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Forced Oscillation of Second-Order Impulsive Differential Equations with Mixed Nonlinearities

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Differential and Difference Equations with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 47))

Abstract

In this paper we give new oscillation criteria for a class of second-order mixed nonlinear impulsive differential equations having fixed moments of impulse actions. The method is based on the existence of a nonprincipal solution of a related second-order linear homogeneous equation.

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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 Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  2. Bainov, D.D., Domshlak, Y.I., Simeonov, P.S.: Sturmian comparison theory for impulsive differential inequalities and equations. Arch. Math. (Basel) 67, 35–49 (1996)

    Google Scholar 

  3. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Essex (1993)

    MATH  Google Scholar 

  4. Cecchi, M., Došlá, Z., Marini, M.: Half-linear equations and characteristic properties of the principal solutions. J. Differ. Equ. 208, 96–507 (2005)

    Article  Google Scholar 

  5. Chen, S.: Asymptotic integrations of nonoscillatory second order differential equations. Tran. Am. Math. Soc. 327, 853–865 (1991)

    Article  MATH  Google Scholar 

  6. Chen, Y.S., Feng, W.Z.: Oscillations of second order nonlinear ode with impulses. J. Math. Anal. Appl. 210, 150–169 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Došlý, O.: Principal solutions and transformations of linear Hamiltonian systems (English summary). Arch. Math. (Brno) 28, 96–120 (1992)

    Google Scholar 

  8. Došlý, O., Řehák, P.: Half-Linear Differential Equations. Elsevier Ltd., Heidelberg (2005)

    MATH  Google Scholar 

  9. 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 

  10. Hartman, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  11. Kartsatos, A.G.: Maintenance of oscillations under the effect of a periodic forcing term. Proc. Am. Math. Soc. 33, 377–383 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Keener, M.S.: Solutions of a linear nonhomogenous second order differential equations. Appl. Anal. 1, 57–63 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  13. Luo, J.: Second-order quasilinear oscillation with impulses. Comput. Math. Appl. 46, 279–291 (2003)

    Article  MathSciNet  Google Scholar 

  14. Morse, M., Leighton, W.: Singular quadratic functionals. Trans. Am. Math. Soc. 40, 252–286 (1936)

    Article  MathSciNet  Google Scholar 

  15. Özbekler, A., Wong, J.S.W., Zafer, A.: Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients. Appl. Math. Lett. 24, 1225–1230 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Özbekler, A., Zafer, A.: A. Sturmian comparison theory for linear and half-linear impulsive differential equations. Nonlinear Anal. 63, 289–297 (2005)

    Google Scholar 

  17. Özbekler, A., Zafer, A.: A. Picone’s formula for linear non-selfadjoint impulsive differential equations. J. Math. Anal. Appl. 319, 410–423 (2006)

    Google Scholar 

  18. Özbekler, A., Zafer, A.: Principal and non-principal solutions of impulsive differential equations with applications. Appl. Math. Comput. 216, 1158–1168 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Özbekler, A., Zafer, A.: Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients. In: Discrete Continuous Dynamical System Series B 2011, Dynamical Systems and Differential Equations. Proceedings of the 8th AIMS International Conference, suppl. (Accepted)

    Google Scholar 

  20. Rainkin, S.M.: Oscillation theorems for second-order nonhomogenous linear differential equations. J. Math. Anal. Appl. 53, 550–553 (1976)

    Article  MathSciNet  Google Scholar 

  21. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    MATH  Google Scholar 

  22. Šimša, J.: Asymptotic integration of second order ordinary differential equations. Proc. Am. Mat. Soc. 101, 96–100 (1987)

    MATH  Google Scholar 

  23. Skidmore, A., Bowers, J.J.: Oscillatory behaviour of solutions of \(y^{{\prime}{\prime}} + p(x)y = f(x)\). J. Math. Anal. Appl. 49, 317–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Skidmore, A., Leighton, W.: On the differential equation \(y^{{\prime}{\prime}} + p(x)y = f(x)\). J. Math. Anal. Appl. 43, 46–55 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  25. Teufel, H.: Forced second order nonlinear oscillations. J. Math. Anal. Appl. 40, 148–152 (1972)

    Article  MathSciNet  Google Scholar 

  26. Trench, W.F.: Linear perturbation of a nonoscillatory second order differential equation. Proc. Am. Mat. Soc. 97, 423–428 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wong, J.S.W.: Oscillation criteria for forced second-order linear differential equation. J. Math. Anal. Appl. 231, 235–240 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zafer, A.: On oscillation and nonoscillation of second order dynamic equations. Appl. Math. Lett. 22, 136–141 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Atilim University, 06836, Incek, Ankara, Turkey

    A. Özbekler

  2. Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey

    A. Zafer

Authors
  1. A. Özbekler
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  2. A. Zafer
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Corresponding author

Correspondence to A. Zafer .

Editor information

Editors and Affiliations

  1. Departamento de Ciências Exactas e Naturais, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. University of Zürich Institute of Mathematics, Zürich, Switzerland

    Michel Chipot

  3. Masaryk University Dept. Mathematics, Brno, Czech Republic

    Zuzana Dosla

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Özbekler, A., Zafer, A. (2013). Forced Oscillation of Second-Order Impulsive Differential Equations with Mixed Nonlinearities. In: Pinelas, S., Chipot, M., Dosla, Z. (eds) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7333-6_13

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