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Oscillation criteria for solution to partial dynamic equations on time scales

Year 2020, Volume: 49 Issue: 5, 1788 - 1797, 06.10.2020
https://doi.org/10.15672/hujms.554157

Abstract

We consider the oscillatory behavior of solutions to partial dynamic equation on time scales. We establish several oscillation criteria by applying a Ricatti transformation. Examples are provided to justify our results.

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References

  • [1] R.P. Agarwal, M. Bohner and S.H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13, 1–18, 2005.
  • [2] C.D. Ahlbrandt and C. Morian, Partial differential equations on time scales, J. Com- put. Appl. Math. 141, 35–55, 2002.
  • [3] M. Bohner and G.S. Guseinov, Partial differentiation on time scales, Dynam. Systems Appl. 13, 351–379, 2004.
  • [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scale, An Introduction with Applications, Birkhäuser, Boston, 2001.
  • [5] M. Bohner and S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34, 1239–1254, 2004.
  • [6] M. Bohner, L. Erbe and A. Peterson, Oscillation for nonlinear second order dynamic equations on time scales, J. Math. Anal. Appl. 301, 491–507, 2005.
  • [7] L. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for second order nonlinear dynamic equations on time scales, J. Lond. Math. Soc. 67 (3), 701–714, 2003.
  • [8] L.C. Evans, Partail Differential Equations, American Math. Society, Graduate Studies in Mathematics Vol. 19, second ed., 2010.
  • [9] P. Hasil and M. Veselý, Oscillation and nonoscillation criteria for linear and half linear difference equations, J. Math. Anal. Appl. 452 (1), 401–428, 2017.
  • [10] J. Hoffacker, Basic partial dynamic equations on time scales, J. Difference Equ. Appl. 8 (4), 307–319, 2002.
  • [11] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2), 391–415, 2006.
  • [12] P. Prakash and S. Harikrishnan, Oscillation of solutions of impulsive vector hyperbolic differential equations with delays, Appl. Anal. 91 (3), 459–473, 2012.
  • [13] P. Prakash, S. Harikrishnan and M. Benchohra, Oscillation of certain nonlinear frac- tional partial differential equation with damping term, Appl. Math. Lett. 43, 72–79, 2015.
  • [14] S.H. Saker, Oscillation criteria of second order half linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2), 375–387, 2005.
  • [15] Y. Shoukaku and N. Yoshida, Osillations of nonlinear hyperbolic equations with func- tional arguments via Riccati method, Appl. Math. Comput. 217, 143–151, 2010.
  • [16] S. Sun, Z. Han and C. Zhang, Oscillation of second order delay dynamic equations on time scales, J. Appl. Math. Comput. 30 (1-2), 459–468, 2009.
  • [17] Q. Zhang, Oscillation of second order half linear delay dynamic equations with damp- ing on time scales, J. Comput. Appl. Math. 235 (5), 1180–1188, 2011.
  • [18] Q. Zhang and L. Gao, Oscillation of second order nonlinear delay dynamic equations with damping on time scales, J. Appl. Math. Comput. 37 (1-2), 145–158, 2011.
  • [19] X. Zhou, C. Liu and W.-S.Wang, Interval oscillation criteria for nonlinear differential equations with impulses and variable delay, Appl. Math. Lett. 85, 150–156, 2018.
Year 2020, Volume: 49 Issue: 5, 1788 - 1797, 06.10.2020
https://doi.org/10.15672/hujms.554157

