Project detail

# Kvalitativní vlastnosti řešení diferenciálních rovnic a jejich aplikace

Duration: 01.01.2011 — 31.12.2015

On the project

Cílem projektu je studium kvalitativních vlastností řešení obyčejných diferenciálních rovnic, včetně rovnic se zpožděným argumentem. Speciální pozornost bude věnována studiu diferenciálních rovnic modelujících reálné problémy

Description in English
Project is a continuation of the projects IAA1163401 GA AV (2004-2006) a 201/08/0469 GA ČR (2008-2010). The main attention will be focused on the following problems of the qualitative theory of differential equations: Asymptotic properties of solutions of differential equations with general Phi-Laplacian; To built relative oscillation theory for half-linear differential equations; Asymptotic and oscillatory properties of solutions of delayed differential equations; Stability of delayed differential systems; Numerical investigation of the first order delayed differential equations; Applications of some derived results to differential equations modelling problems of a technical or scientific nature

Keywords
obyčejné diferenciální rovnice; asymptotické vlastnosti; oscilace; odkloněný argument; diskretizační me

Key words in English
obyčejné diferenciální rovnice; asymptotické vlastnosti; oscilace; odkloněný argument; diskretizační me

Mark

GAP201/11/0768

Default language

Czech

People responsible

Došlá Zuzana, Prof. RNDr., DrSc.
- principal person responsible (2011-01-01 - 2015-12-31)
Čermák Jan, prof. RNDr., CSc.
- fellow researcher (2011-01-01 - 2015-12-31)

Units

Institute of Mathematics
- (2011-01-01 - 2015-12-31)

Funding resources

Czech Science Foundation - Standardní projekty
- whole funder (2011-01-01 - 2015-12-31)

