International Journal of Nonlinear Analysis and Applications
On the $\Psi$-instability of a nonlinear Lyapunov matrix differential equation with integral term as right side
Document Type : Research Paper
Author
Department of Applied Mathematics, University of Craiova, Craiova, Romania
Abstract
The aim of this paper is to give sufficient conditions for $\Psi$-instability of trivial solution of a nonlinear Lyapunov matrix differential equation with integral term as right side.
Keywords
[1] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Company, Inc. New York, 1960.
[2] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, 1965.
[3] A. Diamandescu, On the Ψ-instability of a nonlinear Volterra integro-differential System, Bull. Math. Soc. Sc. Math. Roumanie, Tome 46(94)(3-4) 2003 103–119.
[4] A. Diamandescu, On Ψ-stability of nonlinear Lyapunov matrix differential equations, Elect. J. Qual. Theory Diff. Eq. 54 (2009) 1–18.
[5] A. Diamandescu, On the Ψ-asymptotic stability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. L(1) (2012) 3–25.
[6] A. Diamandescu, On the Ψ-conditional asymptotic stability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. LIII(2) (2015) 29-58.
[2] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, 1965.
[3] A. Diamandescu, On the Ψ-instability of a nonlinear Volterra integro-differential System, Bull. Math. Soc. Sc. Math. Roumanie, Tome 46(94)(3-4) 2003 103–119.
[4] A. Diamandescu, On Ψ-stability of nonlinear Lyapunov matrix differential equations, Elect. J. Qual. Theory Diff. Eq. 54 (2009) 1–18.
[5] A. Diamandescu, On the Ψ-asymptotic stability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. L(1) (2012) 3–25.
[6] A. Diamandescu, On the Ψ-conditional asymptotic stability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. LIII(2) (2015) 29-58.
[7] A. Diamandescu, On the Ψ-boundedness of the solutions of linear nonhomogeneous Lyapunov matrix differential equations, Diff. Geom. Dyn. Syst. 19 (2017) 35–44.
[8] A. Diamandescu, On the Ψ-boundedness of the solutions of a nonlinear Lyapunov matrix differential equation, Appl. Sci. 19 (2017) 31–40.
[9] A. Diamandescu, On the Ψ-instability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. XLIX (1) (2011) 21–37.
[10] A. Diamandescu, Existence of Ψ-bounded solutions for nonhomogeneous Lyapunov matrix differential equations on R, Elect. J. Qual. Theory Diff. Eq., 42 (2010) 1–9.
[11] H. T. Yoneyama and T. Ytoh, Asymptotic stability criteria for nonlinear Volterra integro-differential equations, Funk. Ecva. 33 (1990) 39–57.
[12] J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons Ltd, Chichester, 1999.
[13] M. S. N. Murty and G. Suresh Kumar, On dichotomy and conditioning for two-point boundary value problems associated with first order matrix Lyapunov systems, J. Korean Math. Soc. 45(5) (2008) 1361–1378.
[14] M. S. N. Murty and G. Suresh Kumar, On Ψ-boundedness and Ψ-stability of matrix Lyapunov systems, J. Appl. Math. Comput. 26 (2008) 67–84.
[15] M. S. N. Murty, B. V. Apparao and G. Suresh Kumar, Controllability, observability and realizability of matrix Lyapunov systems, Bull. Korean Math. Soc. 43(1) (2006) 149–159.
[16] M. S. N. Murty and G. Suresh Kumar, On Ψ-bounded solutions for non-homogeneous matrix Lyapunov systems on R, Elect. J. Qual. Theory Diff. Eq. 62 (2009) 1–12.
[17] G. Suresh Kumar, B. V. Appa Rao and M. S. N. Murty, On Ψ-conditional asymptotic stability of first order non-linear matrix Lyapunov systems, Int. J. Nonlinear Anal. Appl. 4(1) (2013) 7–20.
[18] O. Perron, Die Stabilit¨atsfrage bei Differentialgleichungen, Math. Z. 32 (1930) 703–728.
[8] A. Diamandescu, On the Ψ-boundedness of the solutions of a nonlinear Lyapunov matrix differential equation, Appl. Sci. 19 (2017) 31–40.
[9] A. Diamandescu, On the Ψ-instability of nonlinear Lyapunov matrix differential equations, Anal. Univer. Vest, Timi. Seria Mate. Info. XLIX (1) (2011) 21–37.
[10] A. Diamandescu, Existence of Ψ-bounded solutions for nonhomogeneous Lyapunov matrix differential equations on R, Elect. J. Qual. Theory Diff. Eq., 42 (2010) 1–9.
[11] H. T. Yoneyama and T. Ytoh, Asymptotic stability criteria for nonlinear Volterra integro-differential equations, Funk. Ecva. 33 (1990) 39–57.
[12] J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons Ltd, Chichester, 1999.
[13] M. S. N. Murty and G. Suresh Kumar, On dichotomy and conditioning for two-point boundary value problems associated with first order matrix Lyapunov systems, J. Korean Math. Soc. 45(5) (2008) 1361–1378.
[14] M. S. N. Murty and G. Suresh Kumar, On Ψ-boundedness and Ψ-stability of matrix Lyapunov systems, J. Appl. Math. Comput. 26 (2008) 67–84.
[15] M. S. N. Murty, B. V. Apparao and G. Suresh Kumar, Controllability, observability and realizability of matrix Lyapunov systems, Bull. Korean Math. Soc. 43(1) (2006) 149–159.
[16] M. S. N. Murty and G. Suresh Kumar, On Ψ-bounded solutions for non-homogeneous matrix Lyapunov systems on R, Elect. J. Qual. Theory Diff. Eq. 62 (2009) 1–12.
[17] G. Suresh Kumar, B. V. Appa Rao and M. S. N. Murty, On Ψ-conditional asymptotic stability of first order non-linear matrix Lyapunov systems, Int. J. Nonlinear Anal. Appl. 4(1) (2013) 7–20.
[18] O. Perron, Die Stabilit¨atsfrage bei Differentialgleichungen, Math. Z. 32 (1930) 703–728.
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Volume 12, Issue 2
November 2021Pages 99-113
- Receive Date: 13 March 2019
- Accept Date: 17 September 2019