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Generalized Zeros and Nonpositivity of Energy Functionals Associated with Half-Linear Even-Order Difference Equations

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Difference Equations, Discrete Dynamical Systems and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 150))

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Abstract

We investigate the relationship between oscillatory properties of half-linear even order difference equations and nonpositivity of the associated energy functionals. We convert the investigated difference equation into a Hamiltonian type difference system and using this transformation we establish our main result which says that the existence of two (or more) generalized zeros of a solution of the investigated difference equation implies that the corresponding energy functional attains a nonpositive value.

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References

  1. R.P. Agarwal, M. Bohner, S.R. Grace, D. O’Regan, Discrete Oscillation Theory (Hindawi Publishing Corporation, Cairo, 2006)

    MATH  Google Scholar 

  2. C.D. Ahlbrandt, A.C. Peterson, The \((n, n)\) disconjugacy of a 2nth-order linear difference equation. Comput. Appl. Math. 28, 1–9 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Bihari, An oscillation theorem concerning the half-linear differential equation of second order, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, 275–280 (1964)

    MathSciNet  MATH  Google Scholar 

  4. M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. Math. Anal. Appl. 199, 804–826 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bohner, On disconjugacy for Sturm-Liouville difference equations, in Difference Equations: Theory and Applications, (San Francisco, CA, 1995). (J. Differ. Equations Appl. 2 (1996), no. 2, 227–237)

    Google Scholar 

  6. O. Došlý, Oscillation criteria for higher order Sturm-Liouville difference equations. J. Differ. Equ. Appl. 4(5), 425–450 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. O. Došlý, Oscillation and spectral properties of self-adjoint even order differential operators with middle terms, Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations (2004) No. 7, p. 21. (Proc. Colloq. Qual. Theory Differ. Equ. 7, Electron J. Qual Theory Differ. Equ. Szeged

    Google Scholar 

  8. O. Došlý, R. Hilscher, A class of Sturm-Liouville difference equations: (non)oscillation constants and property BD, advances in difference equations, IV. Comput. Math. Appl. 45 no. 6–9, 961–981 (2003)

    Google Scholar 

  9. O. Došlý, P. Řehák, Nonoscillation criteria for half-linear second-order difference equations, advances in difference equations, III. Comput. Math. Appl. 42(3–5), 453–464 (2001)

    MathSciNet  MATH  Google Scholar 

  10. O. Došlý, P. Řehák, Half-Linear Differential Equations. North Holland Mathematics Studies, vol. 202 (Elsevier, Amsterdam, 2005)

    Google Scholar 

  11. O. Došlý, V. Růžička, Nonoscillation of Higher Order Half-Linear Differential Equations. Electron. J. Qual. Theory Differ. Equ. 19, 15 (2015)

    Google Scholar 

  12. W.D. Evans, M.K. Kwong, A. Zettl, Lower bounds for spectrum of ordinary differential operators. J. Differ. Equ. 48, 123–155 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Fišnarová, Oscillation of two-term Sturm-Liouville difference equations. Int. J. Differ. Equ. 1(1), 81–99 (2006)

    MathSciNet  MATH  Google Scholar 

  14. I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations, Jerusalem, 1965 (Daniel Davey & Co. Inc., New York, 1966)

    Google Scholar 

  15. D.B. Hinton, R.T. Lewis, Spectral analysis of second order difference equations. J. Math. Anal. Appl. 63, 421–438 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Hartman, Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. Am. Math. Soc. 246, 1–30 (1978)

    MathSciNet  MATH  Google Scholar 

  17. R. Oinarov, S.Y. Rakhmatulina, Oscillation and nonoscillatorion of two terms linear and half-linear equations of higher order, Electron. J. Qual. Theory Differ. Equ. 2010 No. 49, 1–15 (2010)

    Google Scholar 

  18. S. Peña, Discrete spectra criteria for singular difference operators. Math. Bohem. 124(1), 35–44 (1999)

    MathSciNet  MATH  Google Scholar 

  19. P. Řehák, Oscillation criteria for second order half-linear difference equations. J. Differ. Equ. Appl. 7, 483–505 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Řehák, Oscillatory properties of second order half-linear difference equations. Czechoslov. Math. J. 51(126), 303–321 (2001)

    MathSciNet  MATH  Google Scholar 

  21. P. Řehák, Generalized discrete Riccati equation and oscillation of half-linear difference equations. Math. Comput. Model. 34, 257–269 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Řehák, Comparison theorems and strong oscillation in the half-linear discrete oscillation theory, Rocky Mountain J. Math. 33, 333–352 (2003)

    Google Scholar 

  23. J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258 (Springer-Verlag, Berlin, 1987)

    Google Scholar 

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Acknowledgments

Research supported by the grant P201/10/1032 of the Czech Grant Foundation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, CZ-611 37, Brno, Czech Republic

    Ondřej Došlý

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  1. Ondřej Došlý
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Correspondence to Ondřej Došlý .

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Editors and Affiliations

  1. Missouri University of Science and Techn Dept. of Mathematics and Statistics, Rolla, Montana, USA

    Martin Bohner

  2. Chinese Academy of Sciences Wuhan Institute of Physics and Mathemati, Wuhan, China

    Yiming Ding

  3. Faculty of Science, Masaryk University Faculty of Science, Brno, Czech Republic

    Ondřej Došlý

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Došlý, O. (2015). Generalized Zeros and Nonpositivity of Energy Functionals Associated with Half-Linear Even-Order Difference Equations. In: Bohner, M., Ding, Y., Došlý, O. (eds) Difference Equations, Discrete Dynamical Systems and Applications. Springer Proceedings in Mathematics & Statistics, vol 150. Springer, Cham. https://doi.org/10.1007/978-3-319-24747-2_6

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