Results

DIBLÍK, J.; BAŠTINEC, J.; ŠMARDA, Z. Oscilation and nonoscilation of solution of differential equation with delay. In Dynamical System Modelling and Stability Investigation. Kyjev, Ukrajina: University of Kyiv, UA, 2011. p. 25-26. ISBN: 9667652009.
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DIBLÍK, J.; ŠMARDA, Z.; SVOBODA, Z.; KHUSAINOV, D. Instable trivial solution of autonomous differential systems with quadratic right-hand sides in a cone. Abstract and Applied Analysis, 2011, vol. 2011, no. Article ID 15491, p. 1-23. ISSN: 1085-3375.
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DIBLÍK, J.; ZAFER, A. On stability of linear delay differential equations under Perron's condition. Abstract and Applied Analysis, 2011, vol. 2011, no. 1, p. 1-9. ISSN: 1085-3375.
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ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L. Discrete Mittag-Leffler functions in linear fractional difference equations. Abstract and Applied Analysis, 2011, vol. 2011, no. 2011, p. 1-21. ISSN: 1085-3375.
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BOICHUK, A.; DIBLÍK, J.; KHUSAINOV, D.; RŮŽIČKOVÁ, M. Boundary-value problems for weakly nonlinear delay differential systems. Abstract and Applied Analysis, 2011, vol. 2011, no. 1, p. 1-19. ISSN: 1085-3375.
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DZHALLADOVA, I.; BAŠTINEC, J.; DIBLÍK, J.; KHUSAINOV, D. Estimates of exponential stability for solutions of stochastic control systems with delay. Abstract and Applied Analysis, 2011, vol. 2011, no. 1, p. 1-14. ISSN: 1085-3375.
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DIBLÍK, J.; NOWAK, C. Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem. Abstract and Applied Analysis, 2011, vol. 2011, no. 1, p. 1-15. ISSN: 1085-3375.
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DIBLÍK, J.; KHUSAINOV, D. Representation of solution of the first boundary value problem for delay systems. Bulletin Kiev University, series: physics and Mathematics, 2011, vol. 2011, no. 1, p. 59-62. ISSN: 1812-5409.
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BAŠTINEC, J.; BEREZANSKY, L.; DIBLÍK, J.; ŠMARDA, Z. A final result on the oscillation of solutions of the linear discrete delayed equation \Delta x(n)=-p(n)x(n-k) with a positive coefficient. Abstract and Applied Analysis, 2011, vol. vol. 2011, no. Article ID 58632, p. 1-28. ISSN: 1085-3375.
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DIBLÍK, J.; BAŠTINCOVÁ, A.; BAŠTINEC, J. Oscillation of solution of a linear third-order discrete delayed equation. In NINTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING APPLIED IN COMPUTER AND ECONOMIC ENVIRONMENTS, ICSC 2011. Kunovice: EPI, 2011. p. 95-101. ISBN: 978-80-7314-221-6.
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DIBLÍK, J.; ŠMARDA, Z.; BAŠTINEC, J.; BEREZANSKY, L. Oscillation of solutions of the linear discrete delayed equation and related problems. In Proceedings 2011 World Congress on Engineering and Technology. Shanghai Čína: 2011. p. 457-461. ISBN: 978-1-61284-363-6.
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BAŠTINEC, J.; DIBLÍK, J.; ŠMARDA, Z. An explicit criterion for the existence of positive solutions of the linear delayed equation $\dot x(t)=-c(t)x(t-\tau(t))$. Abstract and Applied Analysis, 2011, vol. 2011, no. 11, p. 1-10. ISSN: 1085-3375.
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DIBLÍK, J.; KHUSAINOV, D.; RŮŽIČKOVÁ, M.; BAŠTINCOVÁ, A. On a dynamical model with delay for the economy. Nonlinear Oscillations, 2011, vol. 14920110, no. 4, p. 556-568. ISSN: 1536-0059.
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SVOBODA, Z. Asymptotic properties of system of the delayed linear differential equations of special type. In Dynamical System Modelling and Stability Investigation Abstracts of Conference Reports. Kyjev: 2011. p. 156-157. ISBN: 9667652009.
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DIBLÍK, J.; KUKHARENKO, O.; MORÁVKOVÁ, B.; KHUSAINOV, D. Delayed exponential functions and their application to representations of solutions of linear equations with constant coefficients and with single delay. In Proceedings of the IEEEAM/NAUN International Conferences. 2011. p. 82-87. ISBN: 978-1-61804-058-9.
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HRABALOVÁ, J. On stability intervals of Euler methods for a delay differential equation. Journal of Applied Mathematics, 2013, vol. 5, no. 2, p. 77-84. ISSN: 1337-6365.
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ČERMÁK, J.; DVOŘÁKOVÁ, S. Boundedness and asymptotic properties of solutions of some linear and sublinear delay difference equations. APPLIED MATHEMATICS LETTERS, 2012, vol. 25, no. 2, p. 813-817. ISSN: 0893-9659.
