Asymptotics for second order delayed differential equations
DOI:
https://doi.org/10.4067/S0716-09172007000100005Keywords:
Second order linear delayed functional differential equations, Asymptotic formula, Haddock-Sacker conjecture.Abstract
In this work we present a way to find asymptotic formulas for some solutions of second order linear differential equations with a retarded functional perturbation.
References
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[2] O. Arino and I. Gyori, Asymptotic integration of delay differential systems, J. Math. Anal. Appl. 138, pp. 311-327, (1989).
[3] O. Arino, I. Gyori and M. Pituk, Asymptotic diagonal delay differential systems, J. Math. Anal. Appl 204, pp. 701-728, (1996).
[4] O. Arino and M. Pituk, More on linear differential systems with small delays, J. Differential Equations 170, pp. 381-407, (2001).
[5] F. Atkinson and J. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl 91, pp. 410-423, (1983).
[6] J. Cassell and Z. Hou, Asymptotically diagonal linear differential equations with retardation, J. London Math. Soc. (2) 47 (1993) 473-483.
[7] S. Castillo, Asymptotic formula for functional dynamic equations in time scale with functional perturbation, Functional Differential Equations, The Research Institute, The College of Judea and Samaria, Ariel Israel Vol 10. No. 1-2, pp. 107-120, (2003).
[8] S. Castillo and M. Pinto, Asymptotic integration of ordinary differential systems. J. Math. Anal. Appl. 218, pp. 1-12, (1998).
[9] S. Castillo and M. Pinto, Levinson theorem for functional differential systems, Nonlinear Analysis, Vol 47/6, pp. 3963-3975, (2001).
[10] S. Castillo and M. Pinto, An asymptotic theory for nonlinear functional differential equations. Comput. Math. Appl. 44, N. 5-6, pp. 763-775, (2002).
[11] S. Castillo and M. Pinto, Asymptotics of Scalar Functional Differential Equations, Functional Differential Equations, The Research Institute, The College of Judea and Samaria, Ariel Israel Vol 11. N 1-2, pp. 29-36, (2004).
[12] K. Cooke, Functional differential equations close to differential equations, Bull. Amer. Math. Soc. 72, pp. 285-288, (1966).
[13] R. Driver, Linear differential systems with small delays, J. Differential Equations 21, pp. 149-167, (1976).
[14] M. Eastham, The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem, Clarendon, Oxford, (1989).
[15] I. Gyori and M. Pituk, L2−Perturbation of a linear delay differential equation, J. Math. Anal. Appl., 195, pp. 415-427, (1995).
[16] J. Haddock and R. Sacker, Stability and asymptotic integration for certain linear systems of functional differential equations, J. Math. Anal. Appl., 76, pp. 328-338, (1980).
[17] W. Harris and D. Lutz, A unified theory of asymptotic integration, J. Math. Anal. Appl. 57, pp. 571-586, (1977).
[18] P. Hartman and A. Wintner, Asymptotic integration of linear differential equations, Amer. J. Math. 77, pp. 48-86 and 932, (1955).
[19] N. Levinson, The asymptotic behavior of system of linear differential equations, Amer. J. Math. 68, pp. 1-6, (1946).
[20] M. Pituk, Asymptotic characterization of solutions of functional differential equations, Boll. Un. Math. Ital. 7-B, pp. 653-689, (1993).
[21] M. Pituk, The Hartman-Wintner theorem for functional-differential equations. J. Differential Equations 155, N. 1, 1-16, (1999).
[2] O. Arino and I. Gyori, Asymptotic integration of delay differential systems, J. Math. Anal. Appl. 138, pp. 311-327, (1989).
[3] O. Arino, I. Gyori and M. Pituk, Asymptotic diagonal delay differential systems, J. Math. Anal. Appl 204, pp. 701-728, (1996).
[4] O. Arino and M. Pituk, More on linear differential systems with small delays, J. Differential Equations 170, pp. 381-407, (2001).
[5] F. Atkinson and J. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl 91, pp. 410-423, (1983).
[6] J. Cassell and Z. Hou, Asymptotically diagonal linear differential equations with retardation, J. London Math. Soc. (2) 47 (1993) 473-483.
[7] S. Castillo, Asymptotic formula for functional dynamic equations in time scale with functional perturbation, Functional Differential Equations, The Research Institute, The College of Judea and Samaria, Ariel Israel Vol 10. No. 1-2, pp. 107-120, (2003).
[8] S. Castillo and M. Pinto, Asymptotic integration of ordinary differential systems. J. Math. Anal. Appl. 218, pp. 1-12, (1998).
[9] S. Castillo and M. Pinto, Levinson theorem for functional differential systems, Nonlinear Analysis, Vol 47/6, pp. 3963-3975, (2001).
[10] S. Castillo and M. Pinto, An asymptotic theory for nonlinear functional differential equations. Comput. Math. Appl. 44, N. 5-6, pp. 763-775, (2002).
[11] S. Castillo and M. Pinto, Asymptotics of Scalar Functional Differential Equations, Functional Differential Equations, The Research Institute, The College of Judea and Samaria, Ariel Israel Vol 11. N 1-2, pp. 29-36, (2004).
[12] K. Cooke, Functional differential equations close to differential equations, Bull. Amer. Math. Soc. 72, pp. 285-288, (1966).
[13] R. Driver, Linear differential systems with small delays, J. Differential Equations 21, pp. 149-167, (1976).
[14] M. Eastham, The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem, Clarendon, Oxford, (1989).
[15] I. Gyori and M. Pituk, L2−Perturbation of a linear delay differential equation, J. Math. Anal. Appl., 195, pp. 415-427, (1995).
[16] J. Haddock and R. Sacker, Stability and asymptotic integration for certain linear systems of functional differential equations, J. Math. Anal. Appl., 76, pp. 328-338, (1980).
[17] W. Harris and D. Lutz, A unified theory of asymptotic integration, J. Math. Anal. Appl. 57, pp. 571-586, (1977).
[18] P. Hartman and A. Wintner, Asymptotic integration of linear differential equations, Amer. J. Math. 77, pp. 48-86 and 932, (1955).
[19] N. Levinson, The asymptotic behavior of system of linear differential equations, Amer. J. Math. 68, pp. 1-6, (1946).
[20] M. Pituk, Asymptotic characterization of solutions of functional differential equations, Boll. Un. Math. Ital. 7-B, pp. 653-689, (1993).
[21] M. Pituk, The Hartman-Wintner theorem for functional-differential equations. J. Differential Equations 155, N. 1, 1-16, (1999).
Published
2017-04-18
How to Cite
[1]
S. Castillo and M. Pinto, “Asymptotics for second order delayed differential equations”, Proyecciones (Antofagasta, On line), vol. 26, no. 1, pp. 91-103, Apr. 2017.
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