Forced oscillation of certain fractional differential equations

The paper deals with the forced oscillation of the fractional differential equation with the initial conditions ( ) and , where is the Riemann-Liouville fractional derivative of order q of x, , is an integer, is the Riemann-Liouville fractional integral of order of x, and ( ) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young’s inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC:34A08, 34C10.