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ČERMÁK, J.; JÁNSKÝ, J.; TOMÁŠEK, P. On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations. JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 2012, vol. 18, no. 11, p. 1781-1800. ISSN: 1023-6198.
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HRABALOVÁ, J. On stability intervals of Euler methods for a delay differential equation. In APLIMAT 11th INTERNATIONAL CONFERENCE. Bratislava: Aplimat, 2012. p. 153-160. ISBN: 978-80-89313-58-7.
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DIBLÍK, J.; KHUSAINOV, D.; KUKHARENKO, O.; SVOBODA, Z. Solution of the first boundary-value problem for a system of autonomous second-order linear partial differential equations of parabolic type with a single delay. Abstract and Applied Analysis, 2012, vol. 2012, no. Article ID 21904, p. 1-27. ISSN: 1085-3375.
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ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L. Stability and asymptotic properties of a linear fractional difference equation. Advances in Difference Equations, 2012, vol. 2012, no. 1, p. 1-14. ISSN: 1687-1847.
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BAŠTINEC, J.; DIBLÍK, J.; HALFAROVÁ, H.; ŠMARDA, Z. An explicit criterion for the existence of positive solutions of the linear delayed equation. In Modelling, Control and Stability MCS-2012. Krym, Simferopol: 2012. p. 26-27. ISBN: 978-966-491-327-7.
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KHAN, Y.; SVOBODA, Z.; ŠMARDA, Z. Solving certain classes of Lane-Emden type equations using differential transformation method. Advances in Difference Equations, 2012, vol. 2012, no. 1, p. 1-13. ISSN: 1687-1847.
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DIBLÍK, J.; DZHALLADOVA, I.; RŮŽIČKOVÁ, M. The stability of nonlinear differential systems with random parameters. Abstract and Applied Analysis, 2012, vol. 2012, no. Article ID924107, p. 1-27. ISSN: 1085-3375.
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STEVIČ, S.; DIBLÍK, J.; IRIČANIN, B.; ŠMARDA, Z. On Some Solvable Difference Equations and Systems of Difference Equations. Abstract and Applied Analysis, 2012, vol. 2012, no. ID 541761, p. 1-11. ISSN: 1085-3375.
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KHAN, Y.; DIBLÍK, J.; FARAZ, N.; ŠMARDA, Z. An efficient new perturbative Laplace method for space-time fractional telegraph equations. Advances in Difference Equations, 2012, vol. 2012, no. 1, p. 1-11. ISSN: 1687-1847.
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ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L. Stability regions for linear fractional differential systems and their discretizations. APPLIED MATHEMATICS AND COMPUTATION, 2013, vol. 219, no. 12, p. 7012-7022. ISSN: 0096-3003.
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ŠMARDA, Z.; KHAN, Y. Singular Initial Value Problem for a System of Integro-Differential Equations. Abstract and Applied Analysis, 2012, vol. 2012, no. ID 918281, p. 1-18. ISSN: 1085-3375.
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SVOBODA, Z.; DIBLÍK, J.; KHUSAINOV, D. SOME PROPERTIES OF SPECIAL DELAYED MATRIX FUNCTIONS IN THEORY OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS AND WITH SINGLE DELAY. In 11-th Intenational conference APLIMAT 12. Bratislava: STU, 2012. p. 205-212. ISBN: 978-80-89313-58-7.
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SVOBODA, Z. Asymptotic properties of delayed matrix functions. In Modelling, Control and Stability MCS-2012. Krym, Simferopol: 2012. p. 40-42. ISBN: 978-966-491-327-7.
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SHATYRKO, A.; DIBLÍK, J.; KHUSAINOV, D.; RŮŽIČKOVÁ, M. Stabilization of Lure-type Nonlinear Control Systems by Lyapunov-Krasovskii Functionals. Advances in Difference Equations, 2012, vol. 2012, no. 1, p. 1-11. ISSN: 1687-1847.
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ČERMÁK, J.; TOMÁŠEK, P. On delay-dependent stability conditions for a three-term linear difference equation. Funkcialaj Ekvacioj Serio Internacia, 2014, vol. 57, no. 1, p. 91-106. ISSN: 0532-8721.
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HRABALOVÁ, J. Stability properties of a discretized neutral delay differential equation. Tatra Mountains Mathematical Publications, 2013, vol. 2013, no. 54, p. 83-92. ISSN: 1210-3195.
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HRABALOVÁ, J.; TOMÁŠEK, P. On stability regions of the modified midpoint method for a linear delay differential equation. Advances in Difference Equations, 2013, vol. 2013, no. 177, p. 1-10. ISSN: 1687-1847.
