The diamond-alpha Riemann integral and mean value theorems on time   scales
  We study diamond-alpha integrals on time scales. A diamond-alpha version of Fermat's theorem for stationary points is also proved, as well as Rolle's, Lagrange's, and Cauchy's mean value theorems on time scales. 
Agnieszka B. Malinowska, Delfim F. M. Torres