A Note on the Asymptotic Stability

Juan Eduardo Napoles Valdes, Luciano M. Lugo Motta Bittencurt, Paulo M. Guzmán

Abstract


In this paper, we stude the influence of the perturbing term in equation x’ = f(t, x) + g(t, x), on the asymptotically behavior of x’ = f(t, x).


Keywords


Lyapunov stability; Perturbation Theory.

Full Text:

PDF

References


Z. Arstein- “Limiting equation and stability of nonautonomous ordinary differential equations”, Appendix A in: LaSalle, “The Stability of dynamical systems”, CBMS Regional Conference Series in Applied Mathematics, vol.25, SIAM, Philadelphia, 1976.

E.A. Barbashin-“Lyapunov´s Functions”, Naúka, Moscú, 1970 (Russian).

T.A. Burton-“Stability and periodic solutions of ordinary and functional differential equations”, Academic Press, Orlando, Florida, 1985.

S.-N. Chow and J.A. Yorke-“Lyapunov theory and perturbing of stable and asymptotically stable systems”,J. Differential Equations 15(1975), 308-321.

C. Corduneau-“Sisteme de ecuatii diferentiale ale caror solutii sint marginate”, Bul. st. al Acad. R.P.R., t.IX, 1957, 315 (Roman).

B.P. Demidovich-“Lectures on Mathematical Theory of Stability”, Moscow, 1967 (Russian).

J. Díblik-“On asymptotic behaviour of solutions of certain classes of ordinary differential equations”, J. Differential Equations 95(1992), 203-217.

S.S. Dragomir-“Some Gronwall type inequalities and applications”, RGMIA Monographs, Victoria University, 2000.

T. Furumochi-“Liapunov functions and equiboundedness in functional differential equations”, Mem. Fac. Sci. Shimane Univ. 19(1985), 11-21.

T. Furumochi-“Uniform asymptotic stability in perturbed differential equations”, J. Math. Anal. Appl. 137(1989), 207-220.

V.E. Germanidze-“On the asymptotic stability by first approximation”, Appl. Math. and Mech., XXI(1957), 133 (Russian).

T. Hara-“On the asymptotic behaviour of solutions of certain nonautonomous differential equations”, Osaka J. Math., 12(1975), 267-282.

T. Hara; T. Yoneyama and Y. Okasari-“On the boundedness of solutions of perturbed linear systems”, J. Math. Anal. Appl., 83(1981), 188-208.

T. Hara; T. Yoneyama and J. Sugie-“Continuability of solutions of perturbed differential equations”, Nonlineal Anal., 8(1984), 963-975.

V. Hutson-“The stability under perturbations of repulsive sets”, J. Differential Equations, 76(1988), 77-90.

V. Lakshmikantham and S. Leela-“On perturbing Liapunov functions”, Mathematical Systems Theory 10(1976), 85-90.

B.S. Klebanov-“On w-limit sets of nonautonomous differential equations”, Comment, Math. Univ. Carolinae 35, 2(1994), 267-281.

J.A. Marlin and R.A. Struble-“Asymptotically equivalence of nonlinear systems”, J. Differential Equations 6(1969), 578-596.

L. Markus-“Asymptotically autonomous differential systems”, Contribution to the Theory on Nonlinear Oscillations, vol.III, Annals of Math. Stud. 36, Princeton University Press, N.J., 1956, 17-29.

N.A. de Molfetta-“Alekseev´s integral formula and applications in stability problems”, A. Acad. Brasil, Ciénc., 41 (3)(1969), 303-308.

L.E. Nápoles-“On the ultimately boundedness of solutions of systems of differential equations”, Revista Integración, 13(1995), 41-47.

J.E. Nápoles-“On the continuability of bidimensional systems”, Extracta Mathematica 11(1996), 366-368.

J.E. Nápoles-“On the boundedness and the asymptotic stability in the whole of solutions of a bidimensional system of differential equations”, Revista Ciencias Matemáticas, XVI(1997), 83-86 (Spanish).

J.E. Nápoles-“On the global stability of nonautonomous systems”, revista Colombiana de Matemática, 33(1999), 1-8.

J.E. Nápoles and J.A. Repilado-“On the boundedness and stability in the whole of the solutions of a system of differential equations”, Revista Ciencias Matemáticas, XVI(1998), 71-74 (Spanish).

J.E. Nápoles and A.I. Ruiz-“Convergence in nonlinear systems with forcing term”, Revista de Matemáticas: Teoría y Aplicaciones, 4(1)(1997), 1-5.

N. Onuchic-“On the uniform stability of a perturbed linear system”, J. Math. Anal. Appl., 6(1963), 457-464.

R. Redheffer-“A new class of Volterra differential equations for wich the solutions are globally asymptotically stable”, J. Differential Equations, 82(1989), 251-268.

H.R. Thieme-“Convergence results and a Poincaré-Bendixon trichotomy for asymptotically autonomous differential equations”, J. Math. Biol. 30(1992), 755-763.

N.P. Vu-“Asymptotic stability of nonlinear time-varying differential equations”, ITCP, Trieste, preprint No.97123, 1997.

J.R. Ward-“A topological method for bounded solutions of non-autonomous ordinary differential equations”, Trans. Amer. Math. Soc. 333(1992), 709-720.

J.R. Ward-“Homotopy and bounded solutions of ordinary differential equations”, J. Differential Equations 107(1994), 428-445.

T. Yoshizawa-“Liapunov´s function and boundedness of solutions”, Funkcial. Ekvac., 2(1959), 95-142.

T. Yoshizawa-“Stability theory by Lyapunov´s second method”, Math. Soc. of Japan, 1966.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Journal of Progressive Research in Mathematics

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright © 2016 Journal of Progressive Research in Mathematics. All rights reserved.

ISSN: 2395-0218.

For any help/support contact us at editorial@scitecresearch.com, jprmeditor@scitecresearch.com.