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TOMÁŠEK, P. An asymptotic estimate for linear delay differential equations with power delayed arguments. Advances in Dynamical Systems and Applications (ADSA), 2013, vol. 8, no. 2, p. 379-386. ISSN: 0973-5321.
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DIBLÍK, J.; KUDELČÍKOVÁ, M. Positive solutions of advanced differential systems. The Scientific World Journal, 2013, vol. 2013, no. Article ID, p. 1-8. ISSN: 1537-744X.
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ČERMÁK, J. Some qualitative properties of linear dynamic equations with multiple delays. Advances in Difference Equations, 2013, vol. 2013, no. 2013, p. 1-12. ISSN: 1687-1847.
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ČERMÁK, J.; DRAŽKOVÁ, J. On stability regions for some delay differential equations and their discretizations. Periodica Mathematica Hungarica, 2014, vol. 68, no. 2, p. 193-206. ISSN: 0031-5303.
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ČERMÁK, J.; JÁNSKÝ, J. On a generalization of the Levin-May Theorem. Carpathian Journal of Mathematics, 2014, vol. 30, no. 1, p. 55-62. ISSN: 1584-2851.
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DIBLÍK, J.; FEČKAN, M.; POSPÍŠIL, M. Forced Fermi-Pasta-Ulam lattice maps. Miskolc Mathematical Notes, 2013, vol. 14, no. 1, p. 63-78. ISSN: 1787-2405.
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DIBLÍK, J.; FEČKAN, M.; POSPÍŠIL, M. Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukrainian Mathematical, 2013, vol. 65, no. 1, p. 64-76. ISSN: 0041-5995.
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DIBLÍK, J.; FEČKAN, M.; POSPÍŠIL, M. Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices. Abstract and Applied Analysis, 2013, vol. 2013, no. 1, p. 1-10. ISSN: 1085-3375.
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DIBLÍK, J.; REBENDA, J.; ŠMARDA, Z. Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations. Abstract and Applied Analysis, 2013, vol. 2013, no. 1, p. 1-12. ISSN: 1085-3375.
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DIBLÍK, J.; DZHALLADOVA, I.; MICHALKOVÁ, M.; RŮŽIČKOVÁ, M. Modeling of applied problems by stochastic systems and their analysis using the moment equations. Advances in Difference Equations, 2013, vol. 2013, no. 1, p. 1-12. ISSN: 1687-1847.
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DIBLÍK, J.; DZHALLADOVA, I.; MICHALKOVÁ, M.; RŮŽIČKOVÁ, M. Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty. Abstract and Applied Analysis, 2013, vol. 2013, no. 1, p. 1-12. ISSN: 1085-3375.
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DIBLÍK, J.; RŮŽIČKOVÁ, M.; CHUPÁČ, R. Unbounded solutions of the equation $\dot y(t)=\sum_{i=1}^{n}\beta_{i}$ (t)\left[y(t-\delta_{i})-y(t-\tau_{i})\right]\$. APPLIED MATHEMATICS AND COMPUTATION, 2013, vol. 2013, no. 221, p. 610-619. ISSN: 0096-3003.
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ČERMÁK, J.; HRABALOVÁ, J. Delay-dependent stability criteria for neutral delay differential and difference equations. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2014, vol. 34, no. 11, p. 4577-4588. ISSN: 1078-0947.
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DIBLÍK, J.; FEČKAN, M.; POSPÍŠIL, M.; ROTHOS, V.; SUSANTO, H. Travelling waves in nonlinear magnetic metamaterials. In Localized Excitations in Nonlinear Complex Systems. Nonlinear Systems and Complexity. 2014. p. 335-358. ISBN: 978-3-319-02056-3.
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DIBLÍK, J.; IRIČANIN, B.; STEVIČ, S.; ŠMARDA, Z. Note on the existence of periodic solutions of a class of systems of differential-difference equations. APPLIED MATHEMATICS AND COMPUTATION, 2014, vol. 2014, no. 232, p. 922-928. ISSN: 0096-3003.
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BAŠTINEC, J.; BEREZANSKY, L.; DIBLÍK, J.; ŠMARDA, Z. On a delay population model with a quadratic nonlinearity without positive steady state. APPLIED MATHEMATICS AND COMPUTATION, 2014, vol. 2014, no. 227, p. 622-629. ISSN: 0096-3003.
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STEVIČ, S.; DIBLÍK, J.; ŠMARDA, Z. On periodic and solutions converging to zero of some systems of differential-difference equations. APPLIED MATHEMATICS AND COMPUTATION, 2014, vol. 2014, no. 227, p. 43-49. ISSN: 0096-3003.
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ČERMÁK, J.; JÁNSKÝ, J. Stability switches in linear delay difference equations. APPLIED MATHEMATICS AND COMPUTATION, 2014, vol. 243, no. 9, p. 755-766. ISSN: 0096-3003.
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ČERMÁK, J.; KISELA, T. Exact and discretized stability of the Bagley-Torvik equation. Journal of Computational and Applied Mathematics, 2014, vol. 269, no. 10, p. 53-67. ISSN: 0377-0427.
