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

Code	Subject	
MME1002	Applied Nonlinear Analysis	

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

Teaching Staff in Charge	Prof. PETRUSEL Adrian Olimpiu, Ph.D., petrusel@math.ubbcluj.ro	
Aims	The aim of this course is to provide basic concepts and tools in fixed point theory and in nonlinear analysis and then to apply this notions and results to the theory of differential and integral equations and to applied functional analysis.

Content	1. Contraction principle and applications (convergence of the sequences, operatorial equations, Cauchy problem, Dirichlet problerm).
2. Generalizations and extensions of the contraction principle (Kannan@s theorem, Maia@s theorem).
3. Caristi-Browder fixed point theorem, graphic contraction theorem.
4. Picard and weakly Picard operators. Examples. data dependence of the fixed point set.
5. Abstract Gronwall lemma. Comparison theorems.
6. Completely continuous operators in Banach spaces. Examples.
7. Schauder@s theorems and applications in Fredholm and Volterra type equations.
8. Nonexpansive operators in Hilbert spaces.
9. Recent trends in nonlinear analysis.
	
Bibliography	1. RUS I.A.: Principii si aplicatii al teoriei punctului fix. Cluj: Ed. Dacia, 1979.
2. SMART D. R.: Fixed point theorems. Cambridge, 1974.
3. GRAMAS A. and DUGUNDIJI J., Fixed point theory. Springer, 2003.
4. RUS I.A.: Generalized contractions and applications, Cluj University Press 2001.
5. PETRUSEL A.: Operatorial Inclusions, House of the Book of Science, 2003.
6. AGARWAL R.P., MEEHAN M. and O@REGAN D.:  Fixed point theory and applications. Cambridge: Univ. Press, 2001.
	
Assessment	80% the mark from the written test during the exams period
20% the mark obtained for the activities during the semester (two written tests and home works)