Estimates for the Green functions of nonautonomous higher order differential equations

Author:
Michael Gil'

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3045-3055

MSC (2000):
Primary 34A30, 34D20

Published electronically:
February 23, 2009

MathSciNet review:
2506463

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the equation

**1.**Benharrat Belaïdi,*Estimation of the hyper-order of entire solutions of complex linear ordinary differential equations whose coefficients are entire functions*, Electron. J. Qual. Theory Differ. Equ. (2002), no. 5, 8 pp. (electronic). MR**1895276****2.**Tomás Caraballo,*On the decay rate of solutions of non-autonomous differential systems*, Electron. J. Differential Equations (2001), No. 5, 17 pp. (electronic). MR**1811778****3.**Carmen Chicone,*Ordinary differential equations with applications*, 2nd ed., Texts in Applied Mathematics, vol. 34, Springer, New York, 2006. MR**2224508****4.**M. De la Sen,*Robust stability of a class of linear time-varying systems*, IMA J. Math. Control Inform.**19**(2002), no. 4, 399–418. MR**1949011**, 10.1093/imamci/19.4.399**5.**Michael I. Gil’,*Operator functions and localization of spectra*, Lecture Notes in Mathematics, vol. 1830, Springer-Verlag, Berlin, 2003. MR**2032257****6.**M. I. Gil’,*A new stability test for nonlinear nonautonomous systems*, Automatica J. IFAC**40**(2004), no. 12, 2161–2165 (2005). MR**2156162**, 10.1016/j.automatica.2004.07.007**7.**Michael I. Gil’,*Explicit stability conditions for continuous systems*, Lecture Notes in Control and Information Sciences, vol. 314, Springer-Verlag, Berlin, 2005. A functional analytic approach. MR**2133008****8.**M. I. Gil’,*Stability of nonlinear systems with differentiable linear parts*, Circuits Systems Signal Process.**24**(2005), no. 3, 243–251. MR**2168010**, 10.1007/s00034-004-0123-2**9.**Michael I. Gil’,*Liapunov exponents for higher-order linear differential equations whose characteristic equations have variable real roots*, Electron. J. Differential Equations (2008), No. 54, 6. MR**2392958****10.**C. J. Harris and J. F. Miles,*Stability of linear systems: some aspects of kinematic similarity*, Mathematics in Science and Engineering, vol. 153, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR**662825****11.**Hoang Nam,*The central exponent and asymptotic stability of linear differential algebraic equations of index 1*, Vietnam J. Math.**34**(2006), no. 1, 1–15. MR**2233315****12.**Gro R. Hovhannisyan,*Asymptotic stability for second-order differential equations with complex coefficients*, Electron. J. Differential Equations (2004), No. 85, 20 pp. (electronic). MR**2075424****13.**O. G. Illarionova,*On the stability of the 𝑘th general exponent of a linear system of differential equations*, Differ. Uravn.**32**(1996), no. 9, 1171–1174, 1293 (Russian, with Russian summary); English transl., Differential Equations**32**(1996), no. 9, 1173–1176 (1997). MR**1600828****14.**A. Ju. Levin,*The non-oscillation of solutions of the equation 𝑥⁽ⁿ⁾+𝑝₁(𝑡)𝑥⁽ⁿ⁻¹⁾+\cdots+𝑝_{𝑛}(𝑡)𝑥=0*, Uspehi Mat. Nauk**24**(1969), no. 2 (146), 43–96 (Russian). MR**0254328****15.**D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,*Inequalities involving functions and their integrals and derivatives*, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. MR**1190927****16.**O. I. Morozov,*A criterion for upper semistability of the highest Lyapunov exponent of a nonhomogeneous linear system*, Differentsial′nye Uravneniya**28**(1992), no. 4, 587–593, 731–732 (Russian, with Russian summary); English transl., Differential Equations**28**(1992), no. 4, 473–478. MR**1188510****17.**Z. Prësdorf,*Linear integral equations*, Current problems in mathematics. Fundamental directions, Vol. 27 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 5–130, 239 (Russian, with Russian summary). MR**1178114****18.**Wilson J. Rugh,*Linear system theory*, Prentice Hall Information and System Sciences Series, Prentice Hall, Inc., Englewood Cliffs, NJ, 1993. MR**1211190****19.**Cemil Tunç,*Stability and boundedness of solutions to certain fourth-order differential equations*, Electron. J. Differential Equations (2006), No. 35, 10 pp. (electronic). MR**2213579**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34A30,
34D20

Retrieve articles in all journals with MSC (2000): 34A30, 34D20

Additional Information

**Michael Gil'**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Email:
gilmi@cs.bgu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-09-09829-3

Keywords:
Ordinary differential equations,
linear nonautonomous equations,
estimates for Green functions

Received by editor(s):
August 12, 2008

Received by editor(s) in revised form:
November 17, 2008

Published electronically:
February 23, 2009

Additional Notes:
This research was supported by the Kamea Fund of Israel

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.