SVIBOR - Project code: 1-01-253

MINISTRY OF SCIENCE AND TECHNOLOGY

Strossmayerov trg 4, HR - 10000 ZAGREB
tel.: +385 1 459 44 44, fax: +385 1 459 44 69
E-mail: ured@znanost.hr

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Project code: 1-01-253


Multi-valued functions, fixed points, and applications


Main researcher: ČERIN, ZVONKO (7994)


Assistants
Type of research: basic
Duration from: 01/01/91. to 12/31/96.
Papers on project (total): 22
Papers on project quoted in Current Contents: 22
Institution name: Prirodoslovno-matematički fakultet, Matematički odjel, Zagreb (37)
Department/Institute: Department of Mathematics Faculty of Natural Sciences University of Zagreb
Address: Bijenička cesta 30
City: 10000 - Zagreb, Croatia
Communication
Phone: 385 (0)1 678 411
Fax: 385 (0)1 432 484
E-mail: zcerin@x400.srce.hr
E-mail: cerin@cromath.math.hr
Summary: In this fundamental research in mathematics we consider some areas in topology in order to contribute to our understanding of this important field of mathematics. The present day mathematics can roughly be divided into three major fields: algebra, analysis, and topology. In algebra we manipulate numbers (adding, multiplying, dividing,...), in analysis we analyse functions (we take derivatives, integrals,.., solve diferential equations,...), while in topology we study the structure of spaces and processes with respect to continuity that plays the key role (we look for limits, investigate continuous functions,...). These three fields are today so much entwined and interdependent that it is all too common that a discovery say in analysis is made using results and techniques of both algebra and topology and vice versa. It is therefore natural and wise to consider all three equally valuable and important for the healthy development of all sciences. Our contribution to topology relates to the existence of fixed points and coincidence points for continuous functions and the application of multi-valued functions in topology. In the period 91-95 working on a project twentytwo original research articles on the above topics were published in renown international journals on over 410 printed pages and several others have been accepted for publication.

Keywords: mathematics, topology, continuous function, multi-valued function, fixed point, coincidence point, universal map (in the sense of Holsztynski), shape theory, proper shape theory, C*-algebra

Research goals: The first goal of this project is to study the class of universal maps that was introduced in sixties by a Polish mathematician Holsztynski. The universality of this maps refers to the fact that they have a coincidence point with every other map. This interesting class of maps has not been studied extensively so far inspite of their close connection with the fixed point theory which is one of the parts of topology that had numerous applications elsewhere. The second goal is to find applications of universal maps in mathematical economics analogous to the way how fixed point theory has been used in equilibrium theory of economies and regulated societies. We hope to contribute to searches for optimal economy models for post-communist countries. The third goal is to improve the existing fixed point theorems (the one of Lefschetz in particular) by enlarging the class of spaces to which this algebraic method of location of fixed points is applicable. We shall also try to obtain versions of these results for multi-valued functions. The fourth and the most important goal is to look for applications of multi-valued functions in topology which will provide new results and lead to improvement of known results. We shall attempt to describe various types of shape theories using multi-valued functions with smaller and smaller images of points analogous to the recent description of shape theory of compact metric spaces by J. Sanjurjo. The expected results is hard to predict. However, I believe that I will menage to prepare several articles on these topics. Whether, where, and when they get published is impossible to forsee.

COOPERATION - INSTITUTIONS


  1. Name of institution: Universita di Torino Dipartimento di Matematica Torino, Italia
    Type of institution: University/Faculty
    Type of cooperation: Joint publishing of scientific papers
    City: 11233 - Torino, Italija

  2. Name of institution: Universita di Messina Dipartimento di Matematica Messina, Italia
    Type of institution: University/Faculty
    Type of cooperation: Joint publishing of scientific papers
    City: 98166 - Messina IT, Italija

OTHER ACHIEVEMENTS


  1. Name: "Proper shape theory" predavanje na znanstvenom skupu "Colloquium on Topology", Szeksard, Mađarska, August 22-28, 1993.
    Type of achievement: Other
    Authors: Zvonko Čerin

  2. Name: "Shape theory intrinsically" predavanje na znanstvenom skupu "11-th Italian International Topology Meeting", Trieste, Italia, Septembar 6-11, 1993.
    Type of achievement: Other
    Authors: Zvonko Čerin

  3. Name: 45 Reviews for Mathematical Reviews
    Type of achievement: Other
    Authors: Zvonko Čerin
Other information about the project.
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