Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 90, pp. 1-18.

Title: Half-linear dynamic equations with mixed derivatives

Authors: Ondrej Dosly (Masaryk Univ., Brno, Czech Republic)
         Daniel Marek (Masaryk Univ., Brno, Czech Republic)

Abstract: 
 We investigate oscillatory properties of the second
 order half-linear dynamic equation on a time scale with mixed
 derivatives
 $$
 (r(t)\Phi(x^{\Delta}))^\nabla+c(t)\Phi(x)=0,\quad
 \Phi(x)=|x|^{p-2}x, \quad p>1.                  
 $$
 In particular, we establish the Roundabout theorem which relates
 oscillatory properties of this equation to the solvability
 of the associated Riccati-type dynamic equation and to the positivity
 of the corresponding energy functional. This result is then used to
 prove (non)oscillation criteria for the above equation. 
  
Submitted March 17, 2005. Published August 15, 2005.
Math Subject Classifications: 39A10.
Key Words: Time scale; half-linear dynamic equations; mixed derivatives;
           Picone's identity; roundabout theorem.