LAUSR

This paper investigates the well‐posedness and iterative formula for fractional oscillator equations with two different fractional orders and time delays. By the mathematical induction, a new generalized Grönwall's inequality in terms of a multivariate Mittag‐Leffler function is derived with an iteration argument. Subsequently, using the inequality, a generalized Mittag‐Leffler estimation of the solution is proposed. Furthermore, the explicit solution and iterative formula, as well as the unique existence, are presented on the basis of the method of steps. Finally, an example is given to illustrate the validity of our main results.