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ČERMÁK, J.; DRAŽKOVÁ, J. On stability sets for numerical discretizations of neutral delay differential equations. Tatra Mountains Mathematical Publications, 2015, vol. 63, no. 2, p. 89-100. ISSN: 1210-3195.
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ČERMÁK, J.; JÁNSKÝ, J. Explicit stability conditions for a linear trinomial delay difference equation. APPLIED MATHEMATICS LETTERS, 2015, vol. 43, no. 5, p. 56-60. ISSN: 0893-9659.
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ČERMÁK, J.; KISELA, T. Introduction to Stability Theory of Linear Fractional Difference Equations. In Fractional Calculus: Theory. Mathematics Research Developments. 2014. p. 117-162. ISBN: 978-1-63463-002-3.
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DIBLÍK, J.; KÚDELČÍKOVÁ, M.; JANGLAJEW, K. An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 2014, vol. 19, no. 2014, p. 2461-2467. ISSN: 1531-3492.
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DIBLÍK, J. A note on explicit criteria for the existence of positive solutions to the linear advanced equation \dot (t) = c(t)x(t + \tau). APPLIED MATHEMATICS LETTERS, 2014, vol. 35, no. 2014, p. 72-76. ISSN: 0893-9659.
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NECHVÁTAL, L. On asymptotics of discrete Mittag-Leffler function. Mathematica Bohemica, 2014, vol. 139, no. 4, p. 667-675. ISSN: 0862-7959.
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ČERMÁK, J.; JÁNSKÝ, J.; TOMÁŠEK, P. Two types of stability conditions for linear delay difference equations. Applicable Analysis and Discrete Mathematics, 2015, vol. 9, no. 4, p. 120-138. ISSN: 1452-8630.
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BATTELLI, F.; DIBLÍK, J.; FEČKAN, M.; PICKTON, J.; POSPÍŠIL, M.; SUSANTO, H. Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss. NONLINEAR DYNAMICS, 2015, vol. 81, no. 1, p. 353-371. ISSN: 0924-090X.
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BEREZANSKY, L.; DIBLÍK, J.; SVOBODA, Z.; ŠMARDA, Z. Simple uniform exponential stability conditions for a system of linear delay differential equations. APPLIED MATHEMATICS AND COMPUTATION, 2015, vol. 2015, no. 250, p. 605-614. ISSN: 0096-3003.
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TOMÁŠEK, P. Asymptotic stability of a full term linear difference equation with two parameters. Tatra Mountains Mathematical Publications, 2015, vol. 63, no. 1, p. 283-290. ISSN: 1210-3195.
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KISELA, T.; ČERMÁK, J. Stability properties of two-term fractional differential equations. NONLINEAR DYNAMICS, 2015, vol. 80, no. 4, p. 1673-1684. ISSN: 0924-090X.
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ČERMÁK, J.; NECHVÁTAL, L.; GYŐRI, I. On explicit stability conditions for a linear fractional difference system. Fractional Calculus and Applied Analysis, 2015, vol. 18, no. 3, p. 651-672. ISSN: 1311-0454.
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ČERMÁK, J. Stability conditions for linear delay difference equations: A survey and perspectives. Tatra Mountains Mathematical Publications, 2015, vol. 63, no. 63, p. 1-29. ISSN: 1210-3195.
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ČERMÁK, J.; KISELA, T.; HORNÍČEK, J. Stability regions for fractional differential systems with a time delay. Communications in Nonlinear Science and Numerical Simulation, 2016, vol. 31, no. 1-3, p. 108-123. ISSN: 1007-5704.
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TOMÁŠEK, P. Asymptotic stability regions for certain two parametric full-term linear difference equation. In Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics. Springer Proceedings in Mathematics and Statistics. New York: Springer, 2016. p. 323-330. ISBN: 978-3-319-32855-3. ISSN: 2194-1009.
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ČERMÁK, J.; JÁNSKÝ, J. Stability and periodic investigations of linear planar difference systems. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2016, vol. 39, no. 18, p. 5343-5354. ISSN: 0170-4214.
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SIEGMUND, S.; DIBLÍK, J.; NOWAK, C. A generalized Picard-Lindelöf theorem. Electronic Journal of Qualitative Theory of Differential Equations, 2016, vol. 2016, no. 28, p. 1-8. ISSN: 1417-3875.
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SVOBODA, Z. Representation of Solutions of Linear Differential Systems of the Second Order with Constant Delays. Journal of Mathematical Sciences, 2017, vol. 222, no. 3, p. 345-358. ISSN: 1072-3374.
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