Скачать или смотреть GIAN DAY 1 : Maths - AMU - THE FIXED POINT PROBLEM: THE OLD AND THE NEW.

Dr. Mohamed Amine Khamsi is a Professor at the University of Texas at El
Paso, U.S.A.

Dr. Javid Ali is an Associate Professor in the Department of Mathematics at
Aligarh Muslim University, India and is the Course Co-ordinator.

Day 1:

Basic relevant results from metric space and functional analysis
Introduction to fixed point theory and Banach fixed point theorem
Problems based on Lecture 1 and Lecture 2, and some open problems

‘The theory of fixed points is one of the most powerful tools of modern
mathematics’ as stated by F. Browder, who played a major role in the development
of modern fixed point theory as a powerful tool in the study of nonlinear functional
analysis. In fact fixed point theory finds its root in the early days of topology
through the work of Lefschetz-Hopf, Leray-Schauder, and Poincare. This theory
finds many applications in analysis where topological considerations play a major
role, including the relationship with degree theory. In many cases, a solution to a
given problem can be translated into a fixed point, such as the existence of closed
periodic orbits in dynamical systems, the existence of models in logic programming,
or the existence of Nash equilibrium with interesting applications in game theory
and economics.
Following the publication of the powerful Banach fixed point theorem, metric fixed
point theory was born. The boundary between metric fixed point theory and the
general theory of topology is bleary and hard to discern. Successive approximations
are an important tool used at the start of metric fixed point theory. They find their
roots in the work of Cauchy, Liouville, Lipschitz, Peano, Fredholm, and Picard.
The fixed point theory for certain important single-valued mappings is very
interesting in its own right because the results have constructive proofs and they
have applications in industrial fields such as image processing engineering, physics,
computer science, economics, and telecommunication.
The theory of multivalued maps is interesting in its own right. It finds applications
in control theory, convex optimization, differential inclusions, and economics for
example. Following the publication of the Banach fixed point theorem, Nadler was
able to give an extension for this fundamental theorem from the single-valued case
to the multivalued one. He was able to do that by introducing the concept of
multivalued contractions. This extension allowed many authors to generalize
Nadler’s result in different directions. Any constructive proof of a fixed-point
theorem makes the result more valuable. These constructive results play a
fundamental role in nonlinear analysis. For example, they are used heavily in
denotational semantics to give meaning to recursive programs. It is very hard to
estimate their applicability and importance in mathematics.