Skip to main content
SpringerLink
Log in

Non-Monotonic Behavior of One-Signed Solutions of Quasilinear Differential Equations

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider a class of the second-order quasilinear differential equations. By deriving relations between certain types of monotonic solutions of the quasilinear equation and corresponding reciprocal half-linear equation on a finite interval (a, b), we obtain criteria for all solutions of the main equation, which do not change sign in (a, b), to be non-monotonic in (a, b). This work is also extended to a perturbed half-linear equation as well as to the half-line \((a,\infty )\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002)

    Book  MATH  Google Scholar 

  2. Almenar, P., Jódar, L.: The distribution of zeroes and critical points of solutions of a second order half-linear differential equation. Abstr. Appl. Anal., 2013, 6 (2013). https://doi.org/10.1155/2013/147192 (Art. ID 147192)

  3. Cecchi, M., Marini, M., Villari, G.: Integral criteria for a classification of solutions of linear differential equations. J. Differ. Equations 99, 381–397 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Došla, Z., Marini, M., Matucci, S.: A Dirichlet problem on the half-line for nonlinear equations with indefinite weight. Ann. Mat. Pura Appl. 196, 51–64 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Došly, O., Řehak, P.: Half-Linear Differential Equations, North-Holland, Mathematics Studies, vol. 202. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  6. El-Sayed, M.A.: An oscillation criterion for a forced second-order linear differential equation. Proc. Am. Math. Soc. 118, 813–817 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Fišnarová, S., Mařík, R.: Local estimates for modified Riccati equation in theory of half-linear differential equation. Electron. J. Qual. Theory Differ. Equations 2012(63), 1–5 (2012)

  8. Hasil, P., Veselý, M.: Non-oscillation of periodic half-linear equations in the critical case. Electron. J. Differ. Equations 2016(120), 1–2 (2016)

  9. Kong, Q.: Oscillation criteria for second order half-linear differential equations. Fields Inst. Commun. 21, 317–323 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Kong, Q.: Nonoscillation and oscillation of second order half-linear differential equations. J. Math. Anal. Appl. 332, 512–522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kong, Q.: A Short Course in Ordinary Differential Equations. Springer, New York (2014)

    Book  MATH  Google Scholar 

  12. Kong, Q., Wang, X.: Nonlinear initial problems with \(p\)-Laplacian. Dyn. Syst. Appl. 19, 33–44 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Ladde, G.S., Lakshmikanthan, V., Vatsala, A.S.: Monotone Iteration Technique for Nonlinear Differential Equations. Pitman, Boston (1985)

    Google Scholar 

  14. Mohapatra, R.N., Vajravelu, K., Yin, Y.: Generalized quasilinearization method and rapid convergence for first order initial value problems. J. Math. Anal. Appl. 207, 206–219 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mařík, R.: A remark on connection between conjugacy of half-linear differential equation and equation with mixed nonlinearities. Appl. Math. Lett. 24, 93–96 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pašić, M.: Sign-changing first derivative of positive solutions of forced second-order nonlinear differential equations. Appl. Math. Lett. 40, 40–44 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pašić, M.: Strong non-monotonic behavior of particle density of solitary waves of nonlinear Schrödinger equation in Bose–Einstein condensates. Commun. Nonlinear Sci. Numer. Simul. 29, 161–169 (2015)

    Article  MathSciNet  Google Scholar 

  18. Pašić, M., Tanaka, S.: Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria. Electron. J. Qual. Theory Differ. Equations 93, 1–25 (2016)

    MathSciNet  Google Scholar 

  19. Potter, R.L.: On self-adjoint differential equations of second order. Pac. J. Math. 3, 467–491 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  20. Řehák, P.: De Haan type increasing solutions of half-linear differential equations. J. Math. Anal. Appl. 412, 236–243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Řehák, P.: Exponential estimates for solutions of half-linear differential equations. Acta Math. Hung. 147, 158–171 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Swanson, C.A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  23. Walter, W.: Ordinary Differential Equations. Springer, New York (1998)

    Book  MATH  Google Scholar 

  24. Zhang, G.-B.: Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation. Appl. Anal. (2016). https://doi.org/10.1080/00036811.2016.1197913

  25. Zheng, W., Sugie, J.: Parameter diagram for global asymptotic stability of damped half-linear oscillators. Monatsh. Math. 179(1), 149–160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115, USA

    Qingkai Kong

  2. Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000, Zagreb, Croatia

    Mervan Pašić

Authors
  1. Qingkai Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mervan Pašić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mervan Pašić.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kong, Q., Pašić, M. Non-Monotonic Behavior of One-Signed Solutions of Quasilinear Differential Equations. Mediterr. J. Math. 15, 184 (2018). https://doi.org/10.1007/s00009-018-1232-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1232-7

Mathematics Subject Classification

Keywords

Search

Navigation