On time transformations for differential equations with state-dependent delay

• Autorzy: Alexander Rezounenko
• Języki publikacji: EN

Abstrakty

EN

Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

Słowa kluczowe

EN

State-dependent delay; Time transformations

Kategorie tematyczne

• 39A22: Growth, boundedness, comparison of solutions
• 34K17: Transformation and reduction of equations and systems, normal forms

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 2
• Strony: 298-307

Daty

wydano

2014-02-01

online

2013-11-21

Twórcy

autor

Alexander Rezounenko

Bibliografia

• [1] Arino O., Hadeler K.P., Hbid M.L., Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations, 1998, 144(2), 263–301 http://dx.doi.org/10.1006/jdeq.1997.3378
• [2] Brunner H., Maset S., Time transformations for delay differential equations, Discrete Contin. Dyn. Syst., 2009, 25(3), 751–775 http://dx.doi.org/10.3934/dcds.2009.25.751
• [3] Brunner H., Maset S., Time transformations for state-dependent delay differential equations, Commun. Pure Appl. Anal., 2010, 9(1), 23–45 http://dx.doi.org/10.3934/cpaa.2010.9.23
• [4] Chepyzhov V.V., Vishik M.I., Appendix: Non-authonomous dynamical systems and their attractors, In: Vishik M.I., Asymptotic Behaviour of Solutions of Evolutionary Equations, Lezioni Lincee, Cambridge University Press, Cambridge, 1992
• [5] Chueshov I.D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Univ. Lektsii Sovrem. Mat., ACTA, Kharkov, 1999 (in Russian); English translation available at http://www.emis.de/monographs/Chueshov/
• [6] Diekmann O., van Gils S.A., Verduyn Lunel S.M., Walther H.-O., Delay Equations, Appl. Math. Sci., 110, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4612-4206-2
• [7] Hale J., Theory of Functional Differential Equations, 2nd ed., Appl. Math. Sci., 3, Springer, Heidelberg-New York, 1977 http://dx.doi.org/10.1007/978-1-4612-9892-2
• [8] Hale J.K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., 25, American Mathematical Society, Providence, 1988
• [9] Hartung F., Krisztin T., Walther H.-O., Wu J., Functional differential equations with state-dependent delays: Theory and applications, In: Handbook of Differential Equations: Ordinary Differential Equations, III, Handb. Differ. Equ., Elsevier/North Holland, 2006, Amsterdam, 435–545 http://dx.doi.org/10.1016/S1874-5725(06)80009-X
• [10] Krisztin T., A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 2003, 9(4), 993–1028 http://dx.doi.org/10.3934/dcds.2003.9.993
• [11] Mallet-Paret J., Nussbaum R.D., Paraskevopoulos P., Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 1994, 3(1), 101–162
• [12] Rezounenko A.V., Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal., 2009, 70(11), 3978–3986 http://dx.doi.org/10.1016/j.na.2008.08.006
• [13] Rezounenko A.V., A condition on delay for differential equations with discrete state-dependent delay, J. Math. Anal. Appl., 2012, 385(1), 506–516 http://dx.doi.org/10.1016/j.jmaa.2011.06.070
• [14] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68, Springer, New York, 1988 http://dx.doi.org/10.1007/978-1-4684-0313-8
• [15] Vorotnikov V.I., Rumyantsev V.V., Stability and Control with Respect to Part of the Coordinates of the Phase Vector of Dynamical Systems: Theory, Methods and Applications, Nauchnyi Mir, Moscow, 2001 (in Russian)
• [16] Walther H.-O., The solution manifold and C 1-smoothness for differential equations with state-dependent delay, J. Differential Equations, 2003, 195(1), 46–65 http://dx.doi.org/10.1016/j.jde.2003.07.001

Identyfikatory

DOI

10.2478/s11533-013-0341-6