBZAS Stefan, Ph.D., scobzas@math.ubbcluj.ro	
Aims	The aim of this course is to present some results  on differential and integral calculus for vector functions  with applications to the differentiability of convex functions defined on normed spaces,  in connection with the geometric properties of these spaces. Applications of these properties to  fixed point theory will be given.	
Content	I. Differential Calculus in Normed  Spaces
 Gateaux si Frechet differentials &#8211; relations between them, examples, properties,  algebraic operations with differentiable functions. The differential of the composed function (chain rule), the differential of the inverse mapping. Mean value theorems and consequences. Directional derivatives and  Gateaux differentiability. The local inversion and the implicit function theorems. Multilinear applications and polynomials, the symmetry of the 2nd differential, higher order differentials, Taylor&#8217;s formula and its converse, evaluations of the rest,  applications. The differentiability of the inversion mapping in Banach algebras, the spectrum in Banach algebras - compactness and nonemptyness. The local inversion and the implicit function theorems for mappings of class Ck , global inversion theorems. Krasnoselski&#8216;s theorem on the compactness of the Frechet differential, a  condition that the Frechet  differential be an isometry - Baker&#8217;s theorem.  Mazur-Ulam&#8217;s theorem on the characterization of surjective isometries. The equivalence between weak and strong holomorphy.

II.  The Differentiability of Convex Functions
 Convex functions &#8211; boundedness, continuity and Lipschitz properties. Semicontinuous functions and the continuity of convex semicontinuous functions. The slope,  directional derivatives, characterizations of convexity. Subgradients and subdifferentials. The Gateaux differentiability of convex functions. Mazur&#8217;s  Theorem on the generic Gateaux differentiability of continuous convex functions on separable Banach spaces.  The Frechet differentiability of convex functions, the continuity of the Frechet differential,  Asplund-Lindenstrauss&#8217; Theorem on the generic Frechet differentiability of continuous convex  functions on Banach spaces with separable dual, Asplund spaces. 

III. Fixed Point Theorems (FPTs)
Brouwer&#8217;s FPT&#8211; proofs, equivalents, extensions. FPTs in infinite dimensional spaces &#8211; the failure of Brouwer&#8217; s FPT  in infinite dimensional Banach spaces, Schauder&#8217;s FPT. Multivalued mappings - Kakutani&#8217;s and  Ky Fan&#8217;s FPTs. Pompeiu-Hausdorff metric, completeness, Nadler&#8217;s  FPT.  FPTs for nonexpansive mappings.
	
Bibliography	Bibliografie la partile I si II
1. Barbu, V.; Precupanu, Th. Convexity and optimization in Banach spaces, D. Reidel Publishing Co., Dordrecht; Editura Academiei Române, Bucuresti, 1986.
2. Fabian, M., et al, Functional analysis and infinite-dimensional geometry, Springer-Verlag, New York, 2001
3. Giles, J. R. , Convex analysis with application in the differentiation of convex functions, Pitman, Boston, Mass.- London, 1982
4. Muntean, I., Analiza functionala &#8211; Capitole speciale, Universitatea Babes-Bolyai, 1990.
5. Phelps, R.R., Convex functions, monotone operators and differentiability,  Springer-Verlag, Berlin, 1993
6. Precupanu, T.,  Spatii liniare topologice  si elemente de analiza convexa, Editura Academiei Române, Bucuresti, 1992

Bibliografie la partea  IIIa
1. Goebel, K.; Kirk, W. A. , Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990
2. Istratescu, V. I., Introducere în teoria punctelor fixe, Editura Academiei Române,Bucuresti, 1973
3. Khamsi, M. A.; Kirk, W. A. , An introduction to metric spaces and fixed point theory, Wiley-Interscience, New York, 2001.
4. Rus, I. A., Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.
	
Assessment	Written examination, seminar activities and talks delivered by students