Abstract

References

  • [1] R.P. Agarwal, M. Bohner and S.H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13, 1–18, 2005.
  • [2] C.D. Ahlbrandt and C. Morian, Partial differential equations on time scales, J. Com- put. Appl. Math. 141, 35–55, 2002.
  • [3] M. Bohner and G.S. Guseinov, Partial differentiation on time scales, Dynam. Systems Appl. 13, 351–379, 2004.
  • [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scale, An Introduction with Applications, Birkhäuser, Boston, 2001.
  • [5] M. Bohner and S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34, 1239–1254, 2004.
  • [6] M. Bohner, L. Erbe and A. Peterson, Oscillation for nonlinear second order dynamic equations on time scales, J. Math. Anal. Appl. 301, 491–507, 2005.
  • [7] L. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for second order nonlinear dynamic equations on time scales, J. Lond. Math. Soc. 67 (3), 701–714, 2003.
  • [8] L.C. Evans, Partail Differential Equations, American Math. Society, Graduate Studies in Mathematics Vol. 19, second ed., 2010.
  • [9] P. Hasil and M. Veselý, Oscillation and nonoscillation criteria for linear and half linear difference equations, J. Math. Anal. Appl. 452 (1), 401–428, 2017.
  • [10] J. Hoffacker, Basic partial dynamic equations on time scales, J. Difference Equ. Appl. 8 (4), 307–319, 2002.
  • [11] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2), 391–415, 2006.
  • [12] P. Prakash and S. Harikrishnan, Oscillation of solutions of impulsive vector hyperbolic differential equations with delays, Appl. Anal. 91 (3), 459–473, 2012.
  • [13] P. Prakash, S. Harikrishnan and M. Benchohra, Oscillation of certain nonlinear frac- tional partial differential equation with damping term, Appl. Math. Lett. 43, 72–79, 2015.
  • [14] S.H. Saker, Oscillation criteria of second order half linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2), 375–387, 2005.
  • [15] Y. Shoukaku and N. Yoshida, Osillations of nonlinear hyperbolic equations with func- tional arguments via Riccati method, Appl. Math. Comput. 217, 143–151, 2010.
  • [16] S. Sun, Z. Han and C. Zhang, Oscillation of second order delay dynamic equations on time scales, J. Appl. Math. Comput. 30 (1-2), 459–468, 2009.
  • [17] Q. Zhang, Oscillation of second order half linear delay dynamic equations with damp- ing on time scales, J. Comput. Appl. Math. 235 (5), 1180–1188, 2011.
  • [18] Q. Zhang and L. Gao, Oscillation of second order nonlinear delay dynamic equations with damping on time scales, J. Appl. Math. Comput. 37 (1-2), 145–158, 2011.
  • [19] X. Zhou, C. Liu and W.-S.Wang, Interval oscillation criteria for nonlinear differential equations with impulses and variable delay, Appl. Math. Lett. 85, 150–156, 2018.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

R. Ramesh This is me 0000-0001-9477-2113

Julio G. Dıx This is me 0000-0002-5294-1529

Harikrishnan S 0000-0003-1238-3523

P. Prakash 0000-0001-5430-1640

Publication Date October 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 5

Cite

APA Ramesh, R., Dıx, J. G., S, H., Prakash, P. (2020). Oscillation criteria for solution to partial dynamic equations on time scales. Hacettepe Journal of Mathematics and Statistics, 49(5), 1788-1797. https://doi.org/10.15672/hujms.554157
AMA Ramesh R, Dıx JG, S H, Prakash P. Oscillation criteria for solution to partial dynamic equations on time scales. Hacettepe Journal of Mathematics and Statistics. October 2020;49(5):1788-1797. doi:10.15672/hujms.554157
Chicago Ramesh, R., Julio G. Dıx, Harikrishnan S, and P. Prakash. “Oscillation Criteria for Solution to Partial Dynamic Equations on Time Scales”. Hacettepe Journal of Mathematics and Statistics 49, no. 5 (October 2020): 1788-97. https://doi.org/10.15672/hujms.554157.
EndNote Ramesh R, Dıx JG, S H, Prakash P (October 1, 2020) Oscillation criteria for solution to partial dynamic equations on time scales. Hacettepe Journal of Mathematics and Statistics 49 5 1788–1797.
IEEE R. Ramesh, J. G. Dıx, H. S, and P. Prakash, “Oscillation criteria for solution to partial dynamic equations on time scales”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, pp. 1788–1797, 2020, doi: 10.15672/hujms.554157.
ISNAD Ramesh, R. et al. “Oscillation Criteria for Solution to Partial Dynamic Equations on Time Scales”. Hacettepe Journal of Mathematics and Statistics 49/5 (October 2020), 1788-1797. https://doi.org/10.15672/hujms.554157.
JAMA Ramesh R, Dıx JG, S H, Prakash P. Oscillation criteria for solution to partial dynamic equations on time scales. Hacettepe Journal of Mathematics and Statistics. 2020;49:1788–1797.
MLA Ramesh, R. et al. “Oscillation Criteria for Solution to Partial Dynamic Equations on Time Scales”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, 2020, pp. 1788-97, doi:10.15672/hujms.554157.
Vancouver Ramesh R, Dıx JG, S H, Prakash P. Oscillation criteria for solution to partial dynamic equations on time scales. Hacettepe Journal of Mathematics and Statistics. 2020;49(5):1788-97.