Hlavním cílem projektu je vyšetřování asymptotických a oscilačních vlastností obyčejných a funkcionálních diferenciálních rovnic. Hlavní pozornost bude zaměřena na asymptotické a oscilační vlastnosti řešení pololineárních a kvazilineárních diferenciálních rovnic 2. řádu, problém existence kladných, ohraničených a globálně prodlužitelných řešení diferenciálních systémů se zpožděním a problém stability řešení dvoudimenzionálních systémů se zpožděním. Pozornost bude rovněž soustředěna na asymptotickou teorii dynamických rovnic 1. řádu, která přirozeným způsobem sjednocuje teorii pro diferenciální a diferenční rovnice.

Description in English
The principal concern of the project are asymptotic and oscillatory properties of solutions of ordinary and functional differential equations. A particular attention will be devoted to the problem of asymptotic and oscillatory properties of second order half-linear and quasilinear differential equations, the existence of bounded, positive and globally continuable solutions for delay differential systems and the stability theory for two-dimensional delay systems. The attention will be also paid to the asymptotic theory of first order dynamic equations as a natural unification of the theory for differential and difference equations.

Keywords
asymptotické vlastnosti, oscilační vlastnosti

Key words in English
asymptotic properties; oscillatory properties

GA201/08/0469

Czech

Došlá Zuzana, Prof. RNDr., DrSc. - principal person responsible
Čermák Jan, prof. RNDr., CSc. - fellow researcher

Institute of Mathematics
- responsible department (1.1.1989 - not assigned)
Institute of Mathematics
- co-beneficiary (1.1.2008 - 31.12.2010)

SVOBODA, Z. Asymptotic properties of one differential equation with unbounded delay. Mathematica Bohemica, 2013, vol. 2012, no. 2, p. 239-248. ISSN: 0862-7959.
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KUNDRÁT, P.; JÁNSKÝ, J. The stability analysis of a discretized pantograph equation. Mathematica Bohemica, 2011, vol. 136, no. 4, p. 385-394. ISSN: 0862-7959.
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ČERMÁK, J. The stability and asymptotic properties of the Theta-methods for the pantograph equation. IMA Journal of Numerical Analysis, 2011, vol. 31, no. 4, p. 1533-1551. ISSN: 0272-4979.
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KUNDRÁT, P. Asymptotic estimate for differential equation with power coefficients and power delays. Tatra Mountains Mathematical Publications, 2011, vol. 48, no. 7, p. 91-99. ISSN: 1210-3195.
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DIBLÍK, J.; BAŠTINCOVÁ, A.; BAŠTINEC, J.; KHUSAINOV, D.; SHATYRKO, A. Estimates of perturbed nonlinear systems of indirect control of neutral type. Cybernetics and computer enginering, 2010, roč. 2010, č. 160, s. 72-85. ISSN: 0452-9901.
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DIBLÍK, J.; BEREZANSKY, L.; BAŠTINEC, J.; ŠMARDA, Z. On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays. Abstract and Applied Analysis, 2010, vol. 2010, no. Article ID 41789, p. 1-20. ISSN: 1085-3375.
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ČERMÁK, J. On a linear difference equation with several infinite lags. Fasciculi Mathematici, 2010, vol. 2010, no. 44, p. 19-28. ISSN: 0044-4413.
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SVOBODA, Z. Asymptotic properties of delayed exponential of matrix. Journal of Applied Mathematics, 2010, vol. 2010, no. III, p. 167-172. ISSN: 1337-6365.
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ČERMÁK, J.; JÁNSKÝ, J. On the asymptotics of the trapezoidal rule for the pantograph equation. Mathematics of Computation, 2009, vol. 78, no. 268, p. 2107-2126. ISSN: 0025-5718.
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ČERMÁK, J.; NECHVÁTAL, L. On (q,h) - analogue of fractional calculus. JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, 2010, vol. 17, no. 1, p. 1-18. ISSN: 1402-9251.
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BAŠTINEC, J.; DIBLÍK, J.; ŠMARDA, Z. Multipoint singular boundary-value problem for systems of nonlinear differential equations. Boundary Value Problems, 2009, vol. 2009, no. 1, p. 1-20. ISSN: 1687-2762.
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ČERMÁK, J. The differential equation with a power delay and its numerical discretizations. International Journal of Qualitative Theory of Differential Equations and Applications, 2009, vol. 2, no. 1, p. 24-32. ISSN: 0973-3590.
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DIBLÍK, J.; ŠMARDA, Z.; BEREZANSKY, L. On connection between second-order delay differential equations and integro-differential equations with delay. Advances in Difference Equations, 2010, vol. 2010, no. 2010, p. 1-8. ISSN: 1687-1839.
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DIBLÍK, J.; KHUSAINOV, D.; LUKÁČOVÁ, J.; RŮŽIČKOVÁ, M. Control of oscillating systems with a single delay. Advances in Difference Equations, 2010, vol. 2010, no. 2010, p. 1-15. ISSN: 1687-1839.
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ČERMÁK, J. Asymptotic bounds for linear difference systems. Advances in Difference Equations, 2010, vol. 2010, no. 2010, p. 1-14. ISSN: 1687-1839.
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BAŠTINEC, J.; DIBLÍK, J. Oscillation of solutions of a linear second order discrete delayed equation. Journal of Applied Mathematics, 2009, vol. II:(2009), no. II., p. 13-18. ISSN: 1337-6365.
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DIBLÍK, J.; ŠMARDA, Z.; BEREZANSKY, L. Positive Solutions of a Second-Order Delay Differential Equations with a Damping Term. Computers and Mathematics with Applications, 2010, vol. 62, no. 2, p. 1332-1342. ISSN: 0898-1221.
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DIBLÍK, J.; SVOBODA, Z.; ŠMARDA, Z. Explicit Criteria for Existence of Positive Solutions for a Scalar Differential Equations with Variable Delay in Critical Case.IF=0,72. Computers and Mathematics with Applications, 2008, vol. 56(2008), no. 1, p. 556-564. ISSN: 0898-1221.
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DIBLÍK, J.; KHUSAINOV, D.; RŮŽIČKOVÁ, M.; BOICHUK, A. Boundary Value Problems for Delay Differential Systems. Advances in Difference Equations, 2010, vol. 2010, no. 1, p. 1-20. ISSN: 1687-1839.
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DIBLÍK, J.; ŠMARDA, Z.; SVOBODA, Z. Retract principle for neutral functional differential equations. Nonlinear Analysis, Theory, Methods and Applications, 2009, vol. 2009, no. 71, p. e1393 (e1400 p.)ISSN: 0362-546X.
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Responsibility: Došlá Zuzana, Prof. RNDr., DrSc.