Home Mathematics Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
Article
Licensed
Unlicensed Requires Authentication

Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales

  • Yaşar Bolat EMAIL logo and Ömer Akin EMAIL logo
Published/Copyright: August 26, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 3

Abstract

In this paper, oscillation criteria are obtained for higher-order half-linear delay difference equations involving generalized difference operator of the form

Δb(pn(Δbm−1xn)α)+qnxn−σβ=0,  n≥n0,

where ∆b is defined by ∆byn = yn+1 - byn, b ∈ ℝ - {0}, p: ℕ → ℝ+, α, β are the ratio of odd positive integers with β ≤ α; m, n, n0, σ are non-negative integers, q: ℕ → ℝ. The cases of b negative and positive and qn ≥ 0, which has important role for oscillation of this equation, are considered. Also we provide some examples to illustrate our main results.

Key words: oscillation; difference equaions; generalized difference; half-linear difference equation
MSC 2010: Primary 39A10

(Communicated by Michal Fečkan)

References

[1] Agarwal, R. P.—Grace, S. R.—O’Regan, D.: Oscillation Theory for Difference an Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000.10.1007/978-94-015-9401-1Search in Google Scholar

[2] Agarwal, R. P.: Difference Equations and Inequalities, Theory, Methods and Applications (2nd ed.), Marcel Dekker, New York, 2000.10.1201/9781420027020Search in Google Scholar

[3] Agarwal, R. P.—Wong, P. J. Y.: Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, 1997.10.1007/978-94-015-8899-7Search in Google Scholar

[4] Bolat, Y.—O. Alzabut, J.: On the oscillation of higher-order half-linear delay difference equations, Appl. Math. Inf. Sci. 6 (2012), 423–427.Search in Google Scholar

[5] Cecchi, M.—Dosla, Z.—Marini, M.: Positive decreasing solutions of quasilinear difference equations, Comput. Math. Appl. 42 (2001), 1401–1410.10.1016/S0898-1221(01)00249-8Search in Google Scholar

[6] Dosly, O.—Rehak, P.: Nonoscillation criteria for half-linear second order difference equations, Comput. Math. Appl. 42 (2001), 453–464.10.1016/S0898-1221(01)00169-9Search in Google Scholar

[7] Dosly, O.—Fisnarov, S.: Linearized Riccati technique and (non-)oscillation criteria for half-linear difference equations, Adv. Difference Equ. 2008, (2008), Article ID 438130, 18 pp.10.1155/2008/438130Search in Google Scholar

[8] Hardy, G. H.—Littlewood, J. E.—Polya, G.: Inequalities (2nd ed.), Cambridge Univ. Press, Cambridge, 1952.Search in Google Scholar

[9] Ladas, G.—Qian, C.: Comparison results and linearized oscillations for higher order diference equations, Int. J. Math. Math. Sci. 15 (1992), 129–142.10.1155/S0161171292000152Search in Google Scholar

[10] El-Morshedy, H. A.: Oscillation and nonoscillation criteria for half-linear second order difference equations, Dynam. Systems Appl. 15 (2006), 429–450.Search in Google Scholar

[11] Parhi, N.: Oscillation and non-oscillation of solutions of second order difference equations involving generalized difference, Appl. Math. Comput. 218 (2011), 458–468.10.1016/j.amc.2011.05.086Search in Google Scholar

[12] Parhi, N.—Panda, A.: Nonoscillation and oscillation of solutions of a class of third order difference equations, J. Math. Anal. Appl. 336 (2007), 213–223.10.1016/j.jmaa.2007.02.054Search in Google Scholar

[13] Popenda, J.: Oscillation and nonoscillation theorems for second-order difference equations, J. Math. Anal. Appl. 123 (1987) 34–38.10.1016/0022-247X(87)90291-5Search in Google Scholar

[14] Popenda, J.—Szmanda, B.: On the oscillation of solutions of certain difference equations, Demonstratio Math. XVII,(1984), 153–164.10.1515/dema-1984-0114Search in Google Scholar

[15] Rehak, P.: Hartman-Wintner type lemma, oscillation and conjugacy criteria for half-linear difference equations, J. Math. Anal. Appl. 252 (2000), 813–827.10.1006/jmaa.2000.7124Search in Google Scholar

[16] Rehak, P.: Generalized discrete Riccati equations and oscillation of half-linear difference equations, Math. Comput. Modelling 34 (2001), 257–269.10.1016/S0895-7177(01)00059-0Search in Google Scholar

[17] Rehak, P.: Oscillatory properties of second-order half-linear delay difference equations, Czechoslovak Math. J. 51(126) (2001), 303–321.10.1023/A:1013790713905Search in Google Scholar

[18] Saker, S. H.: Oscillation criteria of second-order half-linear delay difference equations, Kyungpook Math. J. 45 (2005), 579–594.Search in Google Scholar

[19] Tan, M.—Yang, E.: Ocillation and nonoscillation theorems for second order nonlinear difference equations, J. Math. Anal. Appl. 276 (2002), 239–247.10.1016/S0022-247X(02)00435-3Search in Google Scholar

[20] Thandapani, E.—Ravi, K.: Bounded and monotone properties of solutions of secondorder quasilinear forced difference equations, Comput. Math. Appl. 38 (1999), 113–121.10.1016/S0898-1221(99)00186-8Search in Google Scholar

[21] Thandapani, E.—Ravi, K.: Oscillation of second-order half-linear difference equations, Appl. Math. Lett. 13 (2000), No. 2, 43–49.10.1016/S0893-9659(99)00163-9Search in Google Scholar

[22] Thandapani, E.—Ravi, K.—Graef, J. G.: Oscillation and comparison theorems for half-linear second order difference equations, Comput. Math. Appl. 42 (2001), 953–960.10.1016/S0898-1221(01)00211-5Search in Google Scholar

[23] Thandapani, E.—Vijaya, L. M.: On the oscillation of third order half-linear neutral type difference equations, Electron. J. Qual. Theory Differ. Equ. 76 (2011), 1–13.10.14232/ejqtde.2011.1.76Search in Google Scholar

[24] Wong, P. J. Y.—Agarwal, R. P.: Oscillations and nonoscillation of half-linear difference equations generated by deviating arguments, Comput. Math. Appl. 36 (1998), 11–26.10.1016/S0898-1221(98)80005-9Search in Google Scholar

[25] Zhang, G.—Cheng, S. S.: On two second ordered half-linear difference equations, Fasc. Math. 35 (2005), 163–175.Search in Google Scholar

Received: 2012-11-14
Accepted: 2013-12-9
Published Online: 2016-8-26
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
https://doi.org/10.1515/ms-2015-0169
Keywords for this article
oscillation; difference equaions; generalized difference; half-linear difference equation

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 16.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0169/pdf?lang=en
Scroll to top button