The delta [formula omitted]fractional Gronwall inequality on time scale

Abstract

In this paper, we introduce delta q−fractional integral equations and inequalities. We obtain the equivalence of a Caputo delta q−fractional initial value problem of order 0<α≤1 and delta q−fractional integral equation. Following that, we provide an explicit solution to the linear delta q−fractional integral equation. This enables us to state and prove delta q−fractional Gronwall-type inequality on time scale. By using delta q−exponential type functions and delta q−Mittag Leffler functions, some particular cases are derived. To demonstrate the viability of the resulting inequality, an example is provided.