**Plenary Lecture
On Approximate Solutions of Some Difference Equations**

**Professor Janusz Brzdek**

Pedagogical University of Cracow

Department of Mathematics

Poland

E-mail: jbrzdek@up.krakow.pl

**Abstract: ** In many situations (e.g., in applications) we have to do with functions that satisfy some equations only approximately (in some sense). There arises a natural question what errors we commit when we replace such functions by the exact solutions to those equations. Some tools to evaluate them are provided by the theory of Ulam's (including also the Hyers-Ulam) type stability.

That type of stability for various mathematical objects (e.g., di erential, di erence, functional, integral and operator equations) has been a very popular subject of investigations for the last nearly fifty years. The main motivation for it was given by a problem raised by S.M. Ulam in 1940 and a solution to it published by D.H. Hyers in 1941. Now, that issue is generally understood as follows.

When an approximate (in some sense) solution of an equation is somehow close to a solution of the equation?

It arises in natural ways. For instance, if we cannot determine a solution to an equation, then we can try to nd functions that satisfy the equation approximately (with some particular error) and next show that near (in some sense) each of such functions there is a solution to the equation.

The lecture deals with that stability for some types of di erence equations (e.g., linear of the rst and higher orders and generalizations of them). In many cases, functions satisfying such equations approximately generate the exact solutions to them. That method can be described in many cases in the terms of xed point theory in suitable function spaces. Some examples of simple applications of it will be provided and some outcomes on non-stability will be discussed as well.

**Brief Biography of the Speaker: ** Present permanent employment: Department of Mathematics, Pedagogical University, Kraków, Poland

Major research interests: functional equations and inequalities with their applications, Ulam’s type stability (e.g., of difference, differential, functional, integral and operator equations), real and functional analysis, fixed point theory

Author of more than 100 published papers

Chairman of the Scientific Committee of the series of conferences: International Conference on Functional Equations and Inequalities (ICFEI)

Chairman of the Organizing Committees of 10th (2005), 11th (2006), 12th (2008), 13th (2009), 14th (2011), 15th (2013), and 16th (2015) ICFEIs (http://uatacz.up.krakow.pl/icfei/15ICFEI/prev.php)

Chairman of the Scientific and Organizing Committees of Conference on Ulam’s Type Stability in Ustron (Poland, June 2-6, 2014, http://cuts.up.krakow.pl/2014/) and Chairman of the Scientific Committee of Conference on Ulam’s Type Stability 2016 in Cluj-Napoca (Romania, July 4-9, 2016, http://cuts.up.krakow.pl/)

Member of the Programm or Scientific Committees of numerous other international conferences Editor (jointly with Th.M. Rassias) of the monograph Functional Equations in Mathematical Analysis (nearly 750 pages; collection of 47 papers of 67 authors), volume 52 (2013) of Springer Optimization and Its Applications series, dedicated to the 100th anniversary of S.M. Ulam

Lead Editor of Banach Center Publications volume 99 (2013) titled: Recent Developments in Functional Equations and Inequalities. Selected Topics

Lead Guest Editor of Abstract and Applied Analysis annual special issues: Ulam’s Type Stability (http://www.hindawi.com/journals/aaa/type.stability/) in the years 2012, 2013

Lead Guest Editor of Journal of Function Spaces (formerly: Journal of Function Spaces and Applications) special issue: Ulam’s Type Stability and Fixed Points Methods (http://www.hindawi.com/journals/jfs/si/329604/cfp/)

Lead Guest Editor of Discrete Dynamics in Nature and Society special issue: Approximate and Iterative Methods (http://www.hindawi.com/journals/ddns/si/473241/)

Supervisor of five promoted PhD students

Editor of several international journals

Plenary speaker of several international conferences