Even-order linear dynamic equations with mixed derivatives

  • Authors:

  • Ondřej Došlý;Daniel Marek

  • Affiliations:

  • Department of Mathematics and Statistics, Masaryk University, Janáčkovo nám. 2a, CZ-662 95 Brno, Czech Republic;Department of Mathematics and Statistics, Masaryk University, Janáčkovo nám. 2a, CZ-662 95 Brno, Czech Republic

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2007

Quantified Score

Hi-index 0.09

Visualization

Abstract

We consider the time scale kth-order differential operators D"k^@Dy@?{y^@D^@?^...^@D^@?k even ,y^@D^@?^...^@?^@Dk odd ,D@?"k^@Dy@?{y^@?^@D^...^@?^@Dk even ,y^@D^@?^...^@?^@Dk odd ,D"k^@?y@?{y^@?^@D^...^@?^@Dk even ,y^@?^@D^...^@D^@?k odd ,D@?"k^@?y@?{y^@D^@?^...^@D^@?k even ,y^@?^@D^...^@D^@?k odd , and the higher-order dynamic equations L(y)@?@?@n=0n(-1)^@nD@?"@n^@?(r"@n(t)D"@n^@Dy)=0,M(y)@?@?@n=0n(-1)^@nD@?"@n^@D(r"@n(t)D"@n^@?y)=0. We will show that these equations can be investigated as special cases of the so-called (delta or nabla) symplectic dynamic systems z^@D=S(t)z,z^@?=S(t)z, whose qualitative theory is well developed. We also suggest further perspectives of the investigation of the qualitative properties of higher-order equations with mixed derivatives.