# Research Laboratory for Differential Equations and their Applications

**Members of the research laboratory**

- István Győri, DSc, professor emeritus, head of laboratory
- Ferenc Hartung, DSc, professor
- Mihály Pituk, DSc, professor
- László Horváth, PhD, associate professor

Earlier members of the research laboratory

- David Reynolds, Senior Lecturer, School of Mathematical Sciences, Dublin City University
- Bernát Slezák, PhD, associate professor
- Essam Awwad, PhD student
- Attila Benkő, PhD student
- Beáta Krasznai, assistant professor

**Activity of the research laboratory**

The task of the research laboratory is to study the qualitative and quantitative properties of the solutions of ordinary, functional and partial differential equations, integro-differential equations and difference equations. Similar fields such as integral equations and integral and differential inequalities are also studied. Our aim is to develop new theoretical and numerical methods and applications to practical problems. Our results are mainly motivated by mathematical equations describing biological processes, neural networks and by models of mechanical motions.

**Research results**

Our research is related to the following topics: Asymptotic characterization of solutions; integral equations and inequalities in measure spaces; differential equations with state-dependent delays and abstract differential equations, stability, periodicity, boundedness and applications. In the research period 2012-2016, we have published 34 research papers, including 1 monograph, and 25 have appeared in international journals with impact factor. The total impact factor of these papers is 25.42. We have counted 938 citations of our papers in the last 5 years, including 29 citations of the 34 papers published in the period of the project. We presented 32 invited lectures and 16 contributed talks at international conferences and we gave 7 invited talks in research seminars at national and foreign universities.

**Most important publications**

- J. A. D. Appleby, I. Győri, D. W. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations, J. Difference Equations and Applications, 12 (2006) 1257- 1275.
- O. Arino and M. Pituk: More on linear differential systems with small delays, Journal of Differential Equations 170 (2001), 381-407.
- H. Bereketoglu, I. Győri., Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infininte delay, J. Math. Anal. Appl. 210 (1997) 279-291.
- I. Győri, Connections between compartmental systems with pipes and integro-differential equations, Mathematical Modelling 7 (1986), 1215-1238.
- I. Győri, F. Hartung, Asymptotically exponential solutions in nonlinear integral and differential equations, J. Differential Equations, 249:6 (2010) 1322-1352.
- I. Győri, L. Horváth, A new view of the l
^{p}theory for a sytem of higher order difference equations, Computers and Mathematics with Applications, 59 (2010) 2918-2932. - I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Science Publications, Clarendon Press. Oxford, 1991.
- I. Győri, M. Pituk, Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamic Systems and Applications, 5: (1996) 277-302.
- I. Győri,, I. S. Trofimchuk, Global attractivity and persistence in a discrete population model, J. Difference Equations and Applications, 6 (2000), 647-665.
- F Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J. Computational and Applied Mathematics 174: (2) pp. 201-211.
- F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, J. Dynamics and Differential Equations 25: (4) (2013) 1089-1138.
- F. Hartung, T. Krisztin, H.-O. Walther, and J. Wu, Functional differential equations with state-dependent delay: theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations, volume 3, edited by A. Canada, P. Drábek and A. Fonda, Elsevier, North-Holand, 2006, 435-545.
- L. Horváth, Khuram Ali Khan, Josip Pečarić: Combinatorial Improvements of Jensen's Inequality: Classical and New Refinements of Jensen’s Inequality with Applications, Element d. o. o., 240 p. (Monographs in Inequalities, 8), ISBN: 978-953-197-594-0, 2014
- M. Pituk: Convergence to equilibria in scalar nonquasimonotone functional differential equations, Journal of Differential Equations 193 (2003), 95-130.
- M. Pituk: A Perron type theorem for functional differential equations, Journal of Mathematical Analysis and Applications 316 (2006), 24-41.

Publications of the research laboratory (2000-2016)

Lectures at conferences, workshops and seminars (2000-2011)

**Introduction of the head of laboratory**

István Győri (MSc, University of Szeged, Hungary, 1968; CSc, Hungarian Academy of Sciences, Budapest, 1976; DSc Hungarian Academy of Sciences, Budapest, 1992) worked at A. Szent-Györgyi Medical University, Szeged from 1968 to 1993, where he was the founder and head of Computer Center (1970-1993). He was a Visiting Professor at University of Rhode Island (Providence, USA) from 1987 to 1989. He is a professor and head of Department of Mathematics and Computing, University of Pannonia (formerly University of Veszprém) from 1993 to 2013, and a professor emeritus from 2013. He served as a rector of University of Veszprém 1995-98. He received Grünwald Géza Award, János Bolyai Mathematical Society, Hungary, 1978; Neumann János Award, János Neumann Society of Computer Science, Hungary, 1992; “Prof. Hermann Niemmeyer Fernandez” honorary professorship at University of Santiago, Chile, 1994; Széchenyi Research Scholarship, Ministry of Education, Hungary 1997-2000; Bolyai Farkas scientific award from “Arany János Foundation for the Science” (foundation of the Hungarian Academic of Sciences), 2000;Albert Szent-Györgyi Research Award, Ministry of Education, Hungary, 2000; Order of Merit of the Hungarian Republic, Knight's Cross, 2009, Prima Prize, Veszprém County, in science category (2010). He is a founding member of Information Science & Technology PhD School, University of Pannonia; Editor-in-chief of „International Journal of Qualitative Theory of Differential Equations and Applications” (India) (2007-2016); Member of the editorial board of „Functional Differential Equations” (Izrael); „Advances in Difference Equatons” (USA); „Journal of Advanced Research in Dynamical and Control Systems” (USA); „Communications in Differential and Difference Equations” (India); „International Journal of Difference Equations and Dynamical Systems” (India); „International Journal of Dynamical Systems and Differential Equations” (India); „Far East Journal of Mathematics” (India); „Pacific-Asian Journal of Mathematics and Mathematical Sciences” (USA); „Mathematical Notes” (Miskolc, Hungary); „Alkalmazott Matematikai Lapok” (Hungary). István Győri has published 203 papers including 1 monograph and 141 journal papers, and he has more than 3100 independent references for his work.