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Asymptotic integration of nonoscillatory differential equations: a unified approach

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Abstract

We consider the equation [r(t)x′]′ + f(t)x = 0 as a perturbation of the equation [r(t)y′]′ + g(t)y = 0, where the latter is assumed to be nonoscillatory at infinity. The functions r and g are real-valued, r is positive, and f is complex-valued. The problem of the asymptotic integration of the perturbed equation in comparison with solutions of the unperturbed equation has been studied by many mathematicians, including Hartman and Wintner, Trench, ˇSimˇsa, Chen, and Chernyavskaya and Shuster. Here we apply a unified approach. Working in a matrix setting, we use preliminary and so-called conditioning transformations to bring the system in the form \( \vec{z} = \left[ {\Lambda (t) + R(t)} \right]\vec{z} \), where Λ is a certain diagonal matrix and R is an absolutely integrable perturbation. This allows us to use Levinson’s fundamental theorem to find the asymptotic behavior of solutions and, in addition, to estimate the error involved. This method allows us to derive these known results in a more unified setting and to weaken the hypotheses in some instances.

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References

  1. S. Bodine and D. A. Lutz, Asymptotic solutions and error estimates for linear systems of difference and differential equations. J. Math. Anal. Appl. 290 (2004), 343–362.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Z. Chen, Asymptotic integrations of nonoscillatory second-order differential equations. Trans. Amer. Math. Soc. 327 (1991), 853–865.

    Article  MathSciNet  MATH  Google Scholar 

  3. N. A. Chernyavskaya and L. Shuster, Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner. Proc. Amer. Math. Soc. 125 (1997), 3213–3228.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. A. Coppel, Stability and asymptotic behavior of differential equations. Heath, Boston (1965).

    MATH  Google Scholar 

  5. P. Hartman, Ordinary differential equations. SIAM, Philadelphia (2002).

    Book  MATH  Google Scholar 

  6. P. Hartman and A. Wintner, Asymptotic integrations of linear differential equations, Amer. J. Math. 77 (1955), 45–86, 404.

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Levinson, The asymptotic nature of solutions of linear differential equations. Duke Math. J. 15 (1948), 111–126.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Šimša, Asymptotic integration of a second order ordinary differential equation. Proc. Amer. Math. Soc. 101 (1987), 96–100.

    MathSciNet  MATH  Google Scholar 

  9. S. A. Stepin, Asymptotic integration of nonoscillatory second-order differential equations. Dokl. Math. 82 (2010), 751–754.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. F. Trench, Linear perturbations of a nonoscillatory second order equation. Proc. Amer. Math. Soc. 97 (1986), 423–428.

    Article  MathSciNet  MATH  Google Scholar 

  11. ______, Linear perturbations of a nonoscillatory second order differential equation, II. Proc. Amer. Math. Soc. 131 (2003), 1415–1422.

    Article  MathSciNet  MATH  Google Scholar 

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Bodine, S., Lutz, D.A. Asymptotic integration of nonoscillatory differential equations: a unified approach. J Dyn Control Syst 17, 329–358 (2011). https://doi.org/10.1007/s10883-011-9122-3

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  • DOI: https://doi.org/10.1007/s10883-011-9122-3

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