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A Note on The Asymptotic Solution of Second Order Nonlinear Differential Equation

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We study the existence of solutions of second-order nonlinear differential equations. It is shown that the second-order nonlinear differential equation \(x^{\prime \prime }=f(t,x,x^{\prime })\) has a positive (and so non-oscillatory) solution asymptotic to the function \(\varphi (t)=t\) at \( \infty \) and satisfies the initial condition \(x(0)=0\).

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  1. Agarwal, R.P., Djebali, C., Moussaoui, T., Mustafa, O.G.: On the asymptotic integration of nonlinear differential equations. J. Comput. Appl. Math. 202, 352–376 (2007)

    Article  MathSciNet  Google Scholar 

  2. Aslanov, A.: A note on the positive solutions of second order differential equations. Monatsh. Math. 183, 229–239 (2016)

    Article  MathSciNet  Google Scholar 

  3. Aslanov, A.: On the asymptotic behavior of positive solutions to nonlinear second-order differential equations. Monatsh. Math. 2, 2 (2021).

    Article  Google Scholar 

  4. Bihari, I.: Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Math. Acad. Sci. Hung. 8, 261–278 (1957)

    Article  MathSciNet  Google Scholar 

  5. Boas, M.L., Boas, R.P., Jr., Levinson, N.: The growth of solutions of a differential equation. Duke Math. J. 9, 847–853 (1942)

    Article  MathSciNet  Google Scholar 

  6. Caligo, O.: Comportamento asintotico degli integrali dell’equazione \( y^{\prime \prime }+a(t)y=0\), nell’ipotesi \(\lim _{t\rightarrow \infty }a(t)=0\). Boll. Un. Mat. Ital. 3, 286–295 (1941)

    MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: On the existence of positive solutions of second-order differential equations. Ann. Mat. Pura Appl. 184, 131–138 (2005)

    Article  MathSciNet  Google Scholar 

  8. Park, S., Tan, D.H.: Remarks on the Schauder–Tychonoff fixed-point theorem. Vietnam J. Math. 28, 127–132 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem. Clarendon Press, Oxford (1989)

    MATH  Google Scholar 

  10. Ertem, T., Zafer, A.: Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions. Appl. Math. Comput. 219, 5876–5886 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Ertem, T., Zafer, A.: Existence of solutions for a class of nonlinear boundary value problems on half-line. Boundary Value Prbl. 43, 2 (2012)

    MATH  Google Scholar 

  12. Ertem, T., Zafer, A.: Monotone positive solutions for a class of second-order nonlinear differential equations. J. Comput. Appl. Math. 259, 672–681 (2014)

    Article  MathSciNet  Google Scholar 

  13. Hale, J.K., Onuchic, N.: On the asymptotic behavior of solutions of a class of differential equations. Contrib. Diff. Equ. 2, 61–75 (1963)

    MathSciNet  MATH  Google Scholar 

  14. Haupt, O.: Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math. Z. 48, 289–292 (1942)

    Article  MathSciNet  Google Scholar 

  15. Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  16. Lipovan, O.: On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations. Glasgow Math. J. 45, 179–187 (2003)

    Article  MathSciNet  Google Scholar 

  17. Mingarelli, A.B.: Note on a non-oscilation theorem of Atkinson, Electron. J. Differ. Equ., Vol. (2004), No. 22, pp. 1–6. ISSN: 1072-6691.

  18. Mustafa, O.G., Rogovchenko, Y.V.: Asymptotic integration of a class of nonlinear differential equations. Appl. Math. Lett. 19, 849–853 (2006)

    Article  MathSciNet  Google Scholar 

  19. Yin, Z.: Monotone positive solutions of second-order nonlinear differential equations. Nonlinear Anal. 54, 391–403 (2003)

  20. Zhao, Z.: Positive Solutions of Nonlinear Second Order Ordinary Differential Equations. Proc. Am. Math. Soc. 121, 465–469 (1994)

    Article  MathSciNet  Google Scholar 

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Correspondence to Afgan Aslanov.

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Aslanov, A. A Note on The Asymptotic Solution of Second Order Nonlinear Differential Equation. Mediterr. J. Math. 19, 233 (2022).

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