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Oscillations of difference equations with general advanced argument

  • Research Article
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Central European Journal of Mathematics

Abstract

Consider the first order linear difference equation with general advanced argument and variable coefficients of the form

$$\nabla x(n) - p(n)x(\tau (n)) = 0, n \geqslant 1,$$

where {p(n)} is a sequence of nonnegative real numbers, {τ(n)} is a sequence of positive integers such that

$$\tau (n) \geqslant n + 1, n \geqslant 1,$$

and ▿ denotes the backward difference operator ▿x(n) = x(n) − x(n − 1). Sufficient conditions which guarantee that all solutions oscillate are established. Examples illustrating the results are given.

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References

  1. Berezansky L., Braverman E., Pinelas S., On nonoscillation of mixed advanced-delay differential equations with positive and negative coefficients, Comput. Math. Appl., 2009, 58(4), 766–775

    Article  MathSciNet  MATH  Google Scholar 

  2. Chatzarakis G.E., Koplatadze R., Stavroulakis I.P., Optimal oscillation criteria for first order difference equations with delay argument, Pacific J. Math., 2008, 235(1), 15–33

    Article  MathSciNet  MATH  Google Scholar 

  3. Chatzarakis G.E., Koplatadze R., Stavroulakis I.P., Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal., 2008, 68(4), 994–1005

    Article  MathSciNet  MATH  Google Scholar 

  4. Dannan F.M., Elaydi S.N., Asymptotic stability of linear difference equations of advanced type, J. Comput. Anal. Appl., 2004, 6(2), 173–187

    MathSciNet  MATH  Google Scholar 

  5. El’sgol’ts L.E., Introduction to the Theory of Differential Equations with Deviating Arguments, Holden-Day, San Francisco, 1966

    Google Scholar 

  6. Fukagai N., Kusano T., Oscillation theory of first order functional-differential equations with deviating arguments, Ann. Mat. Pura Appl., 1984, 136(1), 95–117

    Article  MathSciNet  MATH  Google Scholar 

  7. Győri I., Ladas G., Oscillation Theory of Delay Differential Equations, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1991

    Google Scholar 

  8. Koplatadze R.G., Chanturija T.A., Oscillating and monotone solutions of first-order differential equations with deviating argument, Differ. Uravn., 1982, 18(2), 1463–1465 (in Russian)

    MATH  Google Scholar 

  9. Kulenovic M.R., Grammatikopoulos M.K., Some comparison and oscillation results for first-order differential equations and inequalities with a deviating argument, J. Math. Anal. Appl., 1988, 131(1), 67–84

    Article  MathSciNet  MATH  Google Scholar 

  10. Kusano T., On even-order functional-differential equations with advanced and retarded arguments, J. Differential Equations, 1982, 45(1), 75–84

    Article  MathSciNet  MATH  Google Scholar 

  11. Ladas G., Stavroulakis I.P., Oscillations caused by several retarded and advanced arguments, J. Differential Equations, 1982, 44(1), 134–152

    Article  MathSciNet  MATH  Google Scholar 

  12. Li X., Zhu D., Oscillation and nonoscillation of advanced differential equations with variable coefficients, J. Math. Anal. Appl., 2002, 269(2), 462–488

    Article  MathSciNet  MATH  Google Scholar 

  13. Li X., Zhu D., Oscillation of advanced difference equations with variable coefficients, Ann. Differential Equations, 2002, 18(2), 254–263

    MathSciNet  MATH  Google Scholar 

  14. Onose H., Oscillatory properties of the first-order differential inequalities with deviating argument, Funkcial. Ekvac., 1983, 26(2), 189–195

    MathSciNet  MATH  Google Scholar 

  15. Zhang B.G., Oscillation of the solutions of the first-order advanced type differential equations, Sci. Exploration, 1982, 2(3), 79–82

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 14121, Athens, Greece

    George E. Chatzarakis

  2. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    Ioannis P. Stavroulakis

Authors
  1. George E. Chatzarakis
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  2. Ioannis P. Stavroulakis
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Correspondence to George E. Chatzarakis.

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Chatzarakis, G.E., Stavroulakis, I.P. Oscillations of difference equations with general advanced argument. centr.eur.j.math. 10, 807–823 (2012). https://doi.org/10.2478/s11533-011-0137-5

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  • DOI: https://doi.org/10.2478/s11533-011-0137-5

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