Research Article
BibTex RIS Cite
Year 2018, Volume: 1 Issue: 1, 6 - 11, 30.06.2018

Abstract

References

  • [1] V.M. Aleksandrov, and E.V. Kovalenko, Problems in Continuous Mechanics with Mixed Boundary Conditions, Nauka, Moscow 1986. In Russian.
  • [2] J. Appel, A. Kalitvin and P. Zabreiko, Partial Integral Operators and Integrodifferential Equations, Marcel Dekker, New York, 2000.
  • [3] J.A.D. Appleby and D.W. Reynolds, On the non-exponential convergence of asymptotically stable solutions of linear scalar Volterra integro-differential equations, Journal of Integral Equations and Applications, 14, no 2 (2002), 521-543.
  • [4] K. M. Case, P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading Mass. 1967.
  • [5] M.C. Cercignani, Mathematical Methods in Kinetic Theory, Macmillian, New York, 1969.
  • [6] Chuhu Jin and Jiaowan Luo, Stability of an integro-differential equation, Computers and Mathematics with Applications, 57 (2009) 1080–1088.
  • [7] Yu L. Daleckii, and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. 1974.
  • [8] A. Domoshnitsky and Ya. Goltser, An approach to study bifurcations and stability of integro-differential equations. Math. Comput. Modelling, 36 (2002), 663-678.
  • [9] A.D. Drozdov, Explicit stability conditions for integro-differential equations with periodic coefficients, Math. Methods Appl. Sci. 21 (1998), 565-588.
  • [10] A.D. Drozdov and M. I. Gil’, Stability of linear integro-differential equations with periodic coefficients. Quart. Appl. Math. 54 (1996), 609-624.
  • [11] N.T. Dung, On exponential stability of linear Levin-Nohel integro-differential equations, Journal of Mathematical Physics 56, 022702 (2015); doi:10.1063/1.4906811
  • [12] M.I. Gil’, Operator Functions and Localization of Spectra, Lecture Notes In Mathematics vol. 1830, Springer-Verlag, Berlin, 2003.
  • [13] M.I. Gil’, Spectrum and resolvent of a partial integral operator. Applicable Analysis, 87, no. 5, (2008) 555–566.
  • [14] M.I. Gil’, On stability of linear Barbashin type integro-differential equations, Mathematical Problems in Engineering, 2015, Article ID 962565, (2015), 5 pages.
  • [15] M.I. Gil’, A bound for the Hilbert-Schmidt norm of generalized commutators of nonself-adjoint operators, Operators and Matrices, 11, no. 1 (2017), 115-123
  • [16] Ya. Goltser and A. Domoshnitsky, Bifurcation and stability of integrodifferential equations, Nonlinear Anal. 47 (2001), 953-967.
  • [17] Ya. Coltser and A. Domoshnitsky, About reducing integro-differential equations with infinite limits of integration to systems of ordinary differential equations. Advances in Difference Equations 2013:187, (2013) 17 pages.
  • [18] H.G. Kaper, C.G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear Transport Theory, Birkhauser, Basel, 1982.
  • [19] B. G. Pachpatte, On a parabolic integrodifferential equation of Barbashin type, Comment. Math.Univ. Carolin. 52, no. 3 (2011) 391-401
  • [20] W.J. Rugh, Linear System Theory. Prentice Hall, Upper Saddle River, New Jersey, 1996.
  • [21] H. R. Thieme, A differential-integral equation modelling the dynamics of populations with a rank structure, Lect. Notes Biomath. 68 (1986), 496-511
  • [22] J. Vanualailai and S. Nakagiri, Stability of a system of Volterra integro-differential equations J. Math. Anal. Appl. 281 (2003) 602-619
  • [23] B. Zhang, Construction of Liapunov functionals for linear Volterra integrodifferential equations and stability of delay systems, Electron. J. Qual. Theory Differ. Equ. 30 (2000) 1-17.

Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators

Year 2018, Volume: 1 Issue: 1, 6 - 11, 30.06.2018

Abstract

In a Hilbert space $\mathcal{H}$ we consider the equation $dx(t)/dt=(A+B(t))x(t)$ $(t\ge 0),$ where $A$ is a constant bounded operator, and $B(t)$ is a piece-wise continuous function defined on $[0,\8)$ whose values are bounded operators in $\mathcal{H}$. Conditions for the exponential stability are derived in terms of the commutator $AB(t)-B(t)A$. Applications to integro-differential equations are also discussed. Our results are new even in the finite dimensional case.

