A new conformable nabla derivative and its application on arbitrary time scales
AbstractIn this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form $(t-a)^{(\eta )}$ ( t − a ) ( η ) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ y ( t ) = y ( t , f ( t ) ) , and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.
