Asymptotic behavior of solutions of the functional differential equation x'(t) = a(t)x(r(t)) + bx(t)
DOI:
https://doi.org/10.22199/S07160917.1991.0017.00007Keywords:
Ecuaciones diferenciales, Comportamiento asintóticoAbstract
We study the global existence, the stability and the asymptotic behavior of solutions of the functional differential equations x ' ( t) = a ( t) x (r (t)) + bx( t), b ? R where r is a continuous contraction at infinity.
References
R. BELLMAN and K. COOKE, "Differential-Difference Equations", Academic Press, New York, 1963.
N. DE BRUIJN, The Difference-Differential Equations F'(x) = e?x+? F(x-1), i, ii, Neder1. Akad. Wetensch. Proc. Ser. A. 56 (1953) 449-464.
N. DE BRUIJN, The Asymptotically Periodic Behavior of the Solutions of some Linear Functional Equations, Amer. J. Math. 75(1953), 313-330.
R. DRIVER, Ordinary and Oelay Differential Equations", Springer- Verlag, New York, 1977.
R. DRIVER, Existence and Stability of Solutions of a Delay-Differential System, Areh. Rat. Meeh. Anal. 10(5), 1962, 401-426.
L. FOX, D.F. MAYERS, J.R. OCKENDON and A.B. TAYLOR, On a Functional Differential, J. Inst. Math. Applics 8(1971), 271-307.
J. HADDOCK, "Functional Differential Equations for which each constant Function is a Solution: A Narrative", Proc. of the 11th Intern. Conf. on Nonlinear Oscillations, Janos Bolyai Math. Soc. Budapest, 1987, 86-93.
J. HALE, "Functional Differential Equations", Springer-Verlag, New York, 1971.
M. HEARD, Asymptotic Behavior of Solutions of the Functional Differential Equation x'(t) = ax(t) + bx(t?), ? > 1, J. Math. Anal. Appl. 44 (1973), 745-757.
T. KATO and J.B. Mc LEOD, The Functional Difference Equation y' (x) = ay(?x) + by(x), Bull. Amer. Math. Soc. 77 (1971), 891-937.
T. KATO, "Asymptotic Behavoir of Solutions of the Functional Differential Equation y'(x) = ay(?x) + by(x), in Delay and Functional Differential Equations and their Applications", (Klauss Schmitt, Ed.), pp. 197-217, Academic Press, New York, 1972.
N. KRASOVSKI, "Stability of Motion" (Translated from Russian), Stanford, University Press, Stanford, Calif., 1963.
E. LIM, Asymptotic Behavior of Solutions of the Functional Differential Equation x'(t) = Ax(?t) + Bx(t), ? > O, Journal Math. Anal. Appl. 55(1976), 794-806.
K. MAHLER, "On a special Functional Equation", J. London Math. Soc. 15(1940), 115-123.
J.B. Me LEOD, The Functional-Differential Equations y'(x) = ay(?x) + by(x) and Generalizations, Lecture Notes Math. 280 (1972), 308-313.
M. PINTO, Functional Differential Equations with Several Lags, To appear.
M. PINTO, Asymptotic Integration of a Class of Second Order Retarded Differential Equations, To appear.
J. YORKE, Asymptotic Stability for one Dimensional Differential Equations, 7(1969), 189-202.
T. YONEYAMA, On the Stability for the Delay-Differential Equation x'(t) = -a(t)f(x(t-r(t))), J. Math. Anal. Math. 120(1986), 271-275.
N. DE BRUIJN, The Difference-Differential Equations F'(x) = e?x+? F(x-1), i, ii, Neder1. Akad. Wetensch. Proc. Ser. A. 56 (1953) 449-464.
N. DE BRUIJN, The Asymptotically Periodic Behavior of the Solutions of some Linear Functional Equations, Amer. J. Math. 75(1953), 313-330.
R. DRIVER, Ordinary and Oelay Differential Equations", Springer- Verlag, New York, 1977.
R. DRIVER, Existence and Stability of Solutions of a Delay-Differential System, Areh. Rat. Meeh. Anal. 10(5), 1962, 401-426.
L. FOX, D.F. MAYERS, J.R. OCKENDON and A.B. TAYLOR, On a Functional Differential, J. Inst. Math. Applics 8(1971), 271-307.
J. HADDOCK, "Functional Differential Equations for which each constant Function is a Solution: A Narrative", Proc. of the 11th Intern. Conf. on Nonlinear Oscillations, Janos Bolyai Math. Soc. Budapest, 1987, 86-93.
J. HALE, "Functional Differential Equations", Springer-Verlag, New York, 1971.
M. HEARD, Asymptotic Behavior of Solutions of the Functional Differential Equation x'(t) = ax(t) + bx(t?), ? > 1, J. Math. Anal. Appl. 44 (1973), 745-757.
T. KATO and J.B. Mc LEOD, The Functional Difference Equation y' (x) = ay(?x) + by(x), Bull. Amer. Math. Soc. 77 (1971), 891-937.
T. KATO, "Asymptotic Behavoir of Solutions of the Functional Differential Equation y'(x) = ay(?x) + by(x), in Delay and Functional Differential Equations and their Applications", (Klauss Schmitt, Ed.), pp. 197-217, Academic Press, New York, 1972.
N. KRASOVSKI, "Stability of Motion" (Translated from Russian), Stanford, University Press, Stanford, Calif., 1963.
E. LIM, Asymptotic Behavior of Solutions of the Functional Differential Equation x'(t) = Ax(?t) + Bx(t), ? > O, Journal Math. Anal. Appl. 55(1976), 794-806.
K. MAHLER, "On a special Functional Equation", J. London Math. Soc. 15(1940), 115-123.
J.B. Me LEOD, The Functional-Differential Equations y'(x) = ay(?x) + by(x) and Generalizations, Lecture Notes Math. 280 (1972), 308-313.
M. PINTO, Functional Differential Equations with Several Lags, To appear.
M. PINTO, Asymptotic Integration of a Class of Second Order Retarded Differential Equations, To appear.
J. YORKE, Asymptotic Stability for one Dimensional Differential Equations, 7(1969), 189-202.
T. YONEYAMA, On the Stability for the Delay-Differential Equation x'(t) = -a(t)f(x(t-r(t))), J. Math. Anal. Math. 120(1986), 271-275.
Published
2018-04-02
How to Cite
[1]
M. Pinto, “Asymptotic behavior of solutions of the functional differential equation x’(t) = a(t)x(r(t)) + bx(t)”, Proyecciones (Antofagasta, On line), vol. 10, no. 17, pp. 59-76, Apr. 2018.
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