Abstract
We obtain an existence theorem for monotone positive solutions of nonlinear second-order nonlinear differential equations via principal and nonprincipal solutions of the corresponding homogeneous equation.
Similar content being viewed by others
References
Agarwal, R.P., Djebali, S., Moussaoui, T., Mustafa, O.G.: On the asymptotic integration of nonlinear differential equations. J. Comput. Appl. Math. 202, 352–376 (2007)
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)
Boas, M.L., Boas Jr., R.P., Levinson, N.: The growth of solutions of a differential equation. Duke Math. J. 9, 847–853 (1942)
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)
Constantin, A.: On the existence of positive solutions of second-order differential equations. Ann. Mat. Pura Appl. (4) 184(2), 131–138 (2005)
Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem. Clarendon Press, Oxford (1989)
Ertem, T., Zafer, A.: Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions. Appl. Math. Comput. 219(11), 5876–5886 (2013)
Ertem, T., Zafer, A.: Existence of solutions for a class of nonlinear boundary value problems on half-line. Bound. Value Probl. 2012, 43 (2012)
Ertem, T., Zafer, A.: Monotone positive solutions for a class of second-order nonlinear differential equations. J. Comput. Appl. Math. 259, 672–681 (2014)
Hale, J.K., Onuchic, N.: On the asymptotic behavior of solutions of a class of differential equations. Contrib. Differ. Equ. 2, 61–75 (1963)
Haupt, O.: Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math. Z. 48, 289–292 (1942)
Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht (1993)
Lipovan, O.: On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations. Glasg. Math. J. 45, 179–187 (2003)
Mustafa, O.G., Rogovchenko, Y.V.: Asymptotic integration of a class of nonlinear differential equations. Appl. Math. Lett. 19, 849–853 (2006)
Zhao, Z.: Positive solutions of nonlinear second order ordinary differential equations. Proc. Am. Math. Soc. 121, 465–469 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Rights and permissions
About this article
Cite this article
Aslanov, A. A note on the positive solutions of second order differential equations. Monatsh Math 183, 229–239 (2017). https://doi.org/10.1007/s00605-016-0946-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0946-5