References

  • [1] V.M. Aleksandrov, and E.V. Kovalenko, Problems in Continuous Mechanics with Mixed Boundary Conditions, Nauka, Moscow 1986. In Russian.
  • [2] J. Appel, A. Kalitvin and P. Zabreiko, Partial Integral Operators and Integrodifferential Equations, Marcel Dekker, New York, 2000.
  • [3] J.A.D. Appleby and D.W. Reynolds, On the non-exponential convergence of asymptotically stable solutions of linear scalar Volterra integro-differential equations, Journal of Integral Equations and Applications, 14, no 2 (2002), 521-543.
  • [4] K. M. Case, P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading Mass. 1967.
  • [5] M.C. Cercignani, Mathematical Methods in Kinetic Theory, Macmillian, New York, 1969.
  • [6] Chuhu Jin and Jiaowan Luo, Stability of an integro-differential equation, Computers and Mathematics with Applications, 57 (2009) 1080–1088.
  • [7] Yu L. Daleckii, and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. 1974.
  • [8] A. Domoshnitsky and Ya. Goltser, An approach to study bifurcations and stability of integro-differential equations. Math. Comput. Modelling, 36 (2002), 663-678.
  • [9] A.D. Drozdov, Explicit stability conditions for integro-differential equations with periodic coefficients, Math. Methods Appl. Sci. 21 (1998), 565-588.
  • [10] A.D. Drozdov and M. I. Gil’, Stability of linear integro-differential equations with periodic coefficients. Quart. Appl. Math. 54 (1996), 609-624.
  • [11] N.T. Dung, On exponential stability of linear Levin-Nohel integro-differential equations, Journal of Mathematical Physics 56, 022702 (2015); doi:10.1063/1.4906811
  • [12] M.I. Gil’, Operator Functions and Localization of Spectra, Lecture Notes In Mathematics vol. 1830, Springer-Verlag, Berlin, 2003.
  • [13] M.I. Gil’, Spectrum and resolvent of a partial integral operator. Applicable Analysis, 87, no. 5, (2008) 555–566.
  • [14] M.I. Gil’, On stability of linear Barbashin type integro-differential equations, Mathematical Problems in Engineering, 2015, Article ID 962565, (2015), 5 pages.
  • [15] M.I. Gil’, A bound for the Hilbert-Schmidt norm of generalized commutators of nonself-adjoint operators, Operators and Matrices, 11, no. 1 (2017), 115-123
  • [16] Ya. Goltser and A. Domoshnitsky, Bifurcation and stability of integrodifferential equations, Nonlinear Anal. 47 (2001), 953-967.
  • [17] Ya. Coltser and A. Domoshnitsky, About reducing integro-differential equations with infinite limits of integration to systems of ordinary differential equations. Advances in Difference Equations 2013:187, (2013) 17 pages.
  • [18] H.G. Kaper, C.G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear Transport Theory, Birkhauser, Basel, 1982.
  • [19] B. G. Pachpatte, On a parabolic integrodifferential equation of Barbashin type, Comment. Math.Univ. Carolin. 52, no. 3 (2011) 391-401
  • [20] W.J. Rugh, Linear System Theory. Prentice Hall, Upper Saddle River, New Jersey, 1996.
  • [21] H. R. Thieme, A differential-integral equation modelling the dynamics of populations with a rank structure, Lect. Notes Biomath. 68 (1986), 496-511
  • [22] J. Vanualailai and S. Nakagiri, Stability of a system of Volterra integro-differential equations J. Math. Anal. Appl. 281 (2003) 602-619
  • [23] B. Zhang, Construction of Liapunov functionals for linear Volterra integrodifferential equations and stability of delay systems, Electron. J. Qual. Theory Differ. Equ. 30 (2000) 1-17.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Michael Gil'

Publication Date June 30, 2018
Submission Date March 28, 2018
Acceptance Date May 15, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Gil’, M. (2018). Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators. Fundamental Journal of Mathematics and Applications, 1(1), 6-11.
AMA Gil’ M. Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators. Fundam. J. Math. Appl. June 2018;1(1):6-11.
Chicago Gil’, Michael. “Stability Conditions for Non-Autonomous Linear Differential Equations in a Hilbert Space via Commutators”. Fundamental Journal of Mathematics and Applications 1, no. 1 (June 2018): 6-11.
EndNote Gil’ M (June 1, 2018) Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators. Fundamental Journal of Mathematics and Applications 1 1 6–11.
IEEE M. Gil’, “Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators”, Fundam. J. Math. Appl., vol. 1, no. 1, pp. 6–11, 2018.
ISNAD Gil’, Michael. “Stability Conditions for Non-Autonomous Linear Differential Equations in a Hilbert Space via Commutators”. Fundamental Journal of Mathematics and Applications 1/1 (June 2018), 6-11.
JAMA Gil’ M. Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators. Fundam. J. Math. Appl. 2018;1:6–11.
MLA Gil’, Michael. “Stability Conditions for Non-Autonomous Linear Differential Equations in a Hilbert Space via Commutators”. Fundamental Journal of Mathematics and Applications, vol. 1, no. 1, 2018, pp. 6-11.
Vancouver Gil’ M. Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators. Fundam. J. Math. Appl. 2018;1(1):6-11.

Creative Commons License
The published articles in Fundamental Journal of Mathematics and Applications are licensed under a