Oscillation criteria for solution to partial dynamic equations on time scales

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.*********************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

___